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Saturnino L. Sals, Garret J. Etgen, Einar Hille, Calculus: One and Several Variables, 10 edition, Wiley, 2006

 

Section 1.4

Exercises 1–4.  Find the distance between the points.
1. P0 (0, 5),      P1 (6, −3).         2. P0 (2, 2),    P1 (5, 5).
3. P0 (5, −2),    P1 (−3, 2).        4. P0 (2, 7),     P1 (−4, 7).

Exercises 5–8. Find the midpoint of the line segment P0P1 .
5. P0 (2, 4),    P1 (6, 8).             6. P0 (3, −1),    P1 (−1, 5).
7. P0 (2, −3),  P1 (7, −3).          8. P0 (a, 3),       P1 (3, a).

Exercises 9–14. Find the slope of the line through the points.
9. P0 (−2, 5),    P1 (4, 1).          10. P0 (4, −3),    P1 (−2, −7).
11. P(a, b),      Q(b, a).             12. P(4, −1),     Q(−3, −1).
13. P(x0 , 0),   Q(0, y0 ).           14. O(0, 0), P(x0 , y0 ).

Exercises 15–20. Find the slope and y-intercept.
15. y = 2x − 4.              16. 6 − 5x = 0.
17. 3y = x + 6.              18. 6y − 3x + 8 = 0.
19. 7x − 3y + 4 = 0.      20. y = 3.

Exercises 21–24. Write an equation for the line with
21. slope 5 and y-intercept 2.
22. slope 5 and y-intercept −2.
23. slope −5 and y-intercept 2.
24. slope −5 and y-intercept −2.

Exercises  25–26.  Write  an  equation  for  the  horizontal  line 3 units
25. above the x-axis.
26. below the x-axis.

Exercises 27–28. Write an equation for the vertical line 3 units
27. to the left of the y-axis.
28. to the right of the y-axis.

Exercises  29–34.  Find  an  equation  for  the  line  that  passes through the point P(2, 7) and is

29. parallel to the x-axis.

30. parallel to the y-axis.

31. parallel to the line 3y − 2x + 6 = 0.

32. perpendicular to the line y − 2x + 5 = 0.

33. perpendicular to the line 3y − 2x + 6 = 0.

34. parallel to the line y − 2x + 5 = 0.

Exercises 35–38. Determine the point(s) where the line intersects the circle.

35. y = x,   x2 + y2 = 1.

36. y = mx,   x2 + y2 = 4.

37. 4x + 3y = 24,   x2 + y2 = 25.

38. y = mx + b,   x2 + y2 = b2 .

Exercises 39–42. Find the point where the lines intersect.

39. l1 : 4x − y − 3 = 0,   l2 : 3x − 4y + 1 = 0.

40. l1 : 3x + y − 5 = 0,   l2 : 7x − 10y + 27 = 0.

41. l1 : 4x − y + 2 = 0,   l2 : 19x + y = 0.

43. Find the area of the triangle with vertices (1, −2), (−1, 3), (2, 4).

44. Find the area of the triangle with vertices (−1, 1), (3,√2), (√2, −1).

45. Determine the slope of the line that intersects the circle x2 + y2 = 169 only at the point (5, 12).

46. Find an equation for the line which is tangent to the circle x2 + y2 − 2x + 6y − 15 = 0 at the point (4, 1).

Exercises 52–53. The perpendicular bisector of the line segment PQ is the line which is perpendicular to PQ and passes through the midpoint of P Q. Find an equation for the perpendicular bisector of the line segment that joins the two points.
52. P(−1, 3),   Q(3, −4).

Exercises 54–56. The points are the vertices of a triangle. State whether the triangle is isosceles (two sides of equal length), a right triangle, both of these, or neither of these.
54. P0 (−4, 3),    P (−4, −1),    P2 (2, 1).

58. An equilateral triangle is a triangle the three sides of which have the same length. Given that two of the vertices of an equilateral triangle are (0, 0) and (4, 3), find all possible locations for a third vertex. How many such triangles are there?

59. Show that the midpoint M of the hypotenuse of a right triangle is equidistant from the three vertices of the triangle.

60. A median of a triangle is a line segment from a vertex to the midpoint of the opposite side. Find the lengths of the medians of the triangle with vertices (−1, −2), (2, 1), (4, −3).

