ball is dropped from a state of rest at time t = 0. The distance traveled after
t seconds is s(t) = 16t2 ft.
How far does the ball travel during the time interval [2, 2.5]?
Compute the average velocity over [2, 2.5].
Compute the average velocity over time intervals [2, 2.01], [2, 2.005], [2,
2.001], [2, 2.00001]. Use this to estimate the object’s instantaneous velocity
at t = 2.
wrench is released from a state of rest at time t = 0. Estimate the wrench’s
instantaneous velocity at t = 1, assuming that the distance traveled after t
seconds is s(t) = 16t2.
Let v = 20√T as in Example 2. Estimate the instantaneous ROC of v with
respect to T when T = 300 K.
Compute Dy /Dx for the interval [2, 5],
where y = 4x − 9. What is the instantaneous ROC of y with respect to x at
x = 2?
Exercises 5–6, a stone is tossed in the air from ground level with an initial
velocity of 15 m/s. Its height at time t is h(t) = 15t − 4.9t2 m.
Compute the stone’s average velocity over the time interval [0.5, 2.5] and
indicate the corresponding secant line on a sketch of the graph of h(t).
Compute the stone’s average velocity over the time intervals [1, 1.01], [1,
1.001], [1, 1.0001] and [0.99, 1], [ 0.999, 1] , [0.9999, 1] .Use
this to estimate the instantaneous velocity at t = 1.
With an initial deposit of $100, the balance in a bank account after t years is
f(t) = 100(1.08)t dollars.
What are the units of the ROC of f(t)?
Find the average ROC over [0, 0.5] and [0, 1].
Estimate the instantaneous rate of change at t = 0.5 by computing the average
ROC over intervals to the left and right of t = 0.5.
distance traveled by a particle at time t is s(t) = t3 + t. Compute
the average velocity over the time interval [1, 4] and estimate the instantaneous
velocity at t = 1.
Exercises 9–16, estimate the instantaneous rate of change at the point
P(x) = 4x2 − 3; x = 2
The atmospheric temperature T (in ◦F) above a certain point on earth is T
= 59 − 0.00356h, where h is the altitude in feet (valid for h ≤
37,000). What are the average and instantaneous rates of change of T with
respect to h? Why are they the same? Sketch the graph of T for h ≤
The height (in feet) at time t (in seconds) of a small weight oscillating at
the end of a spring is h(t) = 0.5 cos (8t).
Calculate the weight’s average velocity over the time intervals [0, 1] and [3,
Estimate its instantaneous velocity at t = 3.
The number P(t) of E. coli cells at time t (hours) in a petri dish is plotted
in Figure 9.
Calculate the average ROC of P ( t ) over the time interval [ 1 , 3 ] and draw
the corresponding secant line.
Estimate the slope m of the line in Figure 9. What does m represent?
An epidemiologist finds that the percentage N(t) of susceptible children
who were infected on day t during the first three weeks of a measles
outbreak is given, to a reasonable approximation, by the formula
graph of N ( t ) appears in Figure 13.
Draw the secant line whose slope is the average rate of increase in infected
children over the intervals between days 4 and 6 and between days 12 and 14.
Then compute these average rates (in units of percent per day).
Estimate the ROC of N ( t ) on day 12.
The fraction of a city’s population infected by a flu virus is plotted as
a function of time (in weeks) in Figure 14.
Which quantities are represented by the slopes of lines A and B? Estimate these
Is the flu spreading more rapidly at t = 1, 2, or 3?
Is the flu spreading more rapidly at t = 4, 5, or 6?
The fungus fusarium exosporium infects a field of flax plants through
the roots and causes the plants to wilt. Eventually, the entire field is
infected. The percentage f(t) of infected plants as a function of time t (in
days) since planting is shown in Figure 15.
What are the units of the rate of change of f(t) with respect to t? What does
this rate measure?
Use the graph to rank (from smallest to largest) the average infection rates
over the intervals [0, 12], [20, 32], and [40, 52].
Use the following table to compute the average rates of infection over the
intervals [30, 40], [40, 50] ,[30, 50]
Draw the tangent line at t = 40 and estimate its slope. Choose any two points
on the tangent line for the computation.
Let v = 20√T as in Example 2. Is the ROC of v with respect to T greater
at low temperatures or high temperatures? Explain in terms of the graph.
an object moving in a straight line (but with changing velocity) covers Ds feet in Dt seconds, then its average
velocity is v0 = Ds /Dt ft/s. Show that it would cover the same distance if it traveled
at constant velocity v0 over the same time interval of Dt seconds. This is a
justification for calling Ds /Dt the average velocity.
Sketch the graph of f(x) = x(1 − x) over [0, 1] . Refer to the graph
and, without making any computations, find:
The average ROC over [0, 1]
The (instantaneous) ROC at x = 1/2
The values of x at which the ROC is positive
Which graph in Figure 16 has the following property: For all x, the average ROC
over [ 0, x] is greater than the instantaneous ROC at x? Explain.
The height of a projectile fired in the air vertically with initial
velocity 64 ft / s is h (t) = 64t − 16t2 ft.
Compute h(1). Show that h(t) − h(1) can be factored with (t − 1) as
Using part (a), show that the average velocity over the interval [1, t] is
−16(t − 3).
Use this formula to find the average velocity over several intervals [1,
t] with t close to 1. Then estimate the instantaneous velocity at time t = 1.
Let Q(t) = t2. As in the previous exercise, find a formula for
the average ROC of Q over the interval [1, t] and use it to estimate the
instantaneous ROC at t = 1. Repeat for the interval [2 , t] and estimate the
ROC at t = 2.
Show that the average ROC of f(x) = x3 over [1, x] is equal to x2
+ x + 1. Use this to estimate the instantaneous ROC of f (x) at x = 1.
Find a formula for the average ROC of f(x) = x3 over [2, x] and use
it to estimate the instantaneous ROC at x = 2.
for several values of n and then guess the value in general.
59. Show numerically that for b = 3, 5 appears to equal ln 3, ln 5,
where ln x is the natural logarithm. Then make a conjecture (guess) for the
value in general and test your conjecture for two additional values of b.
(m , n) equal to (2, 1), (1, 2), (2, 3), and (3, 2). Then guess the value of
the limit in general and check your guess for at least three additional pairs.
Find the points of discontinuity of f(x) and state whether f (x) is left- or
right-continuous (or neither) at these points.
which point c does f(x) have a removable discontinuity? What value should be
assigned to f(c) to make f continuous at x = c?
Find the point c1 at which f (x) has a jump discontinuity but is
left-continuous. What value should be assigned to f (c1) to make f
right-continuous at x = c1?
(a) For the function shown in Figure 16, determine the one-sided limits at the
points of discontinuity.
Which of these discontinuities is removable and how should f be redefined
to make it continuous at this point?
Exercises 37–50, determine the domain of the function and prove that it is
continuous on its domain using the Laws of Continuity and the facts quoted in
Suppose that f (x) = 2 for x > 0 and f (x) = − 4 for x < 0. What
is f (0) if f is left-continuous at x = 0? What is f (0) if f is
right-continuous at x = 0?
Sawtooth Function Draw the graph of f (x) = x − [x]. At which points is
f discontinuous? Is it left- or right-continuous at those points?
Exercises 53–56, draw the graph of a function on [0, 5] with the given
53. f(x) is not continuous at x = 1,
but and exist and are equal.
f(x) is left-continuous but not continuous at x = 2 and right-continuous but
not continuous at x = 3.
55. f(x) has a removable
discontinuity at x = 1, a jump discontinuity at x = 2, and , .
(x) is right- but not left-continuous at x = 1, left- but not right-continuous
at x = 2, and neither left- nor right-continuous at x = 3.
Each of the following statements is false. For each statement, sketch the graph
of a function that provides a counterexample.
exists, then f (x) is continuous at x = a.
f(x) has a jump discontinuity at x = a, then f(a) is equal to either or .
f (x) has a discontinuity at x = a, then and exist but are not equal.
The one-sided limits and always exist, even if does not exist.
the IVT to show that f (x) = x3 + x takes on the value 9 for some x
in [1, 2].
that takes on
the value 0.499 for some t in [0, 1].
that g(t) = t2 tan t takes on the value ½ for some t in [0 ,
that takes on
the value 0.4.
that cos x = x has a solution in the interval [0, 1].
the IVT to find an interval of length ½ containing a root of f(x)
= x3 + 2x + 1.
Exercises 7–16, use the IVT to prove each of the following statements.
7. for some number c.
all integers n, sin nx = cos x for some x Î [0 ,π] .
√2 exists. Hint: Consider f(x) = x2
positive number c has an nth root for all positive integers n. (This fact is
usually taken for granted, but it requires proof.)
all positive integers k, there exists x such that cos x = xk.
12. 2x = bx has a solution if b > 2.
= b has a solution for all b > 0 (treat b ≥ 1 first).
14. tan x
= x has infinitely many solutions.
equation ex + ln x = 0 has a solution in (0, 1).
= cos−1x has a solution.
out three steps of the Bisection Method for f(x) = 2x − x3
that f(x) has a zero in [1, 1.5].
that f(x) has a zero in [1.25, 1.5].
Determine whether [1.25, 1.375] or [1.375, 1.5] contains a zero.
Figure 4 shows that f(x) = x3 − 8x − 1 has a root in the
interval [2.75, 3]. Apply the Bisection Method twice to find an interval
of length 1/16 containing this root.
an interval of length ¼ in [0, 1] containing a root of x5 − 5x + 1 = 0.
that tan3θ − 8tan2θ + 17tan θ
− 8 = 0 has a root in [0.5, 0.6]. Apply the Bisection Method twice to
find an interval of length 0.025 containing this root.
Exercises 21–24, draw the graph of a function f (x) on [0, 4] with the given
discontinuity at x = 2 and does not satisfy the conclusion of the IVT.
discontinuity at x = 2, yet does satisfy the conclusion of the IVT on [0, 4].
Infinite one-sided limits at x = 2 and does not satisfy the conclusion of
Infinite one-sided limits at x = 2, yet does satisfy the conclusion of
the IVT on [0, 4].
any map (e.g., of the United States) and draw a circle on it anywhere. Prove
that at any moment in time there exists a pair of diametrically opposite points
on that circle corresponding to locations where the temperatures at that moment
are equal. Hint: Let θ be an angular coordinate along the circle and let f
(θ) be the difference in temperatures at the locations corresponding to
θ and θ + π.
Assume that f(x) is continuous and that 0 ≤ f (x) ≤ 1 for 0 ≤
x ≤ 1 (see Figure 5). Show that f(c) = c for some c in [0, 1].
the IVT to show that if f (x) is continuous and one-to-one on an interval [a,
b], then f(x) is either an increasing or a decreasing function.
2. Let f
(x) = 2x2 − 3x − 5. Show that the slope of the secant
line through (2 , f (2)) and (2 + h , f (2 + h)) is 2h + 5. Then
use this formula to compute the slope of:
secant line through (2, f(2)) and (3, f(3))
tangent line at x = 2 (by taking a limit)
find the slope and then an equation of the tangent line to the graph of
f (x) = √x at x = 4.
Exercises 23–40, compute the derivative at x = a using the limit
definition and find an equation of the tangent line.
23. f(x) = 3x2 + 2x, a =
is an equation of the tangent line at x = 3, assuming that f (3) = 5 and f
( 3 ) = 2?
Suppose that y = 5x + 2 is an equation of the tangent line to the graph of y =
f(x) at a = 3. What is f(3)? What is f ' (3)?
Consider the “curve” y = 2x + 8. What is the tangent line at the point (1, 10)
? Describe the tangent line at an arbitrary point.
Suppose that f (x) is a function such that f (2 + h) − f (2) = 3h2
is f ' (2)?
is the slope of the secant line through (2, f (2)) and (6, f(6))?
vapor pressure of water is defined as the atmospheric pressure P at
which no net evaporation takes place. The following table and Figure 13 give P
(in atmospheres) as a function of temperature T in kelvins.
is larger: P ' (300)
or P ' (350)
? Answer by referring to the graph.
Estimate P ' (
T ) for T = 303, 313, 323, 333, 343 using the table and the average of the
difference quotients for h = 10 and − 10:
Exercises 50–51, traffic speed S along a certain road (in mph) varies as
a function of traffic density q (number of cars per mile on the road).
Use the following data to answer the questions:
Estimate S ' (q)
when q = 120 cars per mile using the average of difference quotients at h and
− h as in Exercise 48.
quantity V = q S is called traffic volume. Explain why V is equal to the
number of cars passing a particular point per hour. Use the data to compute
values of V as a function of q and estimate V ' (q) when q = 120.
the graph in Figure 14, determine the intervals along the x-axis on which the
derivative is positive.
Sketch the graph of f (x) = sin x on [ 0 ,π ] and guess the value of f ' (π/2). Then calculate the
slope of the secant line between x = π/2 and x = π/2 + h for at least
three small positive and negative values of h. Are these
calculations consistent with your guess?
Figure 15(A) shows the graph of f (x) =√x. The close-up in (B) shows that
the graph is nearly a straight line near x = 16. Estimate the slope of this
line and take it as an estimate for f ' (16). Then compute f ' (16) and compare with your
the method of Example 6 to f (x) = sin x to determine f ' (p/4) accurately to four decimal places.
the method of Example 6 to f(x) = cos x to determine f ' (π/5) accurately to four
decimal places. Use a graph of f(x) to explain how the method works in this
Exercises 70–72, i ( t ) is the current (in amperes) at time t (seconds)
flowing in the circuit shown in Figure
According to Kirchhoff’s law, i(t) = Cv ' ( t ) + R−1v(t) , where
v(t) is the voltage (in volts) at time t, C the capacitance (in farads), and R
the resistance (in ohms).
Calculate the current at t = 3 if v( t ) = 0.5t + 4 V, C = 0.01 F, and R = 100 W.
the following table to estimate v ¢(10). For a better estimate, take the average of the
difference quotients for h and −h as described in Exercise 48. Then
estimate i(10), assuming C = 0.03 and R = 1000.
Assume that R = 200 W
but C is unknown. Use the following data to estimate v ' (4) as in Exercise 71 and deduce
an approximate value for the capacitance C.
Sketch the graph of f(x) = x − 3x2 and find the values
of x for which the tangent line is horizontal.
the points on the curve y = x2 + 3x − 7 at which the slope of
the tangent line is equal to 4.
Sketch the graphs of f(x) = x2 − 5x + 4 and g(x) = −2x +
3. Find the value of x at which the graphs have parallel tangent lines.
all values of x where the tangent lines to y = x3 and y = x4
that there is a unique point on the graph of the function f(x) = ax2
+ bx + c where the tangent line is horizontal (assume a > 0). Explain graphically.
Determine coefficients a and b such that p(x) = x2 + ax + b
satisfies p(1) = 0 and p ' (1) = 4.
all values of x such that the tangent line to the graph of y = 4x2 +
11x + 2 is steeper than the tangent line to y = x3
f(x) = x3 − 3x + 1. Show that f ' (x) ≥ −3 for all x,
and that for every m > −3, there are precisely two points where f ' (x) = m. Indicate the position of
these points and the corresponding tangent lines for one value of m in a sketch
of the graph of f (x).
that if the tangent lines to the graph of at x = a and at x = b are parallel, then
either a = b or a + b = 2.
