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Jan Stewart, Calculus: Early Transcendentals, 6th Edition.

Section 11.1

17-46. Determine whether the sequence converges or diverges. If it converges, find the limit
17. an = 1 – 0.2n

55. If $1000 is invested at 6% interest, compounded annually, then after n years the investment is worth an = 1000(0.6)n dollars. (a) Find the first five terms of the sequence {an}. (b) Is the sequence convergent or divergent? Explain.

57. For what values of r is the sequence {nrn} convergent?

60–66 Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
60. an = (-2)n+1

Section 11.2


3-8. Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

11-20. Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
11. 3 + 2 + 4/3 + 8/9 + …

21–34. Determine whether the series is convergent or divergent.  If it is convergent, find its sum.
21. ∑ (from n = 1 to ∞) 1/2n

35–40. Determine whether the series is convergent or divergent by expressing sn as a telescoping sum (as in Example 6). If it is convergent, find its sum.
35. ∑ (from n = 2 to ∞) 2/(n2 – 1)

41–46. Express the number as a ratio of integers.
         _
41. 0.2 = 0.2222…

47-51. Find the values of x for which the series converges. Find the sum of the series for those values of x.
47. ∑ (from n = 1 to ∞) xn/3n

Section 11.2


  • Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
  • Find the values of x for which the series converges. Find the sum of the series for those values of x.
  • Use the Integral Test to determine whether the series is convergent or divergent.


  • Find the values of p for which the series is convergent.
  • Find the sum of the series correct to three decimal places.
  • Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.

  • Test the series for convergence or divergence.

  • Approximate the sum of the series correct to four decimal places.

  • Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

  • Find the radius of convergence and interval of convergence of the series.

  • Find a power series representation for the function and determine the interval of convergence. f(x) = 1/(1 + x)

  • Find a power series representation for the function and determine the radius of convergence. f(x) = ln(5 - x)
  • Use a power series to approximate the definite integral to six decimal places.
  • Find the Taylor series for f(x) centered at the given value of a.
    f(x) = x4 – 3x2 + 1, a=1
  • Use the binomial series to expand the function as a power series. State the radius of convergence.
    f(x) = √(1+x)
  • (a) Find the Taylor polynomials up to degree 6 for f(x)=cos x centered at a=0. Graph and these
    polynomials on a common screen.
    (b) Evaluate and these polynomials at x=π/4, π/2, and π.
    (c) Comment on how the Taylor polynomials converge to f(x).
  • A car is moving with speed 20 m/s and acceleration 2 m/s2 at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled during the next minute?
  • Find the Taylor series of f(x)=sin x at a=π/6 .
  • Find an equation of the sphere that passes through the origin and whose center is (1, 2, 3).
  • A woman walks due west on the deck of a ship at 3 mi/h. The ship is moving north at a speed of 22 mi/h. Find the speed and direction of the woman relative to the surface of the water.
  • Ropes 3 m and 5 m in length are fastened to a holiday decoration that is suspended over a town square. The decoration has a mass of 5 kg. The ropes, fastened at different heights, make angles of 52° and 40° with the horizontal. Find the tension in each wire and the magnitude of each tension.
  • A clothesline is tied between two poles, 8 m apart. The line is quite taut and has negligible sag. When a wet shirt with a mass of 0.8 kg is hung at the middle of the line, the midpoint is pulled down 8 cm. Find the tension in each half of the clothesline.
  • The tension T at each end of the chain has magnitude 25 N. What is the weight of the chain?
  • Find the unit vectors that are parallel to the tangent line to the parabola y=x2 at the point (2,4).
  • Find the angle between the vectors.
  • Determine whether the given vectors are orthogonal, parallel, or neither.
    (a) a = <-5, 3, 7>, b = <6, -8, 2>.
  • Find a unit vector that is orthogonal to both i + j and  i + k.
  • Find the scalar and vector projections of b onto a.
  • Find the work done by a force F = 8i – 6j + 9k that moves an object from the point (0,10,8) to the point (6,12,20) along a straight line. The distance is measured in meters and the force in newtons.
  • A tow truck drags a stalled car along a road. The chain makes an angle of 30° with the road and the tension in the chain is 1500 N. How much work is done by the truck in pulling the car 1 km?
  • A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of 30o above the horizontal moves the sled 80 ft. Find the work done by the force.
  • A boat sails south with the help of a wind blowing in the direction S36oE with magnitude 400 lb. Find the work done by the wind as the boat moves 120 ft.
  • Find the angle between a diagonal of a cube and one of its edges.
  • Find the angle between a diagonal of a cube and a diagonal of one of its faces.
  • Find two unit vectors orthogonal to both <1,-1,1> and <0, 4,4>.
  • Use the scalar triple product to determine whether the points A(1,2,3), B(3, –1 ,6), C(5,2,0), and D(3, 6, –4) lie in the same plane.
  • A bicycle pedal is pushed by a foot with a 60-N force as shown. The shaft of the pedal is 18 cm long. Find the magnitude of the torque about P.
  • Find the magnitude of the torque about P if a 36-lb force is applied as shown.
  • A wrench 30 cm long lies along the positive x-axis and grips a bolt at the origin. A force is applied in the direction <0, 3, -4> at the end of the wrench. Find the magnitude of the force needed to supply 100 N∙m of torque to the bolt.
  • The line through the origin and the point (1,2,3).
  • The line through the points (1,3,2) and (–4,3,0).
  • The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1 + t, z = 2 – t.
  • The plane that passes through the point (1,-1,1) and contains the line with symmetric equations
    x = 2y = 3z.
  • The plane that passes through the point (–1,2,1) and contains the line of intersection of the planes
    x + y – z = 2 and 2x – y + 3z = 1.
  • The plane that passes through the line of intersection of the planes x – z = 1 and y + 2z = 3 and is perpendicular to the plane x + y – 2z = 1.
  • Find an equation for the plane consisting of all points that are equidistant from the points (1,0,-2) and (3,4,0).
  • Find an equation of the plane with x-intercept a, y-intercept b, and z-intercept c.
  • Find parametric equations for the line through the point (0, 1,2) that is parallel to the plane
    x + y + z = 2 and perpendicular to the line x = 1 + t, y = 1 – t, z = 2t.
  • Find parametric equations for the line through the point (0, 1, 2) that is perpendicular to the line x = 1 + t, y = 1 – t, z = 2t and intersects this line.
  • Which of the following four planes are parallel? Are any of them identical?
    P1: 4x – 2y + 6z = 3                          P2: 4x – 2y – 2z = 6
    P3: –6x + 3y – 9z = 5                        P4: z = 2x – y – 3
  • Which of the following four lines are parallel? Are any of them identical?
    L1: x = 1 + t, y = t, z = 2 – 5t
    L2: x + 1 = y – 2 = 1 – z
    L3: x = 1 + t, y = 4 + t, z = 1 – t
    L4: r = <2, 1, -3> + t<2, 2, -10>
  • Find the distance from the point to the given plane.
    (1, –2, 4), 3x + 2y + 6z = 5
  • Find the distance between the given parallel planes.