61. The vertices of a triangle are (1, 0), (3, 4), (−1, 6). Find the point(s) where the medians of this triangle intersect.

62. Show that the medians of a triangle intersect in a single point (called the centroid of the triangle).

63. Prove that each diagonal of a parallelogram bisects the other.

64. P1 (x1 , y1 ), P2 (x2 , y2 ), P3 (x3 , y3 ), P4 (x4 , y4 ) are the vertices of a quadrilateral. Show that the quadrilateral formed by joining the midpoints of adjacent sides is a parallelogram.


Section 1.5



63. Express the area of a circle as a function of the circumference.

64. Express the volume of a sphere as a function of the surface area.

65. Express the volume of a cube as a function of the area of one of the faces.

66. Express the volume of a cube as a function of the total surface area.

67. Express the surface area of a cube as a function of the length of the diagonal of a face.

68. Express the volume of a cube as a function of one of the diagonals.

69. Express the area of an equilateral triangle as a function of the length of a side.

70. A right triangle with hypotenuse c is revolved about one of its legs to form a cone. (See the figure.) Given that x is the length of the other leg, express the volume of the cone as a function of x.

71. A Norman window is a window in the shape of a rectangle surmounted by a semicircle. (See the figure.) Given that the perimeter of the window is 15 feet, express the area as a function of the width x.

72. A window has the shape of a rectangle surmounted by an equilateral triangle. Given that the perimeter of the window is 15 feet, express the area as a function of the length of one side of the equilateral triangle.

73. Express the area of the rectangle shown in the accompanying figure as a function of the x-coordinate of the point P.

74. A right triangle is formed by the coordinate axes and a line through the point (2,5). (See the figure.) Express the area of the triangle as a function of the x-intercept.

75. A string 28 inches long is to be cut into two pieces, one piece to form a square and the other to form a circle. Express the total area enclosed by the square and circle as a function of the perimeter of the square.

76. A tank in the shape of an inverted cone is being filled with water. (See the figure.) Express the volume of water in the tank as a function of the depth h.

77. Suppose that a cylindrical mailing container exactly meets the U.S. Postal Service regulations given in Example 3. (See the figure.) Express the volume of the container as a function of the radius of an end.

 

Section 3.1

 

53. Write an equation for the normal line at (c, f (c)) given that the tangent line at this point. (a) is horizontal; (b) has slope f (c) f ′(c) ≠ 0 (c) is vertical.

54. All the normals through a circular arc pass through one point. What is this point?

55. As you saw in Example 7, the line y – 2 = ¼(x – 4) is tangent to the graph of the square-root function at the point (4, 2). Write an equation for the normal line through this point.

57. The lines tangent and normal to the graph of the squaring function at the point (3, 9) intersect the x-axis at points s units apart. What is s?

 

Section 3.2


Exercises 35–38. Find the point(s) where the tangent line is horizontal.
35.  f (x) = (x − 2)(x2 − x − 11).

Exercises 43–44.  Find the points where the tangent to the graph of
43.  f (x) = −x2 − 6 is parallel to the line y = 4x − 1.
44.  f (x) = x3 − 3x is perpendicular to the line 5y − 3x = 8.

49. Find A and B given that the derivative of

f(x) = { Ax3 + Bx + 2               x ≤ 2
           Bx2 – A                       x > 2

is everywhere continuous.

51. Find the area of the triangle formed by the x-axis, the tangent to the graph of  f (x) = 6x − x2 at the point (5, 5), and the normal through this point (the line through this point that is perpendicular to the tangent).

52. Find the area of the triangle formed by the x-axis and the lines tangent and normal to the graph of  f (x) = 9 − x2 at the point (2, 5).

53. Find A, B, C such that the graph of  f (x) = Ax2 + Bx + C passes through the point (1, 3) and is tangent to the line 4x + y = 8 at the point (2, 0).

54. Find A, B, C, D such that the graph of f (x) = Ax3 + Bx2 + Cx + D is tangent to the line y = 3x − 3 at the point (1, 0) and is tangent to the line y = 18x − 27 at the point (2, 9).