Compute the derivative of f (x) = x−2 using the limit
an approximation to m4 using the limit definition and estimate the slope
of the tangent line to y = 4x at x = 0
and x = 2.
f(x) = xex. Use the limit definition to compute f ¢(0) and find the equation of
the tangent line at x = 0.
average speed (in meters per second) of a gas molecule is , where T is the temperature (in
kelvin), M is the molar mass (kg /mol) and R = 8.31. Calculate dvavg
/dT at T = 300 K for oxygen, which has a molar mass of 0.032 kg/mol.
Biologists have observed that the pulse rate P (in beats per minute)
in animals is related to body mass (in kilograms) by the approximate formula P
= 200m−1/4. This is one of many allometric scaling laws
prevalent in biology. Is the absolute value |dP/dm| increasing or decreasing as
m increases? Find an equation of the tangent line at the points on the graph in
Figure 17 that represent goat (m = 33) and man (m = 68).
studies suggest that kidney mass K in mammals (in kilograms) is related to body
mass m (in kilograms) by the approximate formula K = 0.007m0.85.
Calculate dK/dm at m = 68. Then calculate the derivative with respect to m of
the relative kidney-to-mass ratio K/m at m = 68.
relation between the vapor pressure P (in atmospheres) of water and the
temperature T (in kelvin) is given by the Clausius–Clapeyron law:
is a constant. Use the table below and the approximation to estimate dP/dT
for T = 303, 313, 323, 333, 343. Do your estimates seem to confirm the
Clausius–Clapeyron law? What is the approximate value of k? What are the units
69. Let L
be a tangent line to the hyperbola xy = 1 at x = a, where a > 0. Show that
the area of the triangle bounded by L and the coordinate axes does not depend
70. In the notation of Exercise 69, show
that the point of tangency is the midpoint of the segment of L lying in the
the functions (A)–(C) with their derivatives (I)–(III) in Figure 18.
small arches have the shape of parabolas. The first is given by f(x) = 1
− x2 for −1 ≤ x ≤ 1 and the second by g(x) =
4 − (x − 4)2 for 2 ≤ x ≤ 6. A board is placed on top of these arches so it rests on both (Figure 22). What is the slope of the
95. A vase is formed by rotating y = x2
around the y-axis. If we drop in a marble, it will either touch the bottom
point of the vase or be suspended above the bottom by touching the sides
(Figure 23). How small must the marble be to touch the bottom?
Verify the Power Rule for the exponent 1/n, where n is a positive integer,
using the following trick: Rewrite the difference quotient for y = x1/n at
x = b in terms of u = (b + h)1/n and a = b1/n.
Infinitely Rapid Oscillations Define
that f(x) is continuous at x = 0 but f ' (0) does not exist (see Figure 12).
Prove that f(x) = ex is not a polynomial function. Hint:
Differentiation lowers the degree of a polynomial by 1.
Consider the equation ex = λx, where λ is a constant.
which λ does it have a unique solution? For intuition, draw a graph of y =
ex and the line y = λx.
which λ does it have at least one solution?
the ROC of the area of a square with respect to the length of its side s when s
= 3 and s = 5.
the ROC of the volume of a cube with respect to the length of its side s when s
= 3 and s = 5.
the ROC of y = x−1 with respect to x for x = 1, 10.
what rate is the cube root changing with respect to x when x = 1, 8,
Exercises 5–8, calculate the ROC.
where V is the volume of a cylinder whose height is equal to its radius (the
volume of a cylinder of height h and radius r is πr2h)
of the volume V of a cube with respect to its surface area A
of the volume V of a sphere with respect to its radius.
8. , where A is
the surface area of a sphere of diameter D (the surface area of a sphere of
radius r is 4πr2)
Estimate the average velocity over [0.5, 1].
Is average velocity greater over [1, 2] or [2, 3]?
At what time is velocity at a maximum?
the description with the interval (a)–(d).
Average velocity of 50 mph
Figure 11 displays the voltage across a capacitor as a function of time while
the capacitor is being charged. Estimate the ROC of voltage at t = 20 s.
Indicate the values in your calculation and include proper units. Does voltage
change more quickly or more slowly as time goes on? Explain in terms of tangent
Figure 12 to estimate dT/dh at h = 30 and 70, where T is atmospheric
temperature (in degrees Celsius) and h is altitude (in kilometers). Where is
dT/dh equal to zero?
13. A stone is tossed vertically upward
with an initial velocity of 25 ft /s from the top of a 30-ft building.
is the height of the stone after 0.25 s?
the velocity of the stone after 1 s.
does the stone hit the ground?
height (in feet) of a skydiver at time t (in seconds) after opening his parachute
is h(t) = 2000 − 15t ft. Find the skydiver’s velocity after the parachute
temperature of an object (in degrees Fahrenheit) as a function of time (in
minutes) is for 0 ≤ t ≤
20. At what
rate does the object cool after 10 min (give correct units)?
velocity (in centimeters per second) of a blood molecule flowing through
a capillary of radius 0.008 cm is given by the formula v = 6.4 × 10−8 − 0.001r2, where r is the distance from the molecule to the center
of the capillary. Find the ROC of velocity as a function of distance when r = 0.004 cm.
earth exerts a gravitational force of (in Newtons) on an object with a mass of 75 kg, where r is the distance (in meters) from the center of the earth. Find the ROC of force with
respect to distance at the surface of the earth, assuming the radius of the
earth is 6.77 × 106 m
escape velocity at a distance r meters from the center of the earth is vesc = (2.82 × 107)r–1/2 m/s. Calculate the rate at
which vesc changes with respect to distance at the surface of the
power delivered by a battery to an apparatus of resistance R (in ohms) is W. Find the rate of
change of power with respect to resistance for R = 3 and R = 5 W.
position of a particle moving in a straight line during a 5-s trip is s(t) = t2 − t + 10 cm.
is the average velocity for the entire trip?
there a time at which the instantaneous velocity is equal to this average
velocity? If so, find it.
Faraday’s Law, if a conducting wire of length l meters moves at velocity v m/s
perpendicular to a magnetic field of strength B (in teslas), a voltage of
size V = − Blv is induced in the wire. Assume that B = 2 and l = 0.5.
the rate of change dV/dv.
the rate of change of V with respect to time t if v = 4t + 9.
height (in feet) of a helicopter at time t (in minutes) is s(t) =
−3t + 400t for 0 ≤ t ≤ 10.
the graphs of height s(t) and velocity v(t).
the velocity at t = 6 and t = 7.
the maximum height of the helicopter.
population P(t) of a city (in millions) is given by the formula P(t) = 0.00005t2
+ 0.01t + 1, where t denotes the number of years since 1990.
large is the population in 1996 and how fast is it growing?
does the population grow at a rate of 12,000 people per year?
According to Ohm’s Law, the voltage V, current I, and resistance R in a
circuit are related by the equation V = IR, where the units are volts,
amperes, and ohms. Assume that voltage is constant with V = 12 V. Calculate
(specifying the units):
average ROC of I with respect to R for the interval from R = 8 to R = 8.1
ROC of I with respect to R when R = 8
ROC of R with respect to I when I = 1.5
finds that with h hours of tutoring, he is able to answer correctly S (h)
percent of the problems on a math exam. What is the meaning of the derivative S ¢(h)? Which would you expect to be
larger: S ' (3)
or S ' (30)
Suppose θ(t) measures the angle between a clock’s minute and hour hands.
What is θ ' (t)
at 3 o’clock?
2 gives the total U.S. population during each month of 1999 as determined by
the U.S. Department of Commerce.
Estimate P ¢(t)
for each of the months January–November.
these data points for P ¢(t) and connect the points by a smooth curve.
Write a newspaper headline describing the information contained in this plot.
tangent lines to the graph of f(x) = x2 grow steeper as x increases.
At what rate do the slopes of the tangent lines increase?
According to a formula widely used by doctors to determine drug dosages, a
person’s body surface area (BSA) (in meters squared) is given by the formula
BSA = /60,
where is the height in centimeters and w the weight in kilograms. Calculate the
ROC of BSA with respect to weight for a person of constant height h = 180. What
is this ROC for w = 70 and w = 80? Express your result in the correct units.
Does BSA increase more rapidly with respect to weight at lower or higher body
30. A slingshot is used to shoot a
pebble in the air vertically from ground level with an initial velocity 200
m/s. Find the pebble’s maximum velocity and height.
is the velocity of an object dropped from a height of 300 m when it hits the ground?
takes a stone 3 s to hit the ground when dropped from the top of a building.
How high is the building and what is the stone’s velocity upon impact?
33. A ball is tossed up vertically from
ground level and returns to earth 4 s later. What was the initial velocity of
the stone and how high did it go?
object is tossed up vertically from ground level and hits the ground T s later.
Show that its maximum height was reached after T/2 s.
35. A man
on the tenth floor of a building sees a bucket (dropped by a window
washer) pass his window and notes that it hits the ground 1.5 s later. Assuming
a floor is 16 ft high (and neglecting air friction), from which
floor was the bucket dropped?
of the following statements is true for an object falling under the
influence of gravity near the surface of the earth? Explain.
object covers equal distance in equal time intervals.
(b) Velocity increases by equal amounts in equal time intervals.
derivative of velocity increases with time.
that for an object rising and falling according to Galileo’s formula in Eq.
(3), the average velocity over any time interval [t1, t2]
is equal to the average of the instantaneous velocities at t1 and t2.
weight oscillates up and down at the end of a spring. Figure 13 shows the
height y of the weight through one cycle of the oscillation. Make a rough
sketch of the graph of the velocity as a function of time.
Exercises 39–46, use Eq. (2) to estimate the unit change.
Estimate and . Compare your estimates
with the actual values.
Suppose that f(x) is a function with f ¢(x) = 2−x. Estimate f (7)
− f (6). Then estimate f(5), assuming that f (4) = 7.
F(s) = 1.1s + 0.03s2 be the stopping distance as in Example 3.
Calculate F(65) and estimate the increase in stopping distance if speed is
increased from 65 to 66 mph. Compare your estimate with the actual increase.
According to Kleiber’s Law, the metabolic rate P (in kilocalories per day) and
body mass m (in kilograms) of an animal are related by a three-quarter power
law P = 73.3m3/4. Estimate the increase in metabolic rate when body
mass increases from 60 to 61 kg.
dollar cost of producing x bagels is C(x) = 300 + 0.25x − 0.5(x/1000)3.
Determine the cost of producing 2,000 bagels and estimate the cost of the
2001st bagel. Compare your estimate with the actual cost of the 2001st bagel.
Suppose the dollar cost of producing x video cameras is C(x) = 500x −
0.003x2 + 10−8x3.
Estimate the marginal cost at production level x = 5000 and compare it with the
actual cost C(5001) − C(5000).
Compare the marginal cost at x = 5000 with the average cost per camera,
defined as C(x)/x.
demand for a commodity generally decreases as the price is raised. Suppose that
the demand for oil (per capita per year) is D(p) = 900/p barrels, where p is
the price per barrel in dollars. Find the demand when p = $40. Estimate the
decrease in demand if p rises to $41 and the increase if p is decreased to $39.
reproduction rate of the fruit fly Drosophila melanogaster, grown in
bottles in a laboratory, decreases as the bottle becomes more crowded. A
researcher has found that when a bottle contains p flies, the number of
offspring per female per day is f(p) = (34 − 0.612p)p−0.658
Calculate f(15) and f ' (15).
Estimate the decrease in daily offspring per female when p is increased from 15
to 16. Is this estimate larger or smaller than the actual value f(16) −
f (p) for 5 ≤ p ≤ 25 and verify that f(p) is a decreasing function
of p. Do you expect f ' (p) to be positive or negative? Plot f ' (p) and confirm your
47. Let A
= s2. Show that the estimate of A(s + 1) − A(s) provided by
Eq. (2) has error exactly equal to 1. Explain this result using Figure 14.
According to Steven’s Law in psychology, the perceived magnitude of a stimulus
(how strong a person feels the stimulus to be) is proportional to a power of
the actual intensity I of the stimulus. Although not an exact law, experiments
show that the perceived brightness B of a light satisfies B = kI2/3,
where I is the light intensity, whereas the perceived heaviness H of a
weight W satisfies H = kW3/2 (k is a constant that is
different in the two cases). Compute dB/dI and dH/dW and state whether they are
increasing or decreasing functions. Use this to justify the statements:
one-unit increase in light intensity is felt more strongly when I is small than
when I is large.
Adding another pound to a load W is felt more strongly when Wis large than when
W is small.
M(t) be the mass (in kilograms) of a plant as a function of time (in years).
Recent studies by Niklas and Enquist have suggested that for a remarkably wide
range of plants (from algae and grass to palm trees), the growth rate during
the life span of the organism satisfies a three-quarter power law, that
is, dM/dt = CM3/4for some
(a) If a
tree has a growth rate of 6 kg/year when M = 100 kg, what is its growth rate
when M = 125 kg?
(b) If M
= 0.5 kg, how much more mass must the plant acquire to double its growth rate?
50. As an
epidemic spreads through a population, the percentage p of infected individuals
at time t (in days) satisfies the equation (called a differential
equation) dp/dt = 4p − 0.06p2, 0 ≤ p ≤ 100
fast is the epidemic spreading when p = 10% and when p = 70%?
which p is the epidemic neither spreading nor diminishing?
dp/dt as a function of p.
is the maximum possible rate of increase and for which p does this occur?
size of a certain animal population P(t) at time t (in months) satisfies
dP/dt = 0.2(300 − P).
(a) Is P
growing or shrinking when P = 250? when P = 350?
Sketch the graph of dP/dt as a function of P for 0 ≤ P ≤ 300.
of the graphs in Figure 15 is the graph of P(t) if P(0) = 200?
Exercises 53–54, the average cost per unit at production level x is
defined as Cavg (x) = C(x)/x, where C(x) is the cost function.
Average cost is a measure of the efficiency of the production process.
that Cavg(x) is equal to the slope of the line through the origin
and the point (x, C(x)) on the graph of C(x). Using this interpretation,
determine whether average cost or marginal cost is greater at points A, B, C, D
in Figure 16.
cost in dollars of producing alarm clocks is C(x) = 50x3 −
750x2 + 3740x + 3750 where x is in units of 1,000.
Calculate the average cost at x = 4, 6, 8, and 10.
the graphical interpretation of average cost to find the production level
x0 at which average cost is lowest. What is the relation between average
cost and marginal cost at x0 (see Figure 17)?
Find the acceleration at time t = 5 min of a helicopter whose height (in feet)
is h(t) = − 3t3 + 400t.
the acceleration h ²(t) for 0 ≤ t ≤ 6. How does this graph show that the
helicopter is slowing down during this time interval?
an equation of the tangent to the graph of y = f ¢(x) at x = 3, where f(x) = x4.
Figure 5 shows f , f ' , and f ''. Determine which is which.
second derivative f '' is shown in Figure 6. Determine which graph, (A) or (B), is f and
which is f ''.
Figure 7 shows the graph of the position of an object as a function of time.