    2x – 3y + z = 4, 4x – 6y + 2z = 3
  • Find equations of the planes that are parallel to the plane x + 2y – 2z = 1 and two units away from it.
  • Show that the lines with symmetric equations x = y = z and x + 1 = y/2 = z/3 are skew, and find the distance between these lines.
  • (a) Find and identify the traces of the quadric surface –x2 – y2 + z2 = 1 and explain why the graph
    looks like the graph of the hyperboloid of two sheets in Table 1.

        (b) If the equation in part (a) is changed to x2 – y2 + z2, what happens to the graph? Sketch the new
graph.

  • Use traces to sketch and identify the surface
    x = y2 + 4z2
  • Reduce the equation to one of the standard forms, classify the surface, and sketch it.
    z2 = 4x2 + 9y2 + 36
  • Sketch the region bounded by the surfaces z = √(x2 + y2) and x2 + y2 = 1 for 1 ≤ z ≤ 2.
  •  Sketch the region bounded by the paraboloids z = x2 + y2 and z = 2 – x2 – y2
  • Find an equation for the surface obtained by rotating the parabola y = x2 about the y-axis.
  • Find an equation for the surface obtained by rotating the line x = 3y about the x-axis.
  • Find an equation for the surface consisting of all points that are equidistant from the point (–1,0,0) and the plane x = 1. Identify the surface.
  • Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is twice the distance from P to the yz-plane. Identify the surface.
  • A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet (see the photo on page 810). The diameter at the base is 280 m and the minimum diameter, 500 m above the base, is 200 m. Find an equation for the tower.
  • (a) Find an equation of the sphere that passes through the point (6, –2, 3) and has center (–1,2,1).
    (b) Find the curve in which this sphere intersects the yz-plane.
    (c) Find the center and radius of the sphere
    x2 + y2 + z2 – 8x + 2y + 6z + 1 = 0
  • A constant force F = 3i + 5j + 10k moves an object along the line segment from (1,0,2) to (5,3,8). Find the work done if the distance is measured in meters and the force in newtons.
  • A boat is pulled onto shore using two ropes, as shown in the diagram. If a force of 255 N is needed, find the magnitude of the force in each rope.
  • Find the point in which the line with parametric equations x = 2 – t, y = 1 + 3t, z = 4t intersects the plane 2x – y + z = 2.
  • Find the distance from the origin to the line x = 1 + t, y = 2 – t, z = –1 + 2t.

168. Determine whether the lines given by the symmetric equations (x – 1)/2 = (y – 2 )/3 = (z – 3)/4
and (x + 1)/6 = -(y – 3)/1 = (z + 5)/2 are parallel, skew, or intersecting.

  • (a) Show that the planes x + y – z = 1and 2x – 3y + 4z = 5 are neither parallel nor perpendicular.
    (b) Find, correct to the nearest degree, the angle between these planes.
  • Find an equation of the plane through the line of intersection of the planes x – z = 1 and y + 2z = 3 and perpendicular to the plane x + y – 2z = 1.
  • (a) Find an equation of the plane that passes through the points A(2,1,1), B(–1, –1,10) and
    C(1,3, –4).
    (b) Find symmetric equations for the line through B that is perpendicular to the plane in part (a).

        (c) A second plane passes through (2,0,4) and has normal vector <2, -4, -3>. Show that the acute
angle between the planes is approximately 43°.
(d) Find parametric equations for the line of intersection of the two planes.