55. Find  the  point  where  the  line  tangent  to  the  graph  of the quadratic function  f (x) = ax2 + bx + c is horizontal.

56. Find conditions on a, b, c, d which guarantee that the graph of the cubic p(x) = ax3 + bx2 + cx + d has: (a) exactly two horizontal tangents.
(b) exactly one horizontal tangent. (c) no horizontal tangents.

57. Find the points (c, f (c)) where the line tangent to the graph of  f (x) = x3 − x is parallel to the secant line that passes through the points (−1, f (−1)) and (2, f(2)).

58. Find the points (c, f (c)) where the line tangent to the graph of f (x) = x/(x + 1) is parallel to the secant line that passes through the points (1, f (1)) and (3, f (3)).

59. Let f (x) = 1/x, x > 0. Show that the triangle that is formed by each line tangent to the graph of f and the coordinate axes has an area of 2 square units.

60. Find two lines through the point (2, 8) that are tangent to the graph of  f (x) = x3.

61. Find  equations  for  all  the  lines  tangent  to  the  graph  of f (x) = x3 − x that pass through the point (−2, 2).

62. Set  f (x) = x3. (a) Find an equation for the line tangent to the graph of f at (c, f (c)), c ¹ 0. (b) Determine whether the tangent line found in (a) intersects the graph of f at a point other than (c, c3). If it does, find the x-coordinate of the second point of intersection.

63. Given two functions  f  and g, show that if  f  and  f + g are differentiable, then g is differentiable. Give an example to show that the differentiability of  f + g does not imply that f  and g are each differentiable.


Section 3.4


1. Find the rate of change of the area of a circle with respect to the radius r. What is the rate when r = 2?

2. Find the rate of change of the volume of a cube with respect to the length s of a side. What is the rate when s = 4?

3. Find the rate of change of the area of a square with respect to the length z of a diagonal. What is the rate when z = 4?

4. Find the rate of change of  y = 1/x  with respect to  x  at x = −1.

5. Find the rate of change of y = [x(x + 1)]−1 with respect to x at x = 2.

6. Find the values of x  at which the rate of change of  y = x3 − 12x2 + 45x − 1 with respect to x is zero.

7. Find the rate of change of the volume of a sphere with respect to the radius r.

8. Find the rate of change of the surface area of a sphere with respect to the radius r. What is this rate of change when r = r0 ? How must r0 be chosen so that the rate of change is 1?

9. Find x0 given that the rate of change of y = 2x2 + x − 1 with respect to x at x = x0 is 4.

10. Find the rate of change of the area A of a circle with respect to
(a) the diameter d;
(b) the circumference C.

11. Find the rate of change of the volume V of a cube with respect to
(a) the length w of a diagonal on one of the faces.
(b) the length z of one of the diagonals of the cube.

12. The dimensions of a rectangle are changing in such a way that the area of the rectangle remains constant. Find the rate of change of the height h with respect to the base b.

13. The area of a sector in a circle is given by the formula A = 1/2r2θ where r is the radius and θ is the central angle measured in radians.
(a) Find the rate of change of A with respect to θ if r remains constant.
(b) Find the rate of change of A with respect to r if θ remains constant.
(c) Find the rate of change of θ with respect to r if A remains constant.

14. The total surface area of a right circular cylinder is given by the formula A = 2πr(r + h) where r is the radius and h is the height.
(a) Find the rate of change of A with respect to h if r remains constant.
(b) Find the rate of change of A with respect to r if h remains constant.
(c) Find the rate of change of h with respect to r if A remains constant.

15. For what value of x is the rate of change of y = ax2 + bx + c with respect to x the same as the rate of change of z = bx2 + ax + c with respect to x?
Assume that a, b, c are constant with a ≠ b.

16. Find the rate of change of the product  f (x)g(x)h(x) with respect to x at x = 1 given that
f (1) = 0,       g(1) = 2,       h(1) = −2,
f ′(1) = 1       g ′(1) = -1     h ′(1) = 0

 

Section 3.5

66. An equilateral triangle of side length x and altitude h has area A given by A = √3x2/4 where x = 2√3h/3. Find the rate of change of A with respect to h and determine this rate of change when h = 2√3.