Determine the intervals on which the acceleration is positive.
the second derivative of the volume of a cube with respect to the length of a
a polynomial f(x) satisfying the equation xf '' (x) + f (x) = x2.
a value of n such that y = xnex satisfies the
equation xy ' =
(x − 3)y.
of the following descriptions could not apply to Figure 8? Explain.
of acceleration when velocity is constant
of velocity when acceleration is constant
of position when acceleration is zero
servomotor controls the vertical movement of a drill bit that will drill a
pattern of holes in sheet metal. The maximum vertical speed of the drill bit is
4 in ./s, and while drilling the hole, it must move no more than 2.6
in ./s to avoid warping the metal. During a cycle, the bit begins and ends at
rest, quickly approaches the sheet metal, and quickly returns to its initial
position after the hole is drilled. Sketch possible graphs of the drill bit’s
vertical velocity and acceleration. Label the point where the bit enters the
the 100th derivative of p(x) = (x + x5 + x7)10(1
+ x2)11(x3 + x5 + x7)
the Product Rule twice to find a formula for (f g)'' in terms of the first and
second derivative of f and g.
the Product Rule to find a formula for (f g)'' and compare your result with the
expansion of (a + b)3. Then try to guess the general formula for (f
the values of x between 0 and 2 π where the tangent line to the graph of y
= sin x cos x is horizontal.
Calculate the first five derivatives of f(x) = cos x. Then
determine f(8) and f(37).
y(157), where y = sin x.
that no tangent line to the graph of f(x) = tan x has zero slope. What is the
least slope of a tangent line? Justify your response by sketching the graph of
(tan x) '.
height at time t (s) of a weight, oscillating up and down at the end of a
spring, is s(t) = 300 + 40 sin t cm. Find the velocity and acceleration at t =
horizontal range R of a projectile launched from ground level at an angle
θ and initial velocity v0 m/s is . Calculate dR/dθ. If θ = 7π
/24, will the range increase or decrease if the angle is increased slightly?
Base your answer on the sign of the derivative.
you stand 1 m from a wall and mark off points on the wall at equal increments
δ of angular elevation (Figure 4), then these points grow increasingly far
apart. Explain how this illustrates the fact that the derivative of tan θ
the limit definition of the derivative and the addition law for the
cosine to prove that (cos x) ' = −sin x.
that a nonzero polynomial function y = f (x) cannot satisfy the equation y ² = − y. Use this to prove
that neither sin x nor cos x is a polynomial.
that if π / 2 < θ < π , then the distance along the
x-axis between θ and the point where the tangent line intersects the
x-axis is equal to |tan θ| (Figure 5).
average molecular velocity v of a gas in a certain container is given by v =
29√T m/s, where T is the temperature in kelvins. The temperature is
related to the pressure (in atmospheres) by T = 200P.
Assume that f(0) = 2 and f ' (0) = 3. Find the derivatives of (f (x))3 and f
(7x ) at x = 0.
Compute the derivative of h(sin x) at x = π/6, assuming that h ' (0.5) = 10.
F(x) = f(g(x)), where the graphs of f and g are shown in Figure 1. Estimate g ' (2) and f ' (g(2)) from the graph and compute F ' (2).
the Chain Rule to express the second derivative of f ◦ g in terms of the
first and second derivatives of f and g.
Compute the second derivative of sin (g(x)) at x = 2, assuming that g (2) =
π/4, g ' (2)
= 5, and g '' (2)
expanding sphere has radius r = 0.4t cm at time t (in seconds). Let V be the
sphere’s volume. Find dV/dt when (a) r = 3 and (b) t = 3.
power P in a circuit is P = Ri2, where R is resistance and i the
current. Find dP/dt at t = 2 if R = 1000 W and i varies according to i = sin(4πt) (time in
price (in dollars) of a computer component is P = 2C − 18C−1, where C is the manufacturer’s cost to produce it. Assume that cost
at time t (in years) is C = 9 + 3t−1 and determine the ROC of
price with respect to time at t = 3.
force F (in Newtons) between two charged objects is F = 100/r2,
where r is the distance (in meters) between them. Find dF/dt at t = 10 if the
distance at time t (in seconds) is r = 1 + 0.4t2.
According to the U.S. standard atmospheric model, developed by the National
Oceanic and Atmospheric Administration for use in aircraft and rocket design,
atmospheric temperature T (in degrees Celsius), pressure P (kPa = 1000
Pascals), and altitude h (meters) are related by the formulas (valid in the
troposphere h ≤ 11000):
T = 15.04
dP/dh. Then estimate the change in P (in Pascals, Pa) per additional meter of
altitude when h = 3000.
the points on the graph of y2 = x3 − 3x + 1
(Figure 6) where the tangent line is horizontal.
show that 2yy ¢ =
3x2 − 3, where y ¢= dy/dx.
not solve for y ¢.
Rather, set y ¢ =
0 and solve for x. This gives two possible values of x where the slope may be
that the positive value of x does not correspond to a point on the graph.
negative value corresponds to the two points on the graph where the tangent
line is horizontal. Find the coordinates of these two points.
all points on the graph of 3x2 + 4y2 + 3xy = 24 where the
tangent line is horizontal (Figure 7).
differentiating the equation of the curve implicitly and setting y ¢ = 0, show that if the tangent
line is horizontal at (x , y) , then y = −2x.
for x by substituting y = −2x in the equation of the curve.
that no point on the graph of x2 − 3xy + y2 = 1 has
a horizontal tangent line.
Figure 1 shows the graph of y4 + xy = x3 − x + 2.
Find dy/dx at the two points on the graph with x-coordinate 0 and find an
equation of the tangent line at (1, 1).
the derivative dx/dy exists at a point and dx/dy = 0, then the tangent line is
vertical. Calculate dx/dy for the equation y4 + 1 = y2 +
x2 and find the points on the graph where the tangent line is
Differentiate the equation xy = 1 with respect to the variable t and derive the
volumeV and pressure P of gas in a piston (which vary in time t) satisfy PV3/2
= C, where C is a constant. Prove that
of the derivatives is negative. Could you have predicted this from the relation
PV3/2 = C?
all points on the folium x3 + y3 = 3xy at which the
tangent line is horizontal.
that if P lies on the intersection of the two curves x2 − y2
= c and xy = d (c, d constants), then the tangents to the curves at P are
energy E (in joules) radiated as seismic waves from an earthquake of Richter
magnitude M is given by the formula log10E = 4.8 + 1.5M.
Express E as a function of M.
that when M increases by 1, the energy increases by a factor of approximately
fast is the water level rising if water is filling the tub at a rate of
2. At what
rate is water pouring into the tub if the water level rises at a rate of 0.8
radius of a circular oil slick expands at a rate of 2 m/min.
fast is the area of the oil slick increasing when the radius is 25 m?
the radius is 0 at time t = 0, how fast is the area increasing after 3 min?
what rate is the diagonal of a cube increasing if its edges are increasing at a
rate of 2 cm/s?
Exercises 5–8, assume that the radius r of a sphere is expanding at a rate of 14 in./min. The volume of a sphere is .
5. Determine the rate at which the volume is changing with respect to time when
r = 8 in.
Determine the rate at which the volume is changing with respect to time at t =
2 min, assuming that r = 0 at t = 0.
Determine the rate at which the surface area is changing when the radius is r =
Determine the rate at which the surface area is changing with respect to time
at t = 2 min, assuming that r = 3 at t = 0.
9. A road perpendicular to a highway
leads to a farmhouse located 1 mile away (Figure 9). An automobile travels past
the farmhouse at a speed of 60 mph. How fast is the distance between the
automobile and the farmhouse increasing when the automobile is 3 miles past the intersection of the highway and the road?
10. A conical tank has height 3 m and radius 2 m at the top. Water flows in at a rate of 2 m3/min. How fast is the
water level rising when it is 2 m?
Follow the same set-up as Exercise 10, but assume that the water level is
rising at a rate of 0 . 3 m / min when it is 2 m. At what rate is water flowing in?
and Isaac are in motorboats located at the center of a lake. At time t = 0,
Sonya begins traveling south at a speed of 32 mph. At the same time, Isaac takes off, heading east at a speed of 27 mph.
far have Sonya and Isaac each traveled after 12 min?
what rate is the distance between them increasing at t = 12 min?
Answer (a) and (b) in Exercise 12 assuming that Sonya begins moving 1 minute
after Isaac takes off.
14. A 6-ft man walks away from a 15-ft
lamppost at a speed of 3 ft /s (Figure 10). Find the rate at which his shadow
is increasing in length.
15. At a
given moment, a plane passes directly above a radar station at an altitude of 6 miles.
the plane’s speed is 500 mph, how fast is the distance between the plane and
the station changing half an hour later?
fast is the distance between the plane and the station changing when the plane
passes directly above the station?
16. In the setting of Exercise 15,
suppose that the line through the radar station and the plane makes an angle
θ with the horizontal. How fast is θ changing 10 min after the plane
passes over the radar station?
17. A hot air balloon rising vertically
is tracked by an observer located 2 miles from the lift-off point. At a certain moment, the angle between the observer’s line-of-sight and the horizontal
is π/5, and it is changing at a rate
of 0.2 rad/min. How fast is the balloon rising at this moment?
18. As a
man walks away from a 12-ft lamppost, the tip of his shadow moves twice as fast
as he does. What is the man’s height?
Exercises 19–23, refer to a 16-ft ladder sliding down a wall, as in Figures 1
and 2. The variable h is the height of the ladder’s top at time t, and x is the
distance from the wall to the ladder’s bottom.
Assume the bottom slides away from the wall at a rate of 3 ft/s. Find the
velocity of the top of the ladder at t = 2 if the bottom is 5 ft from the wall at t = 0.
Suppose that the top is sliding down the wall at a rate of 4 ft/s. Calculate
dx/dt when h = 12.
Suppose that h(0) = 12 and the top slides down the wall at a rate of 4 ft/s.
Calculate x and dx/dt at t = 2 s.
is the relation between h and x at the moment when the top and bottom of the
ladder move at the same speed?
that the velocity dh / dt approaches infinity as the ladder slides down
to the ground (assuming dx / dt is constant). This suggests that our
mathematical description is unrealistic, at least for small values of h. What
would, in fact, happen as the top of the ladder approaches the ground?
radius r of a right circular cone of fixed height h = 20 cm is increasing at a rate of 2 cm/s. How fast is the volume increasing when r = 10?
Suppose that both the radius r and height h of a circular cone change at a rate
of 2 cm/s. How fast is the volume of the cone increasing when r = 10 and h =
26. A particle moves counterclockwise
around the ellipse 9x2 + 16y2 = 25 (Figure 11).
which of the four quadrants is the derivative dx / dt positive? Explain your
a relation between dx/dt and dy/dt.
what rate is the x-coordinate changing when the particle passes the point (1,
1) if its y-coordinate is increasing at a rate of 6 ft /s?
is dy/dt when the particle is at the top and bottom of the ellipse?
27. A searchlight rotates at a rate of
3 revolutions per minute. The beam hits a wall located 10 miles away and produces a dot of light that moves horizontally along the wall. How fast is this
dot moving when the angle θ between the beam and the line through the
searchlight perpendicular to the wall is π/6?
28. A rocket travels vertically at a
speed of 800 mph. The rocket is tracked through a telescope by an observer
located 10 miles from the launching pad. Find the rate at which the angle
between the telescope and the ground is increasing 3 min after lift-off.
29. A plane traveling at an altitude of
20,000 ft passes directly overhead at time t = 0. One minute later you observe
that the angle between the vertical and your line of sight to the plane is 1.14
rad and that this angle is changing at a rate of 0.38 rad/min. Calculate the
velocity of the airplane.
Calculate the rate (in cm2/s) at which area is swept out by the
second hand of a circular clock as a function of the clock’s radius.
31. A jogger runs around a circular
track of radius 60 ft. Let (x, y) be her coordinates, where the origin is at
the center of the track. When the jogger’s coordinates are (36, 48), her
x-coordinate is changing at a rate of 14 ft/s. Find dy/dt.
32. A car travels down a highway at 55 mph. An observer is standing 500 ft from the highway.
fast is the distance between the observer and the car increasing at the moment
the car passes in front of the observer? Can you justify your answer without
relying on any calculations?
fast is the distance between the observer and the car increasing 1 min later?
Exercises 33–34, assume that the pressure P (in kilopascals) and volume V (in
cubic centimeters) of an expanding gas are related by PVb = C, where
b and C are constants (this holds in adiabatic expansion, without heat gain or
dP/dt if b = 1.2, P = 8 kPa, V = 100 cm2, and dV/dt = 20 cm3/min.
b if P = 25 kPa, dP/dt = 12 kPa/min, V = 100 cm2, and dV/dt = 20 cm3/min.
point moves along the parabola y = x2 + 1. Let l(t) be the distance
between the point and the origin. Calculate l ¢(t) , assuming that the
x-coordinate of the point is increasing at a rate of 9 ft/s.
base x of the right triangle in Figure 12 increases at a rate of 5 cm/s, while
the height remains constant at h = 20. How fast is the angle θ changing
when x = 20?
37. A water tank in the shape of a
right circular cone of radius 300 cm and height 500 cm leaks water from the vertex at a rate of 10 cm3/min. Find the rate at which the
water level is decreasing when it is 200 cm.
parallel paths 50 ft apart run through the woods. Shirley jogs east on one path
at 6 mph, while Jamail walks west on the other at 4 mph. If they pass each other at time t = 0, how far apart are they 3 s later, and how fast is the
distance between them changing at that moment?
is pulling on a rope that passes through a pulley on a 10-ft pole and is
attached to a wagon (Figure 13). Assume that the rope is attached to a loop on
the wagon 2 ft off the ground. Let x be the distance between the loop and the
a formula for the speed of the wagon in terms of x and the rate at which Henry
pulls the rope.
the speed of the wagon when it is 12 ft from the pole, assuming that Henry
pulls the rope at a rate of 1 . 5 ft/s.
40. A roller coaster has the shape of
the graph in Figure 14. Show that when the roller coaster passes the point ( x,
f (x)), the vertical velocity of the roller coaster is equal to f ¢(x) times its horizontal velocity.
a telescope, you track a rocket that was launched 2 miles away, recording the angle θ between the telescope and the ground at half-second
intervals. Estimate the velocity of the rocket if θ(10) = 0 .205 and
θ(10.5) = 0.225.
trains leave a station at t = 0 and travel with constant velocity v along
straight tracks that make an angle θ.
(a) Show that the trains are separating from each other at a rate .
does this formula give for θ = π?
43. A baseball player runs from home
plate toward first base at 20 ft/s. How fast is the player’s distance
from second base changing when the player is halfway to first base? See
the wheel of radius r cm in Figure 16 rotates, the rod of length L attached at
the point P drives a piston back and forth in a straight line. Let x be the
distance from the origin to the point Q at the end of the rod as in the
the Pythagorean Theorem to show that
Differentiate Eq. (8) with respect to t to prove that
Calculate the speed of the piston when θ = π/2, assuming that r = 10 cm, L = 30 cm, and the wheel rotates at 4 revolutions per minute.
45. A spectator seated 300 m away from the center of a circular track of radius 100 m watches an athlete run laps at a speed of 5 m/s. How fast is the distance between the spectator and athlete
changing when the runner is approaching the spectator and the distance between
them is 250 m?