  • Find the distance between the planes 3x + y – 4z = 2 and 3x + y – 4z = 24.
  • An ellipsoid is created by rotating the ellipse 4x2 + y2 = 16 about the x-axis. Find an equation of the ellipsoid.
  • A surface consists of all points such that the distance from P to the plane y = 1 is twice the distance from P to the point (0, –1,0). Find an equation for this surface and identify it.
  • Find the domain of the vector function r(t) = <√(4 – t2), e-3t, ln(t + 1)>.
  • Sketch the curve with the given vector equation. Indicate with an arrow the direction in which increases: r(t) = <sin t, t>.
  • Find a vector equation and parametric equations for the line segment that joins P to Q
    P(0,0,0), Q (1,2,3)
  • Show that the curve with parametric equations x = sin t, y = cos t, z = sin2t lies on the cone
    z2 = x2 + y2, and use this fact to help sketch the curve.

 

  • At what points does the curve r(t) = ti + (2t – t2)k  intersect the paraboloid z = x2 + y2?
  • At what points does the helix r(t) = <sin t, cos t, t> intersect the sphere x2 + y2 + z2 = 5?
  • Find a vector function that represents the curve of intersection of the two surfaces.
    The cylinder x2 + y2 = 4 and the surface z = xy.
  • Two particles travel along the space curves r1(t) = <t, t2, t3>, r2(t) = <1 + 2t, 1 + 6t, 1 + 14t>. Do the particles collide? Do their paths intersect?
  • Find the derivative of the vector function r(t) = <t sin t, t2, t cos 2t>.
  • Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) = <te-t, 2arctan t, 2et>, t = 0
  • Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = 1 + 2√t, y = t3 – t, z = t3 + t; (3,0,2)
  • Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen.
    x = t, y = e-t, z = 2t – t2; (0,1,0)
  • The curves r1(t) = < t, t2, t3>  and r2(t) = < sin t, sin 2t, t> intersect at the origin. Find their angle of intersection correct to the nearest degree.
  • At what point do the curves r1(t) = < t, 1 – t, 3 + t2> and r2(s) = < 3 – s, s – 2, s2> intersect? Find their angle of intersection correct to the nearest degree.
  • If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r ¢(t), show that the curve lies on a sphere with center the origin.
  • Find the length of the curve. r(t) = < 2sin t, 5t, 2cos t>, –10 ≤ t ≤ 1.
  • Graph the curve with parametric equations x = sin t, y = sin 2t, z = sin 3t. Find the total length of this curve correct to four decimal places.

 

  • Let C be the curve of intersection of the parabolic cylinder x2 = 2y and the surface 3z = xy. Find the exact length of C from the origin to the point (6, 18, 36).
  • Find, correct to four decimal places, the length of the curve of intersection of the cylinder
    4x2 + y2 = 4 and the plane x + y + z = 2.
  • Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t.
    r(t) = 2ti + (1 – 3t)j + (5 +4t)k
  • Suppose you start at the point (0,0,3) and move 5 units along the curve x = 3sin t, y = 4t, z = 3cos t in the positive direction. Where are you now?
  • Find the curvature of r(t) = <et cos t, et sin t, t> at the point (1, 1, 1).
  • Find the vectors T, N, and B at the given point. r(t) = <t2, (2/3)t3, t>, (1, 2/3, 1).
  • Find equations of the normal plane and osculating plane of the curve at the given point.
    x = 2sin 3t, y = t, z = 2cos 3t; (0, p, –2).
  • Find equations of the osculating circles of the ellipse 9x2 + 4y2 = 36 at the points (2, 0) and  (0, 3). Use a graphing calculator or computer to graph the ellipse and both osculating circles on the same screen.
  • Find equations of the osculating circles of the parabola y = (1/2)x2 at the points (0, 0) and (1, ½). Graph both osculating circles and the parabola on the same screen.
  • At what point on the curve x = t3, y = 3t, z = t4 is the normal plane parallel to the plane
    6x + 6y – 8z =1?
  • Show that the curvature k is related to the tangent and normal vectors by the equation dT/ds = kN.
  • Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of t.
    r(t) = <-(1/2)t2, t>, t = 2.
  • Find the velocity, acceleration, and speed of a particle with the given position function.
    r(t) = <t2 + 1, t3, t2 – 1>.
  • Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position.
    a(t) = i + 2j, v(0) = k, r(0) = i
  • The position function of a particle is given by r(t) = <t2, 5t, t2 – 16t>. When is the speed a minimum?
  • What force is required so that a particle of mass has the position function r(t) = t3i + t2j + t3k
  • A force with magnitude 20 N acts directly upward from the xy-plane on an object with mass 4 kg. The object starts at the origin with initial velocity v(0) = i – j. Find its position function and its speed at time t.
  • Show that if a particle moves with constant speed, then the velocity and acceleration vectors are orthogonal.
  • A projectile is fired with an initial speed of 500 m/s and angle of elevation 30°. Find (a) the range of the projectile, (b) the maximum height reached, and (c) the speed at impact.
  • A ball is thrown at an angle of 45° to the ground. If the ball lands 90 m away, what was the initial speed of the ball?

Section 13. 4

26. A gun is fired with angle of elevation 30°. What is the muzzle speed if the maximum height of the shell is 500 m?

27. A gun has muzzle speed 150 m/s. Find two angles of elevation that can be used to hit a target 800 m away.

28. A batter hits a baseball 3 ft above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed 115 ft/s at an angle 50° above the horizontal. Is it a home run? (In other words, does the ball clear the fence?)

29. A medieval city has the shape of a square and is protected by walls with length 500 m and height 15 m. You are the commander of an attacking army and the closest you can get to the wall is 100 m. Your plan is to set fire to the city by catapulting heated rocks over the wall (with an initial speed of 80 m/s). At what range of angles should you tell your men to set the catapult? (Assume the path of the rocks is perpendicular to the wall.)

30. A ball with mass 0.8 kg is thrown southward into the air with a speed of 30 m/s at an angle of 30° to the ground. A west wind applies a steady force of 4 N to the ball in an easterly direction. Where does the ball land and with what speed?

31. Water traveling along a straight portion of a river normally flows fastest in the middle, and the  speed slows to almost zero at the banks. Consider a long straight stretch of river flowing north, with parallel banks 40 m apart. If the maximum water speed is 3 m/s, we can use a quadratic function as a basic model for the rate of water flow units from the west bank: f(x) =(3/400)x(40 -x).
(a) A boat proceeds at a constant speed of 5 m/s from a point A on the west bank while maintaining a heading perpendicular to the bank. How far down the river on the opposite bank will the boat touch shore? Graph the path of the boat. (b) Suppose we would like to pilot the boat to land at the point B on the east bank directly opposite A. If we maintain a constant speed of and a constant heading, find the angle at which the boat should head. Then graph the actual path the boat follows. Does the path seem realistic?

32. Another reasonable model for the water speed of the river in Exercise 31 is a sine function:
f(x) = 3sin(px/40). If a boater would like to cross the river from A to B with constant heading and a constant speed of 5 m/s, determine the angle at which the boat should head.

33. Find the tangential and normal components of the acceleration vector. r(t) = (3t – t3)i + 3t2j


Section 14.1

30. Match the function with its graph (labeled I–VI).Give reasons for your choices.  f(x,y) = |x| + |y|

31. A contour map for a function f is shown. Use it to estimate the values of f(–3,3) and f(3, –2). What can you say about the shape of the graph?

32. Two contour maps are shown. One is for a function f whose graph is a cone. The other is for a function g whose graph is a paraboloid. Which is which, and why?

39. Draw a contour map of the function showing several level curves. f(x,y) = (y – 2x)2

61. Describe the level surfaces of the function f(x,y,z) = x + 3y + 5z

Section 14.2

5. Find the limit, if it exists, or show that the limit does not exist.
lim((x,y)→(1,2))(5x3 – x2y2)

29. Determine the set of points at which the function is continuous. F(x, y) = sin(xy)/(ex – y2)

39. Use polar coordinates to find the limit. [If (r, q) are polar coordinates of the point (x,y) with r ³ 0, note that r ® 0+ as (x,y) ®(0,0).]

Section 14.3

9. The following surfaces, labeled a, b, and c, are graphs of a function f and its partial derivatives fx and fy. Identify each surface and give reasons for your choices.

10. A contour map is given for a function f. Use it to estimate fx(2,1) and fy(2,1).

11. If f(x,y) = 16 – 4x2 – y2, find fx(1,2) and fy(1,2) and interpret these numbers as slopes. Illustrate with either hand-drawn sketches or computer plots.

15. Find the first partial derivatives of the function
f(x,y) = y5 – 3xy

43. Use the definition of partial derivatives as limits (4) to find fx(x,y) and fy(x,y) .
f(x,y) = xy2 – x3y

51. Find all the second partial derivatives
f(x,y) = x3y5 + 2x4y

61. Find the indicated partial derivative.

69. Use the table of values of f(x,y) to estimate the values of fx(3,2), fx(3,2.2), and fxy(3,2).

70. Level curves are shown for a function . Determine whether the following partial derivatives are positive or negative at the point P.

81. The total resistance R produced by three conductors with resistances R1, R2, R3 connected in a parallel electrical circuit is given by the formula 1/R = 1/R1 + 1/R2 + 1/R3. Find ∂R/∂R1.

87. You are told that there is a function f whose partial derivatives are fx(x,y) = x + 4y and
fy(x,y) = 3x – y. Should you believe it?

88. The paraboloid z = 6 – x – x2 – 2y2 intersects the plane x = 1 in a parabola. Find parametric equations for the tangent line to this parabola at the point (1, 2, –4). Use a computer to graph the paraboloid, the parabola, and the tangent line on the same screen.

89. The ellipsoid 4x2 + 2y2 + z2 = 16 intersects the plane y = 2 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1, 2, 2).

Section 14.4

1. Find an equation of the tangent plane to the given surface at the specified point.
z = 4x2 – y2 + 2y, (–1, 2, 4)

11. Explain why the function is differentiable at the given point. Then find the linearization of the function at that point.
f(x,y) = x√y, (1,4)

19. Find the linear approximation of the function f(x, y) = √(20 – x2 – 7y2) at (2,1) and use it to approximate f(1.95, 1.08).

22. The wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed. Values of the function h=f(v, t) are recorded in feet in the following table. Use the table to find a linear approximation to the wave height function when v is near 40 knots and t is near 20 hours. Then estimate the wave heights when the wind has been blowing for 24 hours at 43 knots.

23. Use the table in Example 3 to find a linear approximation to the heat index function when the temperature is near 94oF and the relative humidity is near 80%. Then estimate the heat index when the temperature is 95oF and the relative humidity is 78%.

24. The wind-chill index W is the perceived temperature when the actual temperature is T and the wind speed is v, so we can write W=f(T, v). The following table of values is an excerpt from Table 1 in Section 14.1. Use the table to find a linear approximation to the wind-chill index function when is near –15°C and v is near 50 km/h. R Then estimate the wind-chill index when the temperature is –17°C and the wind speed is 55 km/h.