46. A cylindrical tank of radius R and
length L lying horizontally as in Figure 17 is filled with oil to height
that the volume V(h) of oil in the tank as a function of height h is
Suppose that R = 4 ft and L = 30 ft, and that the tank is filled at a
constant rate of 10 ft3/min. How fast is the height h increasing
when h = 5?
CHAPTER REVIEW EXERCISES
Water pours into the tank in Figure 7 at a rate of 20 m3/min. How fast
is the water level rising when the water level is h = 4 m?
minute hand of a clock is 4 in. long and the hour hand is 3 in. long. How fast is the distance between the tips of the hands changing at 3 o’clock?
115. A light moving at 3 ft / s approaches a 6-ft man standing 12 ft from a wall (Figure 8). The light is 3 ft above the ground. How fast is the tip P of the man’s shadow moving when the light is 24 ft from the wall?
116. A bead slides down the curve xy =
10. Find the bead’s horizontal velocity if its height at time t seconds is y =
80 − 16t2 cm.
Side x of the triangle in Figure 9 is increasing at 2 cm/s and side y is
increasing at 3 cm/s. Assume that θ decreases in such a way that the area
of the triangle has the constant value 4 cm2. How fast is θ
decreasing when x = 4, y = 4?
fast is the distance between P and Q changing when x = 2, y = 3?
exercises 1-6 use the linear approximation to estimate Df = f(3.02) – f(3) for the given function
1. f(x) = x2
cube root of 27 is 3. How much larger is the cube root of 27.2? Estimate using
the Linear Approximation.
is larger: √2.1 − √2 or √9.1 − √9? Explain
using the Linear Approximation.
Estimate sin 61◦ − sin 60◦ using the Linear Approximation.
28. A thin silver wire has length L = 18 cm when the temperature is T = 30◦C. Estimate the length when T = 25◦C if the
coefficient of thermal expansion is k = 1.9 × 10−5 ◦C−1.
atmospheric pressure P (in kilopascals) at altitudes h (in kilometers) for 11
≤ h ≤ 25 is approximately P(h) = 128e−0.157h.
the Linear Approximation to estimate the change in pressure at h = 20 when Dh = 0.5.
Compute the actual change and compute the percentage error in the Linear
30. The resistance
R of a copper wire at temperature T = 20◦C is R = 15 W. Estimate the resistance at T =
22◦C, assuming that
side s of a square carpet is measured at 6 ft. Estimate the maximum error in the area A of the carpet if s is accurate to within half an inch.
32. A spherical balloon has a radius of
6 in. Estimate the change in volume and surface area if the radius increases
by 0.3 in.
33. A stone tossed vertically in the
air with initial velocity v ft/s reaches a maximum height of h = v2/64
Estimate Dh if v
is increased from 25 to 26 ft / s.
Estimate Dh if v
is increased from 30 to 31 ft / s.
general, does a 1 ft / s increase in initial velocity cause a greater change in
maximum height at low or high initial velocities? Explain.
the price of a bus pass from Albuquerque to Los Alamos is set at x dollars, a
bus company takes in a monthly revenue of R(x) = 1.5x − 0.01x2
(in thousands of dollars).
Estimate the change in revenue if the price rises from $50 to $53.
Suppose that x = 80. How will revenue be affected by a small increase in price?
Explain using the Linear Approximation.
stopping distance for an automobile (after applying the brakes) is
approximately F(s) = 1.1s + 0.054s2 ft, where s is the speed in
mph. Use the Linear Approximation to estimate the change in stopping distance
per additional mph when s = 35 and when s = 55.
measures the circumference C of a spherical ball at 40 cm and computes the ball’s volume V . Estimate the maximum possible error in V if the error in C
is at most 2 cm. Recall that C = 2πr and , where r is the ball’s radius.
Estimate the weight loss per mile of altitude gained for a 130-lb pilot. At
which altitude would she weigh 129.5 lb? See Example 4.
much would a 160-lb astronaut weigh in a satellite orbiting the earth at an
altitude of 2,000 miles? Estimate the astronaut’s weight loss per additional
mile of altitude beyond 2,000.
volume of a certain gas (in liters) is related to pressure P (in atmospheres)
by the formula PV = 24. Suppose that V = 5 with a possible error of ± 0.5 L.
Compute P and estimate the possible error.
Estimate the maximum allowable error in V if P must have an error of at most
dosage D of diphenhydramine for a dog of body mass w kg is D = k w2/3 mg, where k is a constant. A cocker spaniel has mass w = 10 kg according to a veterinarian’s scale. Estimate the maximum allowable error in w if the
percentage error in the dosage D must be less than 5%.
Exercises 41–50, find the linearization at x = a.
41. y =
cos x sin x, a = 0
Estimate √16.2 using the linearization L(x) of f(x) = √x at a =
16. Plot f(x) and L(x) on the same set of axes and determine if the estimate is
too large or too small.
Estimate 1/√15 using a suitable linearization of f(x) = 1/√x. Plot
f(x) and L(x) on the same set of axes and determine if the estimate is too
large or too small. Use a calculator to compute the percentage error.
Exercises 53–61, approximate using linearization and use a calculator to
compute the percentage error.
f(x) = tan x and its linearization L(x) at a = π/4 on the same set of
the linearization overestimate or underestimate f(x)?
by graphing y = f(x) − L(x) and y = 0.1 on the same set of axes, that
the error |f(x) − L(x)| is at most 0.1 for 0.55 ≤ x ≤ 0.95.
an interval of x-values on which the error is at most 0.05.
Compute the linearization L(x) of f(x) = x2 − x3/2 at a = 2. Then plot f(x) − L(x) and find an interval around a = 1
such that |f(x) − L(x)| ≤ 0.1.
Exercises 64–65, use the following fact derived from Newton’s Laws: An object
released at an angle θ with initial velocity v ft/s travels a total
64. A player located 18.1 ft from a basket launches a successful jump shot from a height of 10 ft (level with the rim of the basket), at an angle θ = 34o and initial velocity of
v = 25 ft/s.
that the distance s of the shot changes by approximately 0.255Dθ ft if the angle changes by
an amount Dθ.
Remember to convert the angles to radians in the Linear Approximation.
(b) Is it
likely that the shot would have been successful if the angle were off by
Estimate the change in the distance s of the shot if the angle changes from
50◦ to 51◦ for v = 25 ft / s and v = 30 ft / s. Is the shot more sensitive to the angle when the velocity is large or small? Explain.
Compute the linearization of f (x) = 3x − 4 at a = 0 and a = 2. Prove
more generally that a linear function coincides with its linearization at x = a
for all a.
According to (3), the error in the Linear Approximation is of “order two” in h.
Show that the Linear Approximation to f(x) =√x at x = 9 yields the
compute the error E for h = 10−n, 1 ≤ n ≤ 4, and
verify numerically that E ≤ 0.006h2.
that the Linear Approximation to f(x) = tan x at x = π/4 yields the
estimate tan (π/4 + h) – 1 » 2h. Compute the error E for h = 10−n, 1 ≤
n ≤ 4, and verify that E satisfies the Error Bound (3) with K =
that for any real number k, (1 + x) k ≈ 1 + kx for small x. Estimate
(1.02)0.7 and (1.02)−0.3.
Show that f(x) = sin x and g(x) = tan x have the same linearization at a = 0.
function is approximated more accurately? Explain using a graph over [0,
Calculate the error in these linearizations at x = π/6. Does the answer
confirm your conclusion in (b)?
71. Let Df = f(5 + h) − f(5) , where
f(x) = x2, and let E = |Df − f ¢(5)h| be the error in the Linear Approximation. Verify
directly that E satisfies (3) with K = 2 (thus E is of order two in h).
following questions refer to Figure 15.
many critical points does f(x) have?
is the maximum value of f(x) on [0, 8]?
are the local maximum values of f(x)?
a closed interval on which both the minimum and maximum values of f(x) occur at
an interval on which the minimum value occurs at an endpoint.
Exercises 3–14, find all critical points of the function.
= x2 − 2x + 4
f(x) = x2 − 4x + 1.
the critical point c of f(x) and compute f(c).
Compute the value of f(x) at the endpoints of the interval [0, 4].
Determine the min and max of f(x) on [0, 4].
the extreme values of f(x) on [0, 1].
the extreme values of 2x3 − 9x2 + 12x on [0, 3] and
the minimum value of y = tan−1(x2 − x).
the critical points of f(x) = sin x + cos x and determine the extreme values on
Compute the critical points of h(t) = (t2 − 1)1/3.
Check that your answer is consistent with Figure 17. Then find the
extreme values of h(t) on [0, 1] and [0, 2].
f(x) = 2√x − x on [0, 4] and determine the maximum value
graphically. Then verify your answer using calculus.
f (x) = ln x − 5 sin x on [0, 2π] (choose an appropriate viewing
rectangle) and approximate both the critical points and extreme values.
Approximate the critical points of g(x) = x arccos x and estimate the maximum
value of g(x).
Exercises 23–56, find the maximum and minimum values of the function on
the given interval.
23. y =
2x2 − 4x + 2, [0, 3].
f(x) = 3x − x3. Check that f (−2) = f(1). What may we conclude
from Rolle’s Theorem? Verify this conclusion.
Exercises 64–67, verify Rolle’s Theorem for the given interval.
64. f(x) = x + x−1,
Rolle’s Theorem to prove that f(x) = x5 + 2x3 + 4x
− 12 has at most one real root.
Rolle’s Theorem to prove that f (x) = x3/6 + x2/2 + x + 1
has at most one real root.
concentration C ( t ) (in mg/cm3) of a drug in a patient’s
bloodstream after t hours is . Find the
maximum concentration and the time at which it occurs.
by considering its minimum, that f(x) = x2 − 2x + 3 takes on
only positive values. More generally, find the conditions on r and s
under which the quadratic function f(x) = x2 + rx + s takes on only
positive values. Give examples of r and s for which f takes on both positive
and negative values.
that if the quadratic polynomial f(x) = x2 + rx + s takes on both
positive and negative values, then its minimum value occurs at the midpoint
between the two roots.
the minimum and maximum values of f(x) = xp(1 − x)q
on [0, 1], where p and q are positive numbers.
Exercises 1–10, find a point c satisfying the conclusion of the MVT for
the given function and interval.
1. y = x−1,
f(x) = x5 + x2. Check that the secant line between x = 0
and x = 1 has slope 2. By the MVT, f ' (c) = 2 for some c Îin the interval (0,1). Estimate c graphically as
follows. Plot f(x) and the secant line on the same axes. Then plot the lines y
= 2x + b for different values of b until you find a value of b for which
it is tangent to y = f(x). Zoom in on the point of tangency to find its
Determine the intervals on which f (x) is positive and negative, assuming that Figure 12 is
the graph of f(x).
Determine the intervals on which f ( x ) is increasing or decreasing, assuming
that Figure 12 is the graph of the derivative f ' (x).
the derivative f ' (x)
of f(x) = 3x5 − 5x3 and describe the sign changes
of f ' (x).
Use this to determine the local extreme values of f(x). Then graph f(x) to
confirm your conclusions.
Exercises 15–18, sketch the graph of a function f(x) whose derivative f ' (x) has the given description.
15. f '(x) > 0 for x > 3 and f '(x) < 0 for x < 3.
Exercises 19–22, use the First Derivative Test to determine whether the
function attains a local minimum or local maximum (or neither) at the given
19. y = 7
+ 4x − x2, c = 2
Exercises 25–52, find the critical points and the intervals on which the
function is increasing or decreasing, and apply the First Derivative Test to
each critical point.
25. y =
−x2 + 7x − 17
that f(x) = x2 + bx + c is decreasing on (−∞,
−b/2) and increasing on (−b/2, ∞).
that f(x) = x3 − 2x2 + 2x is an increasing
conditions on a and b that ensure that f(x) = x3 + ax + b is
increasing on (−∞, ∞) .
57. Sam made two statements that Deborah found dubious.
“Although the average velocity for my trip was 70 mph, at no point in time did my speedometer read 70 mph.”
“Although a policeman clocked me going 70 mph, my speedometer never read 65 mph.”
case, which theorem did Deborah apply to prove Sam’s statement false: the
Intermediate Value Theorem or the Mean Value Theorem? Explain.
Determine where f(x) = (1000 − x)2 + x2 is
increasing. Use this to decide which is larger: 10002 or 9982
that f(x) = 1 − |x| satisfies the conclusion of the MVT on [a, b]
if both a and b are positive or negative, but not if a < 0 and b > 0.
values of c satisfy the conclusion of the MVT on the interval [a, b] if f(x)
is a linear function?
that if f is a quadratic polynomial, then the midpoint c = (a + b)/2
satisfies the conclusion of the MVT on [a, b] for any a and b.
Suppose that f(0) = 4 and f ¢(x) ≤ 2 for x > 0. Apply the MVT to the interval
[0, 3] to prove that f(3) ≤ 10. Prove more generally that f(x) ≤ 4
+ 2x for all x > 0.
Suppose that f(2) = −2 and f ¢( x ) ≥ 5. Show that f(4) ≥ 8.
the minimum value of f(x) = xx for x > 0.
that the cubic function f(x) = x3 + ax2 + bx + c is
increasing on (−∞, ∞) if b > a2/3.
that if f(0) = g(0) and f ' (x) ≤ g ' (x) for x ≥ 0, then f(x) ≤ g(x) for all x
Exercise 66 to prove that sin x ≤ x for x ≥ 0.
Sketch the graph of f(x) = x4 and state whether f has any points of
inflection. Verify your conclusion by showing that f ²(x) does not change sign.
20. Through her website, Leticia has been selling bean bag chairs with monthly
sales as recorded below. In a report to investors, she states, “Sales reached a
point of inflection when I started using pay-per-click advertising.” In
which month did that occur? Explain.
Figure 16 shows the graph of the derivative f ¢(x) on [0, 1.2]. Locate the points
of inflection of f(x) and the points where the local minima and maxima
occur. Determine the intervals on which f(x) has the following properties:
Increasing (b) Decreasing (c) Concave up (d) Concave down
Exercises 23–36, find the critical points of f(x) and use the Second
Derivative Test (if possible) to determine whether each corresponds to a local
minimum or maximum.
23. f(x) = x3 − 12x2
Exercises 37–50, find the intervals on which f is concave up or down, the
points of inflection, and the critical points, and determine whether each
critical point corresponds to a local minimum or maximum (or neither).
37. f(x) = x3 − 2x2
infectious flu spreads slowly at the beginning of an epidemic. The
infection process accelerates until a majority of the susceptible individuals
are infected, at which point the process slows down.
R(t) is the number of individuals infected at time t, describe the concavity of
the graph of R near the beginning and end of the epidemic.
a one-sentence news bulletin describing the status of the epidemic on the day
that R(t) has a point of inflection.
is pumped into a sphere at a constant rate (Figure 17). Let h(t) be the water
level at time t. Sketch the graph of h(t) (approximately, but with the correct
concavity). Where does the point of inflection occur?
is pumped into a sphere at a variable rate in such a way that the water level
rises at a constant rate c (Figure 17). Let V(t) be the volume of water at time
t. Sketch the graph of V(t) (approximately, but with the correct concavity).
Where does the point of inflection occur?
the dimensions of the rectangle of maximum area that can be formed from a
50-in. piece of wire.
is the constraint equation relating the lengths x and y of the sides?
a formula for the area in terms of x alone.
this problem require optimization over an open interval or a closed interval?
the optimization problem.
2. A 100-in. piece of wire is divided
into two pieces and each piece is bent into a square. How should this be done
in order to minimize the sum of the areas of the two squares?
Express the sum of the areas of the squares in terms of the lengths x and y of
the two pieces.
is the constraint equation relating x and y?
this problem require optimization over an open or closed interval?
the optimization problem.
the positive number x such that the sum of x and its reciprocal is as small as
possible. Does this problem require optimization over an open interval or a
legs of a right triangle have lengths a and b satisfying a + b = 10. Which
values of a and b maximize the area of the triangle?
positive numbers x, y such that xy = 16 and x + y is as small as possible.
6. A 20-in. piece of wire is bent into
an L-shape. Where should the bend be made to minimize the distance between the
7. Let S
be the set of all rectangles with area 100.
are the dimensions of the rectangle in S with the least perimeter?
there a rectangle in S with the greatest perimeter? Explain.
8. A box has a square base of side x
and height y.
the dimensions x, y for which the volume is 12 and the surface area is as small
the dimensions for which the surface area is 20 and the volume is as large as
Suppose that 600 ft of fencing are used to enclose a corral in the shape of a
rectangle with a semicircle whose diameter is a side of the rectangle as in
Figure 10. Find the dimensions of the corral with maximum area.
the rectangle of maximum area that can be inscribed in a right triangle with
legs of length 3 and 4 if the sides of the rectangle are parallel to the legs
of the triangle, as in Figure 11.
11. A landscape architect wishes to
enclose a rectangular garden on one side by a brick wall costing $30/ft and on
the other three sides by a metal fence costing $10/ft. If the area of the
garden is 1000 ft2, find the dimensions of the garden that
minimize the cost.
the point on the line y = x closest to the point (1, 0).
the point P on the parabola y = x2 closest to the point (3, 0)
15. A box is constructed out of two
different types of metal. The metal for the top and bottom, which are both square,
costs $1/ft2 and the metal for the sides costs $2/ft2.
Find the dimensions that minimize cost if the box has a volume of 20 ft3.
the dimensions of the rectangle of maximum area that can be inscribed in a
circle of radius r (Figure 14).
of Tartaglia (1500–1557) Among all positive numbers a , b whose sum is 8,
find those for which the product of the two numbers and their difference
the angle θ that maximizes the area of the isosceles triangle whose legs
have length l (Figure 15).
volume of a right circular cone is and its surface area is . Find the dimensions of the cone
with surface area 1 and maximal volume (Figure 16).
production requires both labor and capital investment in equipment and land.
Suppose that if x dollars per acre are invested in labor and y dollars per acre
are invested in equipment and land, then the yield P of rice per acre is
given by the formula P =100√x + 150√y. If a farmer invests $40/
acre, how should he divide the $40 between labor and capital investment in
order to maximize the amount of rice produced?
the dimensions x and y of the rectangle inscribed in a circle of radius r that
maximizes the quantity xy2.
the angle θ that maximizes the area of the trapezoid with a base of length
4 and sides of length 2, as in Figure 17.
Consider a rectangular industrial warehouse consisting of three separate spaces
of equal size as in Figure 18. Assume that the wall materials cost $200 per
linear ft and the company allocates $2,400,000 for the project.
dimensions maximize the total area of the warehouse?
is the area of each compartment in this case?
in the previous exercise, that the warehouse consists of n separate spaces of
equal size. Find a formula in terms of n for the maximum possible area of the
amount of light reaching a point at a distance r from a light source A of intensity
IA is IA /r2. Suppose that a second light
source B of intensity IB = 4IA is located 10 ft from A. Find the point on the segment joining A and B where the total amount of light is at a
the area of the largest rectangle that can be inscribed in the region bounded
by the graph of y = (4 − x)/(2+ x) and the coordinate axes (Figure 19).
According to postal regulations, a carton is classified as “oversized” if
the sum of its height and girth (the perimeter of its base) exceeds 108 in. Find the dimensions of a carton with square base that is not oversized and has maximum
the maximum area of a triangle formed in the first quadrant by the
x-axis, y-axis, and a tangent line to the graph of y = ( x + 1 )−2.
is the area of the largest rectangle that can be circumscribed around a
rectangle of sides L and H? Hint: Express the area of the circumscribed
rectangle in terms of the angle θ (Figure 20).
Optimal Price Let r be the monthly rent per unit in an apartment building
with 100 units. A survey reveals that all units can be rented when r = $900 and
that one unit becomes vacant with each $10 increase in rent. Suppose that the
average monthly maintenance per occupied unit is $100 / month.
that the number of units rented is n = 190 − r/10 for 900 ≤ r
a formula for the net cash intake (revenue minus maintenance) and determine the
rent r that maximizes intake.
8-billion-bushel corn crop brings a price of $2.40/bushel. A commodity broker
uses the following rule of thumb: If the crop is reduced by x percent, then the
price increases by 10x cents. Which crop size results in maximum revenue and
what is the price per bushel?
34. Let P
= (a, b) be a point in the first quadrant.
the slope of the line through P such that the triangle bounded by this line and
the axes in the first quadrant has minimal area.
that P is the midpoint of the hypotenuse of this triangle.
35. A truck gets 10 mpg (miles per
gallon) traveling along an interstate highway at 50 mph, and this is reduced by 0.15 mpg for each mile per hour increase above 50 mph.
the truck driver is paid $30 / hour and diesel fuel costs P = $3/gal, which
speed v between 50 and 70 mph will minimize the cost of a trip along the
highway? Notice that the actual cost depends on the length of the trip but the
optimal speed does not.
cost as a function of v (choose the length arbitrarily) and verify your answer
to part (a).
you expect the optimal speed v to increase or decrease if fuel costs go down to
P = $2/gal? Plot the graphs of cost as a function of v for P = 2 and P = 3 on
the same axis and verify your conclusion.
Figure 21 shows a rectangular plot of size 100 × 200 feet. Pipe is to be laid from A to a point P on side BC and from there to C. The cost of laying
pipe through the lot is $30/ft (since it must be underground) and the cost
along the side of the plot is $15/ft.
f(x) be the total cost, where x is the distance from P to B. Determine f(x),
but note that f is continuous at x = 0 (when x = 0, the cost of the entire
pipe is $15/ft).
is the most economical way to lay the pipe? What if the cost along the sides is
the dimensions of a cylinder of volume 1 m3 of minimal cost if the top and bottom are made of material that costs twice as much as the material
for the side.
38. In Example 6 in this section, find the x-coordinate of the point P where the light beam strikes the mirror
if h1 = 10, h2 = 5, and L = 20.
Exercises 39–41, a box (with no top) is to be constructed from a piece of
cardboard of sides A and B by cutting out squares of length h from the corners
and folding up the sides (Figure 22).
the value of h that maximizes the volume of the box if A = 15 and B = 24. What
are the dimensions of the resulting box?
value of h maximizes the volume if A = B?
Suppose that a box of height h = 3 in. is constructed using 144 in.2 of cardboard (i.e., AB = 144). Which values A and B maximize the volume?
monthly output P of a light bulb factory is given by the formula P = 350LK,
where L is the amount invested in labor and K the amount invested in equipment
(in thousands of dollars). If the company needs to produce 10,000 units per
month, how should the investment be divided among labor and equipment to
minimize the cost of production? The cost of production is L + K.
calculus to show that among all right triangles with hypotenuse of length 1,
the isosceles triangle has maximum area. Can you see more directly why this
must be true by reasoning from Figure 23?
Janice can swim 3 mph and run 8 mph. She is standing at one bank of a river
that is 300 ft wide and wants to reach a point located 200 ft downstream on the other side as quickly as possible. She will swim diagonally across the river
and then jog along the river bank. Find the best route for Janice to take.
Find the radius and height of a cylindrical can of total surface area A whose
volume is as large as possible.
you design a cylinder with total surface area A and minimal total volume?
the area of the largest isosceles triangle that can be inscribed in a circle of
48. A billboard of height b is mounted
on the side of a building with its bottom edge at a distance h from the street.
At what distance x should an observer stand from the wall to maximize the angle
of observation θ (Figure 24)?
x using calculus.
the problem again using geometry (without any calculation!). There is a unique
circle passing through points B and C whichis tangent to the street. Let R be
the point of tangency. Show that θ is maximized at the point R. Hint: The
two angles labeled ψ are, in fact, equal because they subtend equal arcs
on the circle. Let A be the intersection of the circle with PC and show that
ψ = θ + PBA > θ.
that the two answers in (a) and (b) agree.
49. Use the
result of Exercise 48 to show that θ is maximized at the value of x for
which the angles ÐQPB
and ÐQCP are equal.
52. A poster of area 6 ft2 has blank margins of width 6 in. on the top and bottom and 4 in. on the sides. Find the dimensions that maximize the printed area.
the minimum length l of a beam that can clear a fence of height h and touch a
wall located b ft behind the fence (Figure 27).
(a, b) be a fixed point in the first quadrant and let S(d) be the
sum of the distances from (d, 0) to the points (0, 0) , (a, b) , and (a,
the value of d for which S ( d ) is minimal. The answer depends on whether b
< √3a or b ≥√3a.
(b) Let a
= 1. Plot S(d) for b = 0.5, √3, 3 and describe the position of the
minimum force required to drive a wedge of angle α into a block (Figure
28) is proportional to
is a positive constant. Find the angle α for which the least force is
required, assuming f = 0.4.
57. In the setting of Exercise 56, show
that for any f the minimal force required is proportional to .
problem is to put a “roof” of side s on an attic room of height h and width b.
Find the smallest length s for which this is possible. See Figure 29.
the maximum length of a pole that can be carried horizontally around a corner
joining corridors of widths 8 ft and 4 ft (Figure 30).
Exercise 59 for corridors of arbitrary widths a and b.
the isosceles triangle of smallest area that circumscribes a circle of radius 1
(from Thomas Simpson’s The Doctrine and Application of Fluxions, a calculus
text that appeared in 1750). See Figure 31.
Exercises 1–4, use Newton’s Method with the given function and initial value x0
to calculate x1, x2, x3.
1. f(x) = x2 − 2,
x0 = 1
Figure 6 to choose an initial guess x0 to the unique real root of x3 + 2x + 5 = 0. Then compute the first three iterates of Newton’s Method.
6. Use Newton’s Method to find a solution to sin x = cos 2x in the interval [0, π/2]
to three decimal places. Then guess the exact solution and compare with your
7. Use Newton’s Method to find the two solutions of ex = 5x to three decimal
places (Figure 7).
8. Use Newton’s Method to approximate the positive solution to the equation ln(x + 4) = x to
three decimal places.
Exercises 9–12, use Newton’s Method to approximate the root to three decimal
places and compare with the value obtained from a calculator.
13. Use Newton’s Method to approximate the largest positive root of f(x) = x4 −
6x2 + x + 5 to within an error of at most 10−4.
Refer to Figure 5.
Sketch the graph of f(x) = x3 − 4x + 1 and use Newton’s Method to approximate the largest positive root to within an error of at most 10−3.
15. Use a
graphing calculator to choose an initial guess for the unique positive root of
x4 + x2 − 2x − 1 = 0. Calculate the
first three iterates of Newton’s Method.
first positive solution of sin x = 0 is x = π. Use Newton’s Method
to calculate π to four decimal places.
that f(x) = tan2x and g(x) = sec2x have the same
derivative. What can you conclude about the relation between f and g? Verify
this conclusion directly.
by computing derivatives, that for some constant C. Find C by setting x =
71. A particle located at the origin at
t = 0 begins moving along the x-axis with velocity ft/s. Let s(t) be its position at
time t. State the differential equation with initial condition satisfied
by s(t) and find s(t).
Repeat Exercise 71, but replace the initial condition s(0) = 0 with s(2) = 3.
73. A particle moves along the x-axis
with velocity v(t) = 25t −t2 ft/s. Let s(t) be the position at
s(t), assuming that the particle is located at x = 5 at time t = 0.
s(t), assuming that the particle is located at x = 5 at time t = 2.
74. A particle located at the origin at
t = 0 moves in a straight line with acceleration ft / s. Let v(t) be the velocity and s(t)
the position at time t.
and solve the differential equation for v( t ) assuming that the particle is at
rest at t = 0.
75. A car traveling 84 ft/s begins to
decelerate at a constant rate of 14 ft/s2. After how many seconds
does the car come to a stop and how far will the car have traveled before
Beginning at rest, an object moves in a straight line with constant
acceleration a, covering 100 ft in 5 s. Find a.
77. A 900-kg rocket is released from a
spacecraft. As the rocket burns fuel, its mass decreases and its velocity
increases. Let v( m ) be the velocity (in meters per second) as a function of
mass m. Find the velocity when m = 500 if dv/dm = −50m−1/2.
Assume that v(900) = 0.
water flows through a tube of radius R = 10 cm, the velocity of an individual water particle depends on its distance r from the center of the
tube according to the formula dv/dr = −0.06r. Determine v(r), assuming
that particles at the walls of the tube have zero velocity.
constants c1 and c2 such that F(x) = c1x sin x
+ c2 cos x is an antiderivative of f(x) = x cos x.
the general antiderivative of (2x + 9)10.
1. An athlete
runs with velocity 4 mph for half an hour, 6 mph for the next hour, and 5 mph for another half-hour. Compute the total distance traveled and
indicate on a graph how this quantity can be interpreted as an area.
14 shows the velocity of an object over a 3-min interval. Determine the
distance traveled over the intervals [0, 3] and [1, 2.5] (remember to convert
from miles per hour to miles per minute).
3. A rainstorm hit Portland, Maine, in October 1996, resulting in record rainfall. The rainfall rate R(t) on October 21 is recorded, in inches per
hour, in the following table, where t is the number of hours since
midnight. Compute the total rainfall during this 24-hour period and indicate on
a graph how this quantity can be interpreted as an area.
4. The velocity of an object is v(t) = 32t ft/s. Use Eq. (2) and geometry to find the
distance traveled over the time intervals [0, 2] and [2, 5].
5. Compute R6, L6,
and M3 to estimate the distance traveled over [0, 3] if the velocity at half-second intervals is as follows:
6. Use the following table of values to
estimate the area under the graph of f (x) over [0,1] by computing the average of R5
7. Consider f (x) = 2x + 3 on [0, 3].
(a) Compute R6 and L6
over [0, 3].
(b) Find the error in these approximations by
computing the area exactly using geometry.
1. An airplane makes the 350-mile trip from Los Angeles to San Francisco in 1 hour. Assuming that the plane’s velocity at time t is
v(t) mph, what is the value of the integral ?
2. A hot metal object is submerged in cold
water. The rate at which the object cools (in degrees per minute) is a function
f (t) of time. Which quantity is represented by
the integral ?
3. Which of the following quantities would be
naturally represented as derivatives and which as integrals?