25. Find the differential of the function. v = cos xy

31. If z = 5x2 + y2 and (x, y) changes from (1, 2) to (1.05, 2.1), compare the values of Dz and dz.

33. The length and width of a rectangle are measured as 30 cm and 24 cm, respectively, with an error in measurement of at most 0.1 cm in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.

34. The dimensions of a closed rectangular box are measured as 80 cm, 60 cm, and 50 cm, respectively, with a possible error of 0.2 cm in each dimension. Use differentials to estimate the maximum error in calculating the surface area of the box.

35. Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 12 cm if the tin is 0.04 cm thick.

36. Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0.1 cm thick and the metal in the sides is 0.05 cm thick.

37. A boundary stripe 3 in. wide is painted around a rectangle whose dimensions are 100 ft by 200 ft. Use differentials to approximate the number of square feet of paint in the stripe.

38. The pressure, volume, and temperature of a mole of an ideal gas are related by the equation PV=8.31T, where is measured in kilopascals, V in liters, and T in kelvins. Use differentials to find the approximate change in the pressure if the volume increases from 12 L to 12.3 L and the temperature decreases from 310 K to 305 K.

39. If R is the total resistance of three resistors, connected in parallel, with resistances R1, R2, R3  then
1/R = 1/R1 + 1/R2 + 1/R3. If the resistances are measured in ohms as R1 = 25 W, R2 = 40 W, and R3 = 50 W, with a possible error of 0.5% in each case, estimate the maximum error in the calculated value of R.

40. Four positive numbers, each less than 50, are rounded to the first decimal place and then multiplied together. Use differentials to estimate the maximum possible error in the computed product that might result from the rounding.

41. A model for the surface area of a human body is given by S= 0.1091w0.425h0.725, where w is the weight (in pounds), h is the height (in inches), and S is measured in square feet. If the errors in measurement of w and h are at most 2%, use differentials to estimate the maximum percentage error in the calculated surface area.

42. Suppose you need to know an equation of the tangent plane to a surface S at the point P(2,1,3). You don’t have an equation for S but you know that the curves r1(t) = <2 + 3t, 1 – t2, 3 – 4t + t2>, r2(u) = <1 + u2, 2u3 – 1, 2u + 1>, both lie on S. Find an equation of the tangent plane at P.

43. Show that the function is differentiable by finding values of e1 and e2 that satisfy Definition 7.
f(x, y) = x2 + y2

Section 14.5

1. Use the Chain Rule to find dz/dt or dw/dt.  z = x2 + y2 + xy, x = sin t, y = et

21. Use the Chain Rule to find the indicated partial derivatives. z = x2 + xy3, x = uv2 + w3, y = u + vew

35. The temperature at a point (x, y) is T(x, y), measured in degrees Celsius. A bug crawls so that its position after t seconds is given by x = √(1 + t), y = 2 + (1/3)t where x and y are measured in centimeters. The temperature function satisfies Tx(2,3) = 4 and Ty(2,3) = 4. How fast is the temperature rising on the bug’s path after 3 seconds?

36. Wheat production W in a given year depends on the average temperature T and the annual rainfall R. Scientists estimate that the average temperature is rising at a rate of 0.15°C/year and rainfall is decreasing at a rate of 0.1 cm/year. They also estimate that, at current production levels, ∂W/∂T = -2, and ∂W/∂R = 8. (a) What is the significance of the signs of these partial derivatives? (b) Estimate the current rate of change of wheat production, dW/dt.

37. The speed of sound traveling through ocean water with salinity 35 parts per thousand has been modeled by the equation C=1449.2 +4.6T – 0.055T2 + 0.00029T3 + 0.016D where C is the speed of sound (in meters per second), T is the temperature (in degrees Celsius), and D is the depth below the ocean surface (in meters). A scuba diver began a leisurely dive into the ocean water; the diver’s depth and the surrounding water temperature over time are recorded in the following graphs. Estimate the rate of change (with respect to time) of the speed of sound through the ocean water experienced by the diver 20 minutes into the dive. What are the units?

38. The radius of a right circular cone is increasing at a rate of 1.8 in/s while its height is decreasing at a rate of 2.5 in/s. At what rate is the volume of the cone changing when the radius is 120 in. and the height is 140 in.?

39. The length l, width w, and height h of a box change with time. At a certain instant the dimensions are l = 1 m and w = h = 2 m, and l and w are increasing at a rate of 2 m/s while is decreasing at a rate of 3 m/s. At that instant find the rates at which the following quantities are changing. (a) The volume (b) The surface area (c) The length of a diagonal

40. The voltage V in a simple electrical circuit is slowly decreasing as the battery wears out. The  resistance R is slowly increasing as the resistor heats up. Use Ohm’s Law, V=IR, to find how the current is changing at the moment when R=400 W, I=0.08 A, dV/dt =–0.01 V/s, and dR/dt = 0.03 W/s.