(a) Velocity of a train
(b) Rainfall during a 6-month period
(c) Mileage per gallon of an automobile
(d) Increase in the population of Los Angeles from 1970 to 1990
4. Two airplanes take off at t = 0 from the same place and in the same
direction. Their velocities are v1(t) and
v2(t), respectively. What is the physical
interpretation of the area between the graphs of v1(t) and v2(t) over an interval [0, T ]?
1. Water flows into an empty reservoir at a
rate of 3,000 + 5t gal/hour. What is the quantity of water in the
reservoir after 5 hours?
2. Find the displacement of a particle moving
in a straight line with velocity v(t) =
4t − 3 ft/s over the time interval [2, 5].
3. A population of insects increases at a
rate of 200 + 10t + 0.25t2 insects per day.
Find the insect population after 3 days, assuming that there are 35 insects at t
4. A survey shows that a mayoral candidate is
gaining votes at a rate of 2,000t
+ 1,000 votes per day, where t is the
number of days since she announced her candidacy. How many supporters will the
candidate have after 60 days, assuming that she had no supporters at t = 0?
5. A factory produces bicycles at a rate of
95 + 0.1t2 − t bicycles per week (t in weeks). How
many bicycles were produced from day 8 to 21?
6. Find the displacement over the time
interval [1, 6] of a helicopter whose (vertical) velocity at time t is v(t) = 0.02t2 + t ft/s.
7. A cat falls from a tree (with zero initial
velocity) at time t = 0.
How far does the cat fall between t = 0.5 and t = 1 s? Use Galileo’s formula v(t) = −32t ft/s.
8. A projectile is released with initial
(vertical) velocity 100 m/s.
Use the formula v(t) = 100 − 9.8t for velocity to determine the
distance traveled during the first 15 s.
In Exercises 9–12, assume that a particle
moves in a straight line with given velocity. Find the total displacement and
total distance traveled over the time interval, and draw a motion diagram like
Figure 3 (with distance and time labels).
9. 12 − 4t ft/s, [0, 5]
13. The rate (in liters per minute) at which
water drains from a tank is recorded at half-minute intervals. Use the average
of the left- and right-endpoint approximations to estimate the total amount of
water drained during the first 3 min.
14. The velocity of a car is recorded at
half-second intervals (in feet per second). Use the average of the left- and
right-endpoint approximations to estimate the total distance traveled during
the first 4 s.
15. Let a(t) be the acceleration of an object in linear
motion at time t. Explain why is the net change in velocity over [t1, t2]. Find the net change in velocity over [1, 6] if a(t) =
24t − 3t2 ft/s2.
16. Show that if acceleration a is
constant, then the change in velocity is proportional to the length of the time
17. The traffic flow rate past a certain point
on a highway is q(t) =
3,000 + 2,000t − 300t2, where t is
in hours and t = 0
is 8 AM. How many cars pass by during the time interval from 8 to 10 AM?
18. Suppose that the marginal cost of
producing x video recorders is 0.001x2 −0.6x + 350 dollars. What is the cost of producing
300 units if the setup cost is $20,000 (see Example 4)? If production is set at
300 units, what is the cost of producing 20 additional units?
19. Carbon Tax To encourage manufacturers to reduce
pollution, a carbon tax on each ton of CO2 released into the
atmosphere has been proposed. To model the effects of such a tax, policymakers
study the marginal cost of abatement B(x), defined as the cost of increasing CO2
reduction from x to x + 1 tons (in units of ten thousand
tons—Figure 4). Which quantity is represented by ?
20. Power is the rate of energy consumption per
unit time. A megawatt of power is 106 W or 3.6 × 109 J/hour. Figure 5 shows the power supplied by
the California power grid over a typical 1-day period. Which quantity is
represented by the area under the graph?
21. Figure 6 shows the migration rate M(t) of Ireland during the period 1988–1998.
This is the rate at which people (in thousands per year) move in or out of the
(a) What does represent?
(b) Did migration over the 11-year period
1988–1998 result in a net influx or outflow of people from Ireland? Base your answer on a rough estimate of the positive and negative areas involved.
(c) During which year could the Irish prime
minister announce, “We are still losing population but we’ve hit an inflection
point—the trend is now improving.”
22. Figure 7 shows the graph of Q(t), the rate of retail truck sales in the United States (in thousands sold per year).
(a) What does the area under the graph over
the interval [1995, 1997] represent?
(b) Express the total number of trucks sold in
the period 1994–1997 as an integral (but do not compute it).
(c) Use the following data to compute the
average of the right- and left-endpoint approximations as an estimate for the
total number of trucks sold during the 2-year period 1995–1996.
23. Heat Capacity The heat capacity C(T ) of a substance is the amount of energy (in
joules) required to raise the temperature of 1 g by 1◦C at temperature T .
(a) Explain why the energy required to raise
the temperature from T1 to T2 is the area
under the graph of C(T ) over
(b) How much energy is required to raise the
temperature from 50 to 100◦C
if C(T ) = 6 + 0.2√T ?
In Exercises 24 and 25, consider the
following. Paleobiologists have studied the extinction of marine animal
families during the phanerozoic period, which began 544 million years ago. A
recent study suggests that the extinction rate r(t) may be modeled by the function r (t) = 3,130/(t + 262) for 0 ≤ t ≤ 544. Here, t is time elapsed (in
millions of years) since the beginning of the phanerozoic period. Thus, t = 544 refers to the present time, t = 540 is 4 million years ago, etc.
24. Use RN or LN
with N =
10 (or their
average) to estimate the total number of families that became extinct in the
periods 100 ≤
t ≤ 150 and 350 ≤ t ≤ 400.
25. Estimate the total number of extinct
families from t = 0
to the present, using MN with N = 544.
26. Cardiac output is the rate R of
volume of blood pumped by the heart per unit time (in liters per minute).
Doctors measure R by injecting A mg of dye into a vein leading
into the heart at t = 0
and recording the concentration c(t) of dye (in milligrams per liter) pumped
out at short regular time intervals (Figure 8).
(a) The quantity of dye pumped out in a small
time interval [t, t + Δt] is approximately Rc(t) Δt. Explain why.
(b) Show that , where T is large enough that all of
the dye is pumped through the heart but not so large that the dye returns by
(c) Use the following data to estimate R,
assuming that A = 5
27. A particle located at the origin at t = 0 moves along the x-axis with
velocity v(t) = (t + 1)−2. Show that the particle will never pass
the point x =
28. A particle located at the origin at t = 0 moves along the x-axis with
velocity v(t) = (t + 1)−1/2.
Will the particle be at the point x = 1 at any time t? If so, find t.
1. Two quantities increase exponentially with
growth constants k = 1.2 and k = 3.4, respectively. Which quantity doubles
2. If you are given both the doubling time
and the growth constant of a quantity that increases exponentially, can you
determine the initial amount?
3. A cell population grows exponentially
beginning with one cell. Does it take less time for the population to increase
from one to two cells than from 10 million to 20 million cells?
4. Referring to his popular book A Brief
History of Time, the renowned physicist Stephen Hawking said, “Someone told
me that each equation I included in the book would halve its sales.” If this is
so, write a differential equation satisfied by the sales function S(n), where n is the number of
equations in the book.
5. Carbon dating is based on the assumption
that the ratio R of C14 to C12 in the atmosphere
has been constant over the past 50,000 years. If R were actually smaller
in the past than it is today, would the age estimates produced by carbon dating
be too ancient or too recent?
6. Which is preferable: an interest rate of
12% compounded quarterly, or an interest rate of 11% compounded continuously?
7. Find the yearly multiplier if r = 9% and interest is compounded (a)
continuously and (b) quarterly.
8. The PV of N dollars received at
time T is (choose the correct answer):
(a) The value at time T of N dollars
(b) The amount you would have to invest today
in order to receive N dollars at time T
9. A year from now, $1 will be received. Will
its PV increase or decrease if the interest rate goes up?
10. Xavier expects to receive a check for
$1,000 1 year from today. Explain, using the concept of PV, whether he will be
happy or sad to learn that the interest rate has just increased from 6% to 7%.
1. A certain bacteria population P obeys
the exponential growth law P(t) =
2,000e1.3t (t in hours).
(a) How many bacteria are present initially?
(b) At what time will there be 10,000
2. A quantity P obeys the exponential
growth law P(t) =
e5t (t in years).
(a) At what time t is P = 10?
(b) At what time t is P = 20?
(c) What is the doubling time for P?
3. A certain RNA molecule replicates every 3
minutes. Find the differential equation for the number N(t) of molecules present at time t (in
minutes). Starting with one molecule, how many will be present after 10 min?
4. A quantity P obeys the exponential
growth law P(t) =
(t in years).
Find the formula for P(t),
assuming that the doubling time is 7 years and P(0) = 100.
5. The decay constant of Cobalt-60 is 0.13 years−1. What is its halflife?
6. Find the decay constant of Radium-226,
given that its half-life is 1,622 years.
7. Find all solutions to the differential
equation y ' = −5y.
Which solution satisfies the initial condition y(0) = 3.4?
8. Find the solution to y ' = √2y satisfying y(0) = 20.
9. Find the solution to y ' = 3y satisfying y(2) = 4.
10. Find the function y = f (t) that satisfies the differential equation y
' = −0.7y and initial condition y(0) = 10.
11. The population of a city is P(t) = 2 · e0.06t (in millions), where t is measured
(a) Calculate the doubling time of the
(b) How long does it take for the population
to triple in size?
(c) How long does it take for the population
to quadruple in size?
12. The population of Washington state
increased from 4.86 million in 1990 to 5.89 million in 2000. Assuming exponential
(a) What will the population be in 2010?
(b) What is the doubling time?
13. Assuming that population growth is
approximately exponential, which of the two sets of data is most likely to
represent the population (in millions) of a city over a 5-year period?
14. Light Intensity The intensity of light passing
through an absorbing medium decreases exponentially with the distance traveled.
Suppose the decay constant for a certain plastic block is k = 2 when the distance is measured in feet.
How thick must the block be to reduce the intensity by a factor of one-third?
15. The Beer–Lambert Law is used in
spectroscopy to determine the molar absorptivity α or the concentration c
of a compound dissolved in a solution at low concentrations (Figure 12).
The law states that the intensity I of light as it passes through the
solution satisfies ln(I /I0) =
where I0 is the initial intensity and x is the
distance traveled by the light. Show that I satisfies
a differential equation dI/dx =
−kx for some constant k.
16. An insect population triples in size after
5 months. Assuming exponential growth, when will it quadruple in size?
17. A 10-kg quantity of a radioactive isotope
decays to 3 kg after 17 years. Find the decay constant of the isotope.
18. Measurements showed that a sample of
sheepskin parchment discovered by archaeologists had a C14 to C12
ratio equal to 40% of that found in the atmosphere. Approximately how old is
20. A paleontologist has discovered the
remains of animals that appear to have died at the onset of the Holocene ice
age. She applies carbon dating to test her theory that the Holocene age started
between 10,000 and 12,000 years ago. What range of C14 to C12
ratio would she expect to find in the animal remains?
21. Atmospheric Pressure The atmospheric pressure P(h) (in pounds per square inch) at a height h
(in miles) above sea level on earth satisfies a differential equation P
' = −kP for some positive constant k.
(a) Measurements with a barometer show that P(0) = 14.7 and P(10) = 2.13. What is the decay constant k?
(b) Determine the atmospheric pressure 15
miles above sea level.
22. Inversion of Sugar When cane sugar is dissolved in
water, it converts to invert sugar over a period of several hours. The
percentage f (t) of
unconverted cane sugar at time t decreases exponentially. Suppose that f
' = −0.2f. What percentage of cane sugar
remains after 5 hours? After 10 hours?
23. A quantity P increases
exponentially with doubling time 6 hours. After how many hours has P increased
24. Two bacteria colonies are cultivated in a
laboratory. The first colony has a doubling time of 2 hours and the second a
doubling time of 3 hours. Initially, the first colony contains 1,000 bacteria
and the second colony 3,000 bacteria. At what time t will sizes of the
colonies be equal?
25. Moore’s Law In 1965, Gordon Moore predicted that the
number N of transistors on a microchip would increase exponentially.
(a) Does the table of data below confirm Moore’s prediction for the period from 1971 to 2000? If so, estimate the growth constant k.
(b) Plot the data in the table.
(c) Let N(t) be the number of transistors t years
after 1971. Find an approximate formula N(t) ≈ Cekt, where t is the number
of years after 1971.
(d) Estimate the doubling time in Moore’s Law for the period from 1971 to 2000.
(e) If Moore’s Law continues to hold until the
end of the decade, how many transistors will a chip contain in 2010?
(f) Can Moore have expected his prediction to
26. Assume that in a certain country, the rate
at which jobs are created is proportional to the number of people who already
have jobs. If there are 15 million jobs at t = 0 and 15.1 million jobs 3 months later,
how many jobs will there be after two years?
28. To model mortality in a population of 200
laboratory rats, a scientist assumes that the number P(t) of rats alive at time t (in months)
satisfies the Gompertz equation with M = 204 and k = 0.15 months−1 (Figure 13). Find P(t) [note that P(0) = 200] and determine the population after 20
29. A certain quantity increases
quadratically: P(t) =
(a) Starting at time t0 = 1, how long will it take for P to
double in size? How long will it take starting at t0 = 2 or 3?
(b) In general, starting at time t0,
how long will it take for P to double in size?
30. Verify that the half-life of a quantity
that decays exponentially with decay constant k is equal to ln 2/k.
31. Compute the balance after 10 years if
$2,000 is deposited in an account paying 9% interest and interest is compounded
(a) quarterly, (b) monthly, and (c) continuously.
32. Suppose $500 is deposited into an account
paying interest at a rate of 7%, continuously compounded. Find a formula for
the value of the account at time t. What is the value of the account
after 3 years?
33. A bank pays interest at a rate of 5%. What
is the yearly multiplier if interest is compounded
(a) yearly? (b) three times a year? (c) continuously?
34. How long will it take for $4,000 to double
in value if it is deposited in an account bearing 7% interest, continuously
35. Show that if interest is compounded
continuously at a rate r, then an account doubles after (ln 2)/r years.
36. How much must be invested today in order
to receive $20,000 after 5 years if interest is compounded continuously at the
rate r = 9%?
37. An investment increases in value at a
continuously compounded rate of 9%. How large must the initial investment be in
order to build up a value of $50,000 over a seven-year period?
38. Compute the PV of $5,000 received in 3
years if the interest rate is (a) 6% and (b) 11%. What is the PV in these two
cases if the sum is instead received in 5 years?
39. Is it better to receive $1,000 today or
$1,300 in 4 years? Consider r = 0.08 and r = 0.03.
40. Find the interest rate r if the PV
of $8,000 to be received in 1 year is $7,300.
41. If a company invests $2 million to upgrade
its factory, it will earn additional profits of $500,000/year for 5 years. Is the investment
worthwhile, assuming an interest rate of 6% (assume that the savings are
received as a lump sum at the end of each year)?
42. A new computer system costing $25,000 will
reduce labor costs by $7,000/year for 5 years.
(a) Is it a good investment if r = 8%?
(b) How much money will the company actually
43. After winning $25 million in the state
lottery, Jessica learns that she will receive five yearly payments of $5
million beginning immediately.
(a) What is the PV of Jessica’s prize if r = 6%?
(b) How much more would the prize be worth if
the entire amount were paid today?
44. An investment group purchased an office
building in 1998 for $17 million and sold it 5 years later for $26 million.
Calculate the annual (continuously compounded) rate of return on this
45. Use Eq. (3) to compute the PV of an income
stream paying out R(t) =
$5,000/year continuously for 10 years and r = 0.05.
46. Compute the PV of an income stream if
income is paid out continuously at a rate R(t) = $5,000e0.1t /year for 5 years and r = 0.05.
47. Find the PV of an investment that produces
income continuously at a rate of $800/year for 5 years, assuming an interest
rate of r =
48. The rate of yearly income generated by a
commercial property is $50,000/year at t = 0 and increases at a continuously
compounded rate of 5%. Find the PV of the income generated in the first four
years if r =
49. Show that the PV of an investment that
pays out R dollars/year
continuously for T years is R(1 − e−rT )/r, where r is the interest rate.
50. Explain this statement: If T is very
large, then the PV of the income stream described in Exercise 49 is
approximately R/r .
51. Suppose that r = 0.06. Use the result of Exercise 50 to
estimate the payout rate R needed to produce an income stream whose PV
is $20,000, assuming that the stream continues for a large number of years.
53. Use Eq. (6) to compute the PV of an
investment that pays out income continuously at a rate R(t) = (5,000 + 1,000t)e0.02t dollars/year
for 10 years and r = 0.08.
Banker’s Rule of 70 Bankers have a rule of thumb that if you receive R percent
interest, continuously compounded, then your money doubles after approximately
70/R years. For example, at R = 5%, your money doubles after 70/5 or 14 years. Use the concept of doubling
time to justify the Banker’s Rule.
55. Isotopes for Dating Which of the following isotopes
would be most suitable for dating extremely old rocks: Carbon-14 (half-life
5,570 years), Lead-210 (half-life 22.26 years), and Potassium-49 (half-life 1.3
billion years)? Explain why.
56. Let P = P(t) be a quantity that obeys an exponential
growth law with growth constant k. Show that P
increases m-fold after an interval of (lnm)/k years.
57. Average Time of Decay Physicists use the radioactive
decay law R =
R0e−kt to compute the average or mean time M until
an atom decays. Let F(t) =
R/R0 = e−kt be the fraction of atoms that have survived to time t without
(a) Find the inverse function t (F).
(b) The error in the following approximation
tends to zero as N →∞:
39. On a typical day, a city consumes water at
the rate of r (t) =
100 + 72t − 3t2 (in thousands of
gallons per hour), where t is the number of hours past midnight. What is
the daily water consumption? How much water is consumed between 6 PM and
40. The learning curve for producing bicycles
in a certain factory is L(x) =
12x−1/5 (in hours per bicycle), which means that
it takes a bike mechanic L(n) hours
to assemble the nth bicycle. If 24 bicycles are produced, how long does
it take to produce the second batch of 12?
41. Cost engineers at NASA have the task of
projecting the cost P of major space projects. It has been found that
the cost C of developing a projection increases with P at the
rate dC/dP ≈ 21P−0.65, where C is in thousands of
dollars and P in millions of dollars. What is the cost of developing a
projection for a project whose cost turns out to be P = $35 million?
42. The cost of jet fuel increased
dramatically in 2005. Figure 6 displays Department of Transportation estimates
for the rate of percentage price increase R(t) (in units of percentage per year) during
the first 6 months of the year. Express the total percentage price increase I
during the first 6 months as an integral and calculate I . When
determining the limits of integration, keep in mind that t is in years
since R(t) is a yearly rate.
1. Find the area of the region between y = 3x2 +12 and y = 4x +4 over
[−3, 3] (Figure 8).
2. Compute the area of the region in Figure
9(A), which lies between y = 2 −
x2 and y = −2 over [−2, 2].
3. Let f (x) = x and g(x) = 2 − x2 [Figure 9(B)].
(a) Find the points of intersection of the
(b) Find the area enclosed by the graphs of f
4. Let f (x) = 8x − 10 and g(x) = x2 − 4x + 10.
(a) Find the points of intersection of the
(b) Compute the area of the region below the
graph of f and above the graph of g.
19. Find the area of the region enclosed by
the curves y =
x3 − 6x and y = 8 −
20. Find the area of the region enclosed by
the semicubical parabola y2 = x3 and the line x = 1.
23. Find the area of the region lying to the
right of x =
y2 + 4y − 22 and the left of x = 3y + 8.
24. Find the area of the region lying to the
right of x =
y2 − 5 and the left of x = 3 − y2.
25. Calculate the area enclosed by x = 9− y2
and x = 5 in two ways: as an integral
along the y-axis and as an integral along the x-axis.
26. Figure 15 shows the graphs of x = y3 − 26y + 10
and x = 40 − 6y2 − y3. Match the equations with the curve and
compute the area of the shaded region.
50. Find the area enclosed by the curves y = c − x2 and y = x2 − c as
a function of c. Find the value of c for which this area is equal
57. Find the line y = mx that divides the area under the curve y
= x(1 − x) over [0, 1] into two regions of equal area.
58. Let c be the number such that the
area under y =
sin x over [0,π] is divided in half by the line y = cx (Figure 18). Find an equation for c and
solve this equation numerically using a computer algebra system.
1. What is the average value of f (x) on [1, 4] if the area between the graph of f (x) and the x-axis is equal to 9?
2. Find the volume of a solid extending from y
= 2 to y = 5 if the cross section at y has
area A(y) = 5 for all y.
1. Let V be the volume of a pyramid of
height 20 whose base is a square of side 8.
(a) Use similar triangles as in Example 1 to
find the area of the horizontal cross section at a height y.
(b) Calculate V by integrating the
2. Let V be the volume of a right
circular cone of height 10 whose base is a circle of radius 4 (Figure 16).
(a) Use similar triangles to find the area of
a horizontal cross section at a height y.
(b) Calculate V by integrating the
3. Use the method of Exercise 2 to find the
formula for the volume of a right circular cone of height h whose base
is a circle of radius r (Figure 16).
4. Calculate the volume of the ramp in Figure
17 in three ways by integrating the area of the cross sections:
(a) Perpendicular to the x-axis
(b) Perpendicular to the y-axis
(c) Perpendicular to the z-axis
5. Find the volume of liquid needed to fill a
sphere of radius R to height h (Figure 18).
6. Find the volume of the wedge in Figure
19(A) by integrating the area of vertical cross sections.
7. Derive a formula for the volume of the
wedge in Figure 19(B) in terms of the constants a, b, and c.
8. Let B be the solid whose base is
the unit circle x2 + y2
= 1 and whose vertical cross
sections perpendicular to the x-axis are equilateral triangles. Show
that the vertical cross sections have area and compute the volume of B.
In Exercises 9–14, find the volume of the
solid with given base and cross sections.
9. The base is the unit circle x2
+ y2 = 1 and the cross sections perpendicular to the x-axis are
triangles whose height and base are equal.
10. The base is the triangle enclosed by x + y = 1, the x-axis, and the y-axis.
The cross sections perpendicular to the y-axis are semicircles.
11. The base is the semicircle , where −3 ≤ x ≤ 3. The cross sections perpendicular to the
x-axis are squares.
12. The base is a square, one of whose sides
is the interval [0,l] along the x-axis. The cross
sections perpendicular to the x-axis are rectangles of height f (x) = x2.
13. The base is the region enclosed by y = x2 and y = 3. The cross sections perpendicular to the
y-axis are squares.
14. The base is the region enclosed by y = x2 and y = 3. The cross sections perpendicular to the
y-axis are rectangles of height y3.
15. Find the volume of the solid whose base is
the region |x| +|y| ≤ 1 and whose vertical cross sections
perpendicular to the y-axis are semicircles (with diameter along the
16. Show that the volume of a pyramid of
height h whose base is an equilateral triangle of side s is equal
17. Find the volume V of a regular tetrahedron
whose face is an equilateral triangle of side s (Figure 20).
18. The area of an ellipse is π ab, where a and b are
the lengths of the semimajor and semiminor axes (Figure 21). Compute the volume
of a cone of height 12 whose base is an ellipse with semimajor axis a = 6 and
semiminor axis b = 4.
20. A plane inclined at an angle of 45◦
passes through a diameter of the base of a cylinder of radius r. Find the
volume of the region within the cylinder and below the plane (Figure 23).
21. Figure 24 shows the solid S obtained by intersecting
two cylinders of radius r whose axes are perpendicular.
(a) The horizontal cross section of each cylinder at
distance y from the central axis is a rectangular strip. Find the strip’s
(b) Find the area of the horizontal cross section of S at
(c) Find the volume of S as a function of r.
22. Let S be the solid obtained by intersecting two
cylinders of radius r whose axes intersect at an angle θ . Find the volume
of S as a function of r and θ.
23. Calculate the volume of a cylinder inclined at an angle
θ = 30◦ whose height is 10 and whose base is a circle of radius 4
24. Find the total mass of a 1-m rod whose linear density
function is ρ(x) = 10(x + 1)−2 kg/m for 0 ≤ x
25. Find the total mass of a 2-m rod whose linear density
function is ρ(x) = 1 + 0.5sin(πx) kg/m for 0 ≤ x ≤ 2.
26. A mineral deposit along a strip of length 6 cm has density s(x) = 0.01x(6 − x) g/cm for 0 ≤ x ≤ 6. Calculate the total mass
of the deposit.
27. Calculate the population within a 10-mile radius of the
city center if the radial population density is ρ(r) = 4(1 + r2)1/3
(in thousands per square mile).
28. Odzala National Park in the Congo has a high density of
gorillas. Suppose that the radial population density is ρ(r) = 52(1 + r2)−2
gorillas per square kilometer, where r is the distance from a large grassy
clearing with a source of food and water. Calculate the number of gorillas
within a 5-km radius of the clearing.
9. Table 1 lists the population density (in people per
squared kilometer) as a function of distance r (in kilometers) from the center
of a rural town. Estimate the total population within a 2-km radius of the
center by taking the average of the left- and right-endpoint approximations.
10. Find the total mass of a circular plate of radius 20 cm whose mass density is the radial function ρ(r) = 0.03 + 0.01cos (πr2) g/cm2.
31. The density of deer in a forest is the radial
function ρ(r) = 150(r2 + 2)−2 deer
per km2, where r is the distance (in kilometers) to a small
meadow. Calculate the number of deer in the region 2 ≤ r ≤ 5 km.
32. Show that a circular plate of radius 2 cm with radial mass density ρ(r) = 4/r g/cm has finite total mass, even though the
density becomes infinite at the origin.
33. Find the flow rate through a tube of radius 4 cm, assuming that the velocity of fluid particles at a distance r cm from the center is v(r)
= 16 − r2 cm/s.
34. Let v(r) be the velocity of blood in an arterial
capillary of radius R = 4 × 10−5 m. Use Poiseuille’s Law
(Example 6) with k = 106 (m-s)−1to determine the
velocity at the center of the capillary and the flow rate (use correct
35. A solid rod of radius 1cm is placed in a
pipe of radius 3cm so that their axes are aligned. Water flows through
the pipe and around the rod. Find the flow rate if the velocity of the
water is given by the radial function v(r) = 0.5(r − 1)(3 − r)
36. To estimate the volume V of Lake Nogebow, the Minnesota
Bureau of Fisheries created the depth contour map in Figure 26 and determined
the area of the cross section of the lake at the depths recorded in the table
below. EstimateV by taking the average of the right- and left-endpoint
approximations to the integral of cross-sectional area.
53. The temperature T(t) at time t (in hours) in an art
museum varies according to T(t) = 70 + 5cos(πt/12). Find the average over
the time periods [0, 24] and [2, 6].
54. A ball is thrown in the air vertically from
ground level with initial velocity 64 ft/s. Find the average height of the ball
over the time interval extending from the time of the ball’s release to its
return to ground level. Recall that the height at time t is h(t) = 64t −
5. What is the average area of the circles whose radii
vary from 0 to 1?
6. An object with zero initial velocity accelerates at a
constant rate of 10 m/s2. Find its average velocity during the
first 15 s.
57. The acceleration of a particle is a(t) = t − t3
m/s2 for 0 ≤ t ≤ 1. Compute the average acceleration and
average velocity over the time interval [0, 1], assuming that the particle’s
initial velocity is zero.
58. Let M be the average value of f(x) = x4 on
[0, 3] . Find a value of c in [0, 3] such that f(c) = M.
both the Shell and Disk Methods to calculate the volume of the solid obtained
by rotating the region under the graph of f(x) = 8 − x3 for 0
≤ x ≤ 2 about:
x-axis (b) the y-axis
the Shell Method to calculate the volume V of the “bead” formed by removing a
cylinder of radius r from the center of a sphere of radius R (compare with
Exercise 51 in Section 6.3).
surface area of a sphere of radius r is 4πr2. Use this to
derive the formula for the volume V of a sphere of radius R in a new way.
that the volume of a thin spherical shell of inner radius r and thickness Dx is approximately 4πr2Dx.
Approximate V by decomposing the sphere of radius R into N thin spherical
shells of thickness Dx
that the approximation is a Riemann sum which converges to an integral.
Evaluate the integral.
much work is done raising a 4-kg mass to a height of 16 m above ground?
2. How much
work is done raising a 4-lb mass to a height of 16 ft above ground?
Exercises 3–6, compute the work (in joules) required to stretch or compress a
spring as indicated, assuming that the spring constant is k = 150 kg/s2.
Stretching from equilibrium to 12 cm past equilibrium
Compressing from equilibrium to 4 cm past equilibrium
Stretching from 5 to 15 cm past equilibrium
Compressing the spring 4 more cm when it is already compressed 5 cm
7. If 5 J
of work are needed to stretch a spring 10 cm beyond equilibrium, how much work is required to stretch it 15 cm beyond equilibrium?
8. If 5 J
of work are needed to stretch a spring 10 cm beyond equilibrium, how much work is required to compress it 5 cm beyond equilibrium?
9. If 10
ft-lb of work are needed to stretch a spring 1 ft beyond equilibrium, how far will the spring stretch if a 10-lb weight is attached to its end?
that the work required to stretch a spring from position a to position b is , where k is the spring
constant. How do you interpret the negative work obtained when |b| < |a|?
Exercises 11–14, calculate the work against gravity required to build the
structure out of brick using the method of Examples 2 and 3. Assume that brick
has density 80 lb/ft3.
11. A tower of height 20 ft and square base of side 10 ft
12. A cylindrical tower of height 20 ft and radius 10 ft
13. A 20-ft-high tower in the shape of
a right circular cone with base of radius 4 ft
14. A structure in the shape of a
hemisphere of radius 4 ft
around 2600 BCE, the Great Pyramid of Giza in Egypt is 485 ft high (due to erosion, its current height is slightly less) and has a square base of side 755.5 ft (Figure 6). Find the work needed to build the pyramid if the density of the stone is
estimated at 125 lb/ft3.