  • The pressure of 1 mole of an ideal gas is increasing at a rate of 0.05 kPa/s and the temperature is increasing at a rate of 0.15 K/s. Use the equation in Example 2 to find the rate of change of the volume when the pressure is 20 kPa and the temperature is 320 K.
  • Car A is traveling north on Highway 16 and car B is traveling west on Highway 83. Each car is approaching the intersection of these highways. At a certain moment, car A is 0.3 km from the intersection and traveling at 90 km/h while car B is 0.4 km from the intersection and traveling at 80 km/h. How fast is the distance between the cars changing at that moment?
  • One side of a triangle is increasing at a rate of and a second side is decreasing at a rate of 2 cm/s. If the area of the triangle remains constant, at what rate does the angle between the sides change when the first side is 20 cm long, the second side is 30 cm, and the angle is p/6?
  • Level curves for barometric pressure (in millibars) are shown for 6:00 AM on November 10, 1998. A deep low with pressure 972 mb is moving over northeast Iowa. The distance along the red line from K (Kearney, Nebraska) to S (Sioux City, Iowa) is 300 km. Estimate the value of the directional derivative of the pressure function at Kearney in the direction of Sioux City. What are the units of the directional derivative?
  • The contour map shows the average maximum temperature for November 2004 (in oC). Estimate the value of the directional derivative of this temperature function at Dubbo, New South Wales, in the direction of Sydney. What are the units?
  • Find the directional derivative of f at the given point in the direction indicated by the angle q.
    f(x,y)= x2y3 – y4, (2,1), q = p/4
  • Find the directional derivative of the function at the given point in the direction of the vector v.
    f(x,y) = ln(x2 + y2), (2,1), v = á–1,2ñ
  • Find the directional derivative of f(x,y,z) = xy +yz + zx at P(1,-1,3) in the direction of  Q(2,4,5).
  • Find the maximum rate of change of f at the given point and the direction in which it occurs.
    f(x,y) = y2/x,          (2,4)
  • (a) Show that a differentiable function decreases most f rapidly at in the direction opposite to the gradient vector, that is, in the direction of –Ñf(x).
    (b) Use the result of part (a) to find the direction in which the function f(x,y) = x4y – x2y3 decreases fastest at the point (2, –3).
  • Find the directions in which the directional derivative of f(x,y) = ye-xy at the point (0,2) has the value 1.
  • Find all points at which the direction of fastest change of the function f(x,y) = x2 + y2 – 2x – 4y  is
    i + j.
  • Near a buoy, the depth of a lake at the point with coordinates (,y) is z = 200 + 0.02x2 – 0.001y3 , where x, y and z are measured in meters. A fisherman in a small boat starts at the point (80, 60) and moves toward the buoy, which is located at (0,0). Is the water under the boat getting deeper or shallower when he departs? Explain.

  • The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1,2,2) is 120o. (a) Find the rate of change of T at (1,2,2) in the direction toward the point (2,1,3). (b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points toward the origin.
  • Suppose that over a certain region of space the electrical potential V is given by
    V(x,y,z) = 5x2 – 3xy + xyz. (a) Find the rate of change of the potential at P(3,4,5) in the direction of the vector v = i + j – k. (b) In which direction does V change most rapidly at P? (c) What is the maximum rate of change at P?
  • Suppose you are climbing a hill whose shape is given by the equation z = 1000 – 0.005x2 – 0.01y2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (60,40,966). The positive x-axis points east and the positive y-axis points north. (a) If you walk due south, will you start to ascend or descend? At what rate? (b) If you walk northwest, will you start to ascend or descend? At what rate? (c) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?
  • Let f be a function of two variables that has continuous partial derivatives and consider the points A(1,3), B(3,3), C(1,7), and D(6,15). The directional derivative of f at A in the direction of the vector
    AB is 3 and the directional derivative at A in the direction of  AC is 26. Find the directional derivative of at in the direction of the vector AD.
  • Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.
    2(x – 2)2 + (y – 1)2 + (z – 3)2 = 10,               (3,3,5)
  • If f(x,y) = xy, find the gradient vector Ñf(3,2) and use it to find the tangent line to the level curve f(x,y) = 6 at the point (3,2). Sketch the level curve, the tangent line, and the gradient vector.
  • Find the equation of the tangent plane to the hyperboloid x2/a2 + y2/b2 + z2/c2 = 1 at the point
    (xo, yo, zo) and express it in a form similar to the one in Exercise 49.
  • At what point on the paraboloid y = x2 + z2 is the tangent plane parallel to the plane
    x + 2y + 3z = 1?
  • Are there any points on the hyperboloid x2 – y2 -  z2 = 1 where the tangent plane is parallel to the plane z = x + y?
  • Show that every plane that is tangent to the cone x2 + y2 = z2 passes through the origin.
  • Show that every normal line to the sphere x2 + y2 + z2 = 1 passes through the center of the sphere.
  • Show that the sum of the x-, y-, and z-intercepts of any tangent plane to the surface √x + √y + √z = √c is a constant.
  • Show that the pyramids cut off from the first octant by any tangent planes to the surface xyz= 1 at points in the first octant must all have the same volume.
  • (a) The plane y + z = 3 intersects the cylinder x2 + y2 = 5 in an ellipse. Find parametric equations
    for the tangent line to this ellipse at the point (1,2,1). (b) Graph the cylinder, the plane, and the tangent line on the same screen.
  • Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.
    f(x,y) = 9 – 2x + 4y – x2 – 4y2
  • Show that f(x,y) = x2 + 4y2 – 4xy + 2  has an infinite number of critical points and that D = 0 at each one. Then show that f has a local (and absolute) minimum at each critical point.
  • Find the absolute maximum and minimum values of f on the set D.
    f(x,y) = 1 + 4x – 5y, D is the closed triangular region with vertices (0,0), (2,0), and (3,0).
  • Find the shortest distance from the point (2,1,– 1) to the plane x + y – z = 1.
  • Find the point on the plane x – y + z = 4 that is closest to the point (1,2,3).
  • Find the points on the cone z2 = x2 + y2 that are closest to the point (4, 2, 0).
  • Find the points on the surface y2 = 9 + xz that are closest to the origin.
  • Find three positive numbers whose sum is 100 and whose product is a maximum.
  • Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.
  • Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.
  • Find the dimensions of the box with volume 1000 cm3 that has minimal surface area.
  • Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 6.
  • Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm2.
  • Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c.
  • The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials.
  • A cardboard box without a lid is to have a volume of 32000 cm3. Find the dimensions that minimize the amount of cardboard used.
  • A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of 10 units/m2 per day, the north and south walls at a rate of 8 units/m2 per day, the floor at a rate of 1 units/m2 per day, and the roof at a rate of 5 units/m2 per day. Each wall must be at least 30 m long, the height must be at least 4 m, and the volume must be exactly 4000 m3.
    (a) Find and sketch the domain of the heat loss as a function of the lengths of the sides.
    (b) Find the dimensions that minimize heat loss. (Check both the critical points and the points on the boundary of the domain.)
    (c) Could you design a building with even less heat loss if the restrictions on the lengths of the walls were removed?
  • If the length of the diagonal of a rectangular box must be L, what is the largest possible volume?
  • Find an equation of the plane that passes through the point (1, 2, 3) and cuts off the smallest volume in the first octant.
  • Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).
    f(x,y) = x2 + y2;         xy = 1
  • Find the extreme values of f on the region described by the inequality.
    f(x,y) = 2x2 + 3y2 – 4x – 5,               x2 + y2 £ 16
  • The total production P of a certain product depends on the amount of labor L used and the amount K of capital investment. In Sections 14.1 and 14.3 we discussed how the Cobb-Douglas model
    P = bLaK1-a  follows from certain economic assumptions, where b and a are positive constants and
    a <1. If the cost of a unit of labor is m and the cost of a unit of capital is n, and the company can spend only dollars p as its total budget, then maximizing the production is subject to the constraint
    mL + nK = p. Show that the maximum production occurs when L = αp/m and k = (1 – α)p/n.
  • Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square.
  • Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral.
  • Find the maximum and minimum volumes of a rectangular box whose surface area is 1500 cm2 and whose total edge length is 200 cm.
  • The plane x + y + 2z = 2 intersects the paraboloid z = x2 + y2 in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.
  • The plane 4x – 3y + 8z = 5 intersects the cone in an ellipse.
    (a) Graph the cone, the plane, and the ellipse.
    (b) Use Lagrange multipliers to find the highest and lowest points on the ellipse.
  • Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.
    z = 3x2 – y2 + 2x. (1, –2,1)
  • Find the points on the hyperboloid x2 + y2 – z2 = 4 where the tangent plane is parallel to the plane
    2x + 2y + z = 5.
  • The two legs of a right triangle are measured as 5 m and 12 m with a possible error in measurement of at most 0.2 cm in each. Use differentials to estimate the maximum error in the calculated value of (a) the area of the triangle and (b) the length of the hypotenuse.
  • The length x of a side of a triangle is increasing at a rate of 3 in/s, the length y of another side is decreasing at a rate of 2 in/s, and the contained angle q is increasing at a rate 0.05 of radian/s. How fast is the area of the triangle changing when x=40 in, y=50 in, and q=p/6?
  • Find the gradient of the function f(x,y,z)=z2ex√y.
  • (a) When is the directional derivative of f a maximum? (b) When is it a minimum? (c) When is it 0? (d) When is it half of its maximum value?
  • Find the directional derivative of f at the given point in the indicated direction. f(x,y) = 2√x – y2, (1,5) in the direction toward the point (4,1).
  • Find the maximum rate of change of f(x,y)=x2y + √x at the point (2,1). In which direction does it occur?
  • Find the direction in which f(x, y, z)=zexy increases most rapidly at the point (0,1,2). What is the maximum rate of increase?
  • The contour map shows wind speed in knots during Hurricane Andrew on August 24, 1992. Use it to estimate the value of the directional derivative of the wind speed at Homestead, Florida, in the direction of the eye of the hurricane.
  • Find parametric equations of the tangent line at the point (–2,2,4) to the curve of intersection of the surface z = 2x2 – y2 and the plane z=4.
  • Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.
    f(x,y) = x2 – xy + y2 + 9x – 6y + 10
  • Find the absolute maximum and minimum values of on the set D.
    f(x,y) = 4xy2 – x2y2 – xy3; D is the closed triangulat region in the xy-plane with vertices (0,0),(0,6),
    and (6,0).
  • Use Lagrange multipliers to find the maximum and minimum values of subject to the given constraint(s).
    f(x,y) = x2y;               x2 + y2 = 1
  • Find the points on the surface xy2z3 = 2 that are closest to the origin.
  • A package in the shape of a rectangular box can be mailed by the US Postal Service if the sum of its length and girth (the perimeter of a cross-section perpendicular to the length) is at most 108 in. Find the dimensions of the package with largest volume that can be mailed.
  • A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. If the pentagon has fixed perimeter , find the lengths of the sides of the pentagon that maximize the area of the pentagon.
  • A particle of mass m moves on the surface z=f(x,y). Let x=x(t) and y=y(t) be the x- and
    y-coordinates of the particle at time t.
    (a) Find the velocity vector v and the kinetic energy K = (1/2)m|v|2 of the particle. (b) Determine the acceleration vector a. (c)Let z = x2 + y2 and x(t) = t cos(t), y(t) = t sin t. Find the velocity vector, the kinetic energy, and the acceleration vector.
  • (a) Estimate the volume of the solid that lies below the surface z = xy and above the rectangle R = {(x, y) | 0 £ x £ 6, 0 £ y £ 4}. Use a Riemann sum with m = 3, n = 2, and take the sample point to be the upper right corner of each square.
    (b) Use the Midpoint Rule to estimate the volume of the solid in part (a).
  • If R = [–1, 3] ´ [0, 2] use a Riemann sum with m=4, n=2 to estimate the value of . Take the sample points to be the upper left corners of the squares.
  • (a) Use a Riemann sum with m=n=2 to estimate the value of , where
    R = [0,p] ´ [0,p]. Take the sample points to be lower left corners.
    (b) Use the Midpoint Rule to estimate the integral in part (a).
  • (a) Estimate the volume of the solid that lies below the surface z = x + 2y2 and above the rectangle R = [0,2] ´ [0,4]. Use a Riemann sum with m=n=2 and choose the sample points to be lower right corners.
    (b) Use the Midpoint Rule to estimate the volume in part (a).
  • A table of values is given for a function f(x,y) defined on R = [1,3] ´ [0,4].
    (a) Estimate using the Midpoint Rule with m = n = 2.
    (b) Estimate the double integral with by choosing the sample points to be the points farthest from the origin.
  •  A 20-ft-by-30-ft swimming pool is filled with water. The depth is measured at 5-ft intervals, starting at one corner of the pool, and the values are recorded in the table. Estimate the volume of water in the pool.
  • Let be the volume of the solid that lies under the graph of f(x, y) = √(52 – x2 – y2) and above the rectangle given by 2 £ x £ 4, 2 £ y £ 6. We use the lines x = 3 and y = 4 to divide R into subrectangles. Let L and U be the Riemann sums computed using lower left corners and upper right corners, respectively. Without calculating the numbers V, L, and U, arrange them in increasing order and explain your reasoning.
  • The figure shows level curves of a function f in the square R = [0,2] ´ [0,2] . Use the Midpoint Rule with m=n=2 to estimate . How could you improve your estimate?
  • A contour map is shown for a function on the square R = [0,4] ´ [0,4].
    (a) Use the Midpoint Rule with m=n=2 to estimate the value of .
    (b) Estimate the average value of f.
  • The contour map shows the temperature, in degrees Fahrenheit, at 4:00 PM on February 26, 2007, in Colorado. (The state measures 388 mi east to west and 276 mi north to south.) Use the Midpoint Rule with m=n=4 to estimate the average temperature in Colorado at that time.
  • Evaluate the double integral by first identifying it as the volume of a solid.
    ,     R = {(x,y) | –2 £ x £ 2, 1 £ y £ 6}
  • Find the volume of the solid that lies under the plane 3x + 2y + z = 12 and above the rectangle
    R = {(x,y) | 0 £ x £ 1, –2 £ y £ 3}
  • Find the volume of the solid that lies under the hyperbolic paraboloid z = 4 + x2 – y2 and above the square R = [–1,1] ´ [0,2].
  • Find the volume of the solid lying under the elliptic paraboloid x2/4 + y2/9 + z = 1 and above the rectangle R = [–1,1] ´ [–2,2].
  • Find the volume of the solid enclosed by the surface z =1 + ex and the planes x = ±1, y = 0, y = p, and z = 0.
  • Find the volume of the solid enclosed by the surface z =x sec2y and the planes z=0, x=0, x=2, y=0, and y=p/4.
  • Find the volume of the solid in the first octant bounded by the cylinder z=16 – x2 and the plane y=5.
  • Find the volume of the solid enclosed by the paraboloid z = 2 + x2 + (y – 2)2 and the planes z = 1,
    x = 1, x = –1, y = 0, and  y = 4.