Exercises 16–20, calculate the work (in joules) required to pump all of the
water out of the tank. Assume that the tank is full, distances are measured in
meters, and the density of water is 1000 kg/m3.
box in Figure 7; water exits from a small hole at the top.
hemisphere in Figure 8; water exits from the spout as shown.
conical tank in Figure 9; water exits through the spout as shown.
horizontal cylinder in Figure 10; water exits from a small hole at the top.
trough in Figure 11; water exits by pouring over the sides.
the work W required to empty the tank in Figure 7 if it is half full of water.
Assume the tank in Figure 7 is full of water and let W be the work required to
pump out half of the water. Do you expect W to equal the work computed in
Exercise 21? Explain and then compute W.
the work required to empty the tank in Figure 9 if it is half full of water.
Assume the tank in Figure 9 is full of water and find the work required
to pump out half of the water.
much work is done lifting a 25-ft chain over the side of a building (Figure
12)? Assume that the chain has a density of 4 lb/ft.
much work is done lifting a 3-m chain over the side of a building if the chain
has mass density 4 kg/m?
8-ft chain weighs 16 lb. Find the work required to lift the chain over the side
of a building.
29. A 20-ft chain with mass density 3
lb/ft is initially coiled on the ground. How much work is performed in lifting
the chain so that it is fully extended (and one end touches the ground)?
much work is done lifting a 20-ft chain with mass density 3 lb/ft (initially
coiled on the ground) so that its top end is 30 ft above the ground?
31. A 1,000-lb wrecking ball hangs
from a 30-ft cable of density 10 lb/ft attached to a crane. Calculate the
work done if the crane lifts the ball from ground level to 30 ft in the air by drawing in the cable.
Exercises 32–34, use Newton’s Universal Law of Gravity, according to which the
gravitational force between two objects of mass m and M separated by a
distance r has magnitude GMm/r2, where G = 6.67 × 10−11 m3kg−1s−1. Although the
Universal Law refers to point masses, Newton proved that it also holds for
uniform spherical objects, where r is the distance between their centers.
spheres of mass M and m are separated by a distance r1. Show that
the work required to increase the separation to a distance r2 is equal
the result of Exercise 32 to calculate the work required to place a 2,000-kg
satellite in an orbit 1,200 km above the surface of the earth. Assume that the
earth is a sphere of mass Me = 5.98 × 1024 kg and radius re = 6.37 × 106 m. Treat the satellite as a point mass.
the result of Exercise 32 to compute the work required to move a 1,500-kg
satellite from an orbit 1,000 to 1,500 km above the surface of the earth.
Assume that the pressure P and volume V of the gas in a 30-in. cylinder of
radius 3 in. with a movable piston are related by PV1.4 = k,
where k is a constant (Figure 13). When the cylinder is full, the gas pressure
is 200 lb/in.2.
Calculate the force on the piston as a function of the length x of the column
of gas (the force is PA, where A is the piston’s area).
Calculate the work required to compress the gas column from 30 to 20 in.
36. A 20-ft chain with linear mass
density ρ(x) = 0.02x(20 − x) lb/ft lies on the ground.
much work is done lifting the chain so that it is fully extended (and one end
touches the ground)?
much work is done lifting the chain so that its top end has a height of 30 ft?
38. A model train of mass 0.5 kg is placed at one end of a straight 3-m electric track. Assume that a force F(x) = 3x − x2
N acts on the train at distance x along the track. Use the Work-Kinetic Energy
Theorem (Exercise 37) to determine the velocity of the train when it reaches
the end of the track.
what initial velocity v0 must we fire a rocket so it attains a maximum
height r above the earth? Hint: Use the results of Exercises 32 and 37. As the
rocket reaches its maximum height, its KE decreases from to zero.
40. With what initial velocity must we fire a rocket so it attains a maximum
height of r = 20 km above the surface of the earth?
airplane’s velocity is recorded at 5-min intervals during a 1-hour period with
the following results, in mph: 550, 575, 600, 580, 610, 640, 625, 595,
590, 620, 640, 640, 630
Simpson’s Rule to estimate the distance traveled during the hour.
Simpson’s Rule to determine the average temperature in a museum over a 3-hour
period, if the temperatures (in degrees Celsius), recorded at 15-min intervals,
are 21, 21.3, 21.5, 21.8, 21.6, 21.2, 20.8, 20.6, 20.9, 21.2, 21.1, 21.3, 21.2.
the average height of a point on the semicircle for − 1 ≤ x ≤ 1.
the volume of the solid obtained by revolving the graph of over [0, 1] about the y-axis.
the volume of the solid obtained by revolving the region between the graph of y2 − x2 = 1 and the line y = 2 about the line y = 2.
the volume of revolution for the region in Exercise 56, but revolve around y =
58. A charged wire creates an electric
field at a point P located at a distance D from the wire (Figure 7). The
component E^ of the field perpendicular to the wire (in volts) is
investment pays a dividend of $250/year continuously forever. If the interest
rate is 7%, what is the present value of the entire income stream generated by
76. An investment
is expected to earn profits at a rate of 10000e0.01t
dollars/year forever. Find the present value of the income stream if the
interest rate is 4%.
Compute the present value of an investment that generates income at a rate of
5000te0.01t dollars/year forever, assuming an interest rate of 6%.
the volume of the solid obtained by rotating the region below the graph of y =
e−x about the x-axis for 0 ≤ x < ∞.
Express the arc length of the curve y = x4 between x = 2 and x = 6
as an integral (but do not evaluate).
Exercises 5–10, calculate the arc length over the given interval.
5. y = 3x
+ 1, [0, 3]
1. A box of height 6 ft and square base of side 3 ft is submerged in a pool of water. The top of the box is 2 ft below the surface of the water.
Calculate the fluid force on the top and bottom of the box.
a Riemann sum that approximates the fluid force on a side of the box by
dividing the side into N horizontal strips of thickness Dy = 6/N.
(c) To which
integral does the Riemann sum converge?
Compute the fluid force on a side of the box.
2. A plate in the shape of an
isosceles triangle with base 1 ft and height 2 ft is submerged vertically in a tank of water so that its vertex touches the surface of the water
that the width of the triangle at depth y is f(y) = y/2.
Consider a thin strip of thickness Dy at depth y. Explain why the fluid force on a side of this
strip is approximately equal to where w = 62.5 lb/ft3.
an approximation for the total fluid force F on a side of the plate as a
Riemann sum and indicate the integral to which it converges.
Exercise 2, but assume that the top of the triangle is located 3 ft below the surface of the water.
thin plate R in Figure 8, bounded by the parabola y = x2 and y = 1,
is submerged vertically in water. Let F be the fluid force on one side of
that the width of R at height y is f(y) = 2√y and the fluid force
on a side of a horizontal strip of thickness Dy at height y is approximately .
a Riemann sum that approximates F and use it to explain why .
5. Let F
be the fluid force (in Newtons) on a side of a semicircular plate of
radius r meters, submerged in water so that its diameter is level with the
water’s surface (Figure 9).
that the width of the plate at depth y is .
Calculate F using Eq. (2).
Calculate the force on one side of a circular plate with radius 2 ft, submerged vertically in a tank of water so that the top of the circle is tangent to the water
7. A semicircular plate of radius r,
oriented as in Figure 9, is submerged in water so that its diameter is located
at a depth of m feet. Calculate the force on one side of the plate in terms of
m and r.
10 shows the wall of a dam on a water reservoir. Use the Trapezoidal Rule and
the width and depth measurements in the figure to estimate the total
force on the wall.
Calculate the total force (in Newtons) on a side of the plate in Figure 11(A),
submerged in water.
Calculate the total force (in Newtons) on a side of the plate in Figure 11(B),
submerged in a fluid of mass density ρ = 800 kg/m3.
plate in Figure 12 is submerged in water with its top level with the surface of
the water. The left and right edges of the plate are the curves y = x1/3
and y = − x1/3. Find the fluid force on a side of the
12. Let R
be the plate in the shape of the region under y = sin x for 0 ≤ x ≤
π/2 in Figure 13(A). Find the fluid force on a side of R if it is
rotated counterclockwise by 90◦ and submerged in a fluid of density
140 lb/ft3 with its top edge level with the surface of the
fluid as in (B).
13. In the notation of Exercise 12,
calculate the fluid force on a side of the plate R if it is oriented as
in Figure 13(A). You may need to use Integration by Parts and trigonometric
14. Let A
be the region under the graph of y = ln x for 1 ≤ x ≤ e (Figure
14). Calculate the fluid force on one side of a plate in the shape of
region A if the water surface is at y = 1.
Calculate the fluid force on one side of the “infinite” plate B in
16. A square plate of side 3 m is submerged in water at an incline of 30◦ with the horizontal. Its top edge is located at
the surface of the water. Calculate the fluid force (in Newtons) on one
side of the plate.
Repeat Exercise 16, but assume that the top edge of the plate lies at a depth
of 6 m.
Figure 15(A) shows a ramp inclined at 30◦ leading into a swimming pool.
Calculate the fluid force on the ramp.
Calculate the fluid force on one side of the plate (an isosceles
triangle) shown in Figure 15(B).
trough in Figure 16 is filled with corn syrup, whose density is 90 lb/ft3.
Calculate the force on the front side of the trough.
Calculate the fluid pressure on one of the slanted sides of the trough in
Figure 16, filled with corn syrup as in Exercise 20.
Figure 17 shows an object whose face is an equilateral triangle with 5-ft
sides. The object is 2 ft thick and is submerged in water with its vertex 3 ft below the water surface. Calculate the fluid force on both a triangular face and a slanted
rectangular edge of the object.
end of the trough in Figure 18 is an equilateral triangle of side 3. Assume
that the trough is filled with water to height y. Calculate the
fluid force on each side of the trough as a function of the level y and
the length l of the trough.
24. A rectangular plate of side l is
submerged vertically in a fluid of density w, with its top edge at depth
h. Show that if the depth is increased by an amount Dh, then the force on a side of the plate
increases by wADh,
where A is the area of the plate.
that the force on the side of a rectangular plate of area A submerged
vertically in a fluid is equal to p0A, where p0 is
the fluid pressure at the center point of the rectangle.
the density of a fluid varies with depth, then the pressure at depth y is
a function p ( y ) (which need not equal w y as in the case of constant
density). Use Riemann sums to argue that the total force F on the flat
side of a submerged object submerged vertically is , where f(y) is the width of the
side at depth y.
particles are located at points (1, 1), (1, 2), (4, 0), (3, 1)
the moments Mx and My and the center of mass of the
system, assuming that the particles have equal mass m.
the center of mass of the system, assuming the particles have mass 3, 2, 5, and
the center of mass for the system of particles of mass 4, 2, 5, 1 located at
(1, 2), (−3, 2), (2, −1), (4, 0).
masses of equal size are placed at the vertices of the triangle with
coordinates (a, 0), (b, 0), and (0, c). Show that the center of mass of the
system of masses has coordinates .
masses of mass m1, m2, and m3 are placed at
the points (−1, 0), ( 3, 0), and (0, 4).
Suppose that m1 = 6. Show that there is a unique value of m2 such
that the center of mass lies on the y-axis.
Suppose that m1 = 6 and m2 = 4. Find the value of m3
such that yCM = 2.
the lamina S of constant density ρ = 3 g/cm2 occupying the
region beneath the graph of y = x2 for 0 ≤ x ≤ 3.
formulas (1) and (2) to compute Mx and My.
the area and the center of mass of S.
Eqs. (1) and (3) to find the moments and center of mass of the lamina S
of constant density ρ = 2 g/cm2 occupying
the region between y = x2 and y = 9x over [0, 3]. Sketch S,
indicating the location of the center of mass.
the moments and center of mass of the lamina of uniform density ρ
occupying the region underneath y = x3 for 0 ≤ x ≤ 2.
Calculate Mx (assuming ρ = 1) for the region underneath the graph
of y = 1 − x2 for 0 ≤ x ≤ 1 in two ways, first using Eq. (2) and then using Eq. (3).
9. Let T
be the triangular lamina in Figure 17.
that the horizontal cut at height y has length and use Eq. (2) to compute Mx
(with ρ = 1).
the Symmetry Principle to show that My = 0 and find the center
Exercises 10–17, find the centroid of the region lying underneath the
graph of the function over the given interval.
10. f(x) = 6 − 2x, [0, 3]
Calculate the moments and center of mass of the lamina occupying the region
between the curves y = x and y = x2 for 0 ≤ x ≤ 1.
Sketch the region between y = x + 4 and y = 2 − x for 0 ≤ x ≤
2. Using symmetry, explain why the centroid of the region lies on the line y =
3. Verify this by computing the moments and the centroid.
Exercises 34–36, use the additivity of moments to find the COM of the
Isosceles triangle of height 2 on top of a rectangle of base 4 and height 3
ice cream cone consisting of a semicircle on top of an equilateral triangle of
side 6 (Figure 20)
Exercises 1–14, calculate the Taylor polynomials T2(x) and T3(x)
centered at x = a for the given function and value of a.
1. f(x) = sin x, a = 0
that the nth Maclaurin polynomial for f(x) = ex is
y = ex together with the Maclaurin polynomials Tn(x)
for n = 1, 3, 5 and then for n = 2, 4, 6 on the interval [−3, 3]. What
difference do you notice between the even and odd Maclaurin polynomials?
the Error Bound to find the maximum possible size of |cos 0.3 − T5(0.3)|,
where T5(x) is the Maclaurin polynomial. Verify your result with a
32. Let and let Tn(x)
be the Taylor polynomial centered at a = 8.
T3(x) and calculate T3(8.02).
the Error Bound to find a bound for |T3(8.02) −
n such that |Tn(1.3) − ln(1.3)| ≤ 10−4,
where Tn is the Taylor polynomial for f(x) = ln x at a = 1.
Verify that the third Maclaurin polynomial for f(x) = exsin x is
equal to the product of the third Maclaurin polynomials of ex and
sin x (after discarding terms of degree greater than 3 in the product).
the fourth Maclaurin polynomial for f(x) = sin x cos x by multiplying the
fourth Maclaurin polynomials for f(x) = sin x and f(x) = cos x.
the Maclaurin polynomials Tn(x) for f(x) = cos (x2). You
may use the fact that Tn(x) is equal to the sum of the terms up to
degree n obtained by substituting x2for x in the nth Maclaurin
polynomial of cos x.
45. A cylindrical tank filled with water has height 10 ft and a base of
area 30 ft2. Water leaks through a hole in the bottom of area
1/3 ft2. How long does it take (a) for half of the water to leak out and (b) for
the tank to empty?
46. A conical tank filled with water has height 12 ft [Figure 7(A)]. Assume that the top is a circle of radius 4 ft and that water leaks through
a hole in the bottom of area 2 in2. Let y ( t ) be the water level at time t.
(a) Show that the cross-sectional area of the tank at height y is
A ( y ) = (π/9)y2.
(b) Find the differential equation satisfied by y ( t ) and solve for y ( t ).
Use the initial condition y(0) = 12.
(c) How long does it take for the tank to empty?
47. The tank in Figure 7(B) is a cylinder of radius 10 ft and length
40 ft. Assume that the tank is half-filled with water and that water
leaks through a hole in the bottom of area B = 3 in2. Determine the
water level y(t) and the time te when the tank is empty.
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