    Find the volume of the given solid.
  • Under the plane x + 2y – z = 0 and above the region bounded by y = x and y = x4.
  • Under the surface z = 2x + y2 and above the region bounded by x = y2 and x = y3.
  • Under the surface z = xy and above the triangle with vertices (1,1), (4,1), and (1,2).
  • Enclosed by the paraboloid z = x2 + 3y2 and the planes x = 0, y =1, y = x, z = 0
  • Bounded by the coordinate planes and the plane 3x + 2y + z = 6
  • Bounded by the planes z = x, y = x, x + y = 2, and z = 0.
  • Enclosed by the cylinders z = x2, y = x2 and the planes z = 0, y = 4
  • Bounded by the cylinder x2 + y2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octant
  • Bounded by the cylinder x2 + y2 = 1 and the planes y = z, x = 0, z = 0 in the first octant
  • Bounded by the cylinders x2 + y2 = r2 and y2 + z2 = r2
  • The solid enclosed by the parabolic cylinders y = 1 – x2 , y = x2 – 1 and the planes x + y + z = 2,
    2x + 2y – z + 10 = 0
  • The solid enclosed by the parabolic cylinder y = x2 and the planes z = 3y, z = 2 + y
  • Sketch the solid whose volume is given by the iterated integral.
Sketch the region of integration and change the order of integration.

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