67. As air is pumped into a spherical balloon, the radius increases at the constant rate of 2 centimeters per  second. What is the rate of change of the balloon’s volume when the radius is 10 centimeters?

68. Air is pumped into a spherical balloon at the constant rate of 200 cubic centimeters per second. How fast is the surface area of the balloon changing when the radius is 5 centimeters?

69. Newton’s law of gravitational attraction states that if two bodies are at a distance r apart, then the force F exerted by one body on the other is given by F(r) = -k/r2 where k is a positive constant. Suppose that, as a function of time, the distance between the two bodies is given by r(t) = 49t − 4.9t2, 0 ≤ t ≤ 10.
(a) Find the rate of change of F with respect to t.
(b) (F ○ r)′ (3) = -(F ○ r)′ (7)

 

Section 3.6

 

58. The double-angle formula for the sine function takes the form: sin 2x = 2sin x cos x. Differentiate this formula to obtain a double-angle formula for the cosine function.

72. A simple pendulum consists of a mass m swinging at the end of a rod or wire of negligible mass. The figure shows a simple pendulum of length L. The angular displacement θ at time t is given by a trigonometric expression: θ(t) = A sin(ωt + φ) where A, ω, φ are constants.
(a) Show that the function θ satisfies the equation d2θ/dt2 + ω2θ = 0
(b) Show that θ can be written in the form θ(t) = A sin ωt + B cos ωt where A, B, ω are constants.

73. An isosceles triangle has two sides of length c. The angle between them is x radians. Express the area A of the triangle as a function of x and find the rate of change of A with respect to x.

74. A triangle has sides of length a and b, and the angle between them is x radians. Given that a and b are kept constant, find the rate of change of the third side c with respect to x.


Section 3.7

 

43. Show that all normals to the circle x2 + y2 = r2 pass through the center of the circle.

44. Determine the x-intercept of the tangent to the parabola y2 = x at the point where x = a.

45. At what angles do the parabolas y2 = 2px + p2 and y2 = p2 − 2px intersect?

46. At what angles does the line y = 2x intersect the curve x2 − xy + 2y2 = 28?

47. The curves y = x2 and x = y3 intersect at the points (1, 1) and (0, 0). Find the angle between the curves at each of these points.

48. Find the angles at which the circles (x − 1)2 + y2 = 10 and x2 + (y − 2)2 = 5 intersect.

53. Find equations for the lines tangent to the ellipse 4x2 + y2 = 72 that are perpendicular to the line
x + 2y + 3 = 0.

54. Find equations  for  the  lines  normal  to  the  hyperbola 4x2 − y2 = 36 that are parallel to the line
2x + 5y − 4 = 0.

55. The curve (x2 + y2 )2 = x2 − y2 is called a lemniscate. The curve is shown in the figure. Find the four points of the curve at which the tangent line is horizontal.

57. Show that the sum of the x- and  y-intercepts of any line tangent to the graph of x1/2 + y1/2 = c1/2 is constant and equal to c.

58. A circle of radius 1 with center on the y-axis is inscribed in the parabola y = 2x2. See the figure. Find the points of contact.

 

Section 4.1

 

11. Determine whether the function f(x) = √(1 – x2)/(3 + x2) satisfies the conditions of Rolle’s theorem on the interval [−1, 1]. If so, find the numbers c for which f ¢(c) = 0.

12. The function  f (x) = x2/3 − 1 has zeros at x = −1 and at x = 1.
(a)  Show that  f ¢ has no zeros in (−1, 1).
(b)  Show that this does not contradict Rolle’s theorem.

13. Does there exist a differentiable function f with f (0) = 2, f (2) = 5, and f ¢(x) ≤ 1 for all x in (0, 2)? If not, why not?

14. Does there exist a differentiable function f  with  f (x) = 1 only at x = 0, 2, 3, and f ¢(x) = 0 only at x = −1, 3/4, 3/2? If not, why not?

16. Find a point on the graph of  f (x) = x2 + x + 3, x between −1 and 2, where the tangent line is parallel to the line through (−1, 3) and (2, 9).

23. Show that the equation 6x4 − 7x + 1 = 0 does not have more than two distinct real roots.

24. Show that the equation 6x5 + 13x + 1 = 0 has exactly one real root.

25. Show that the equation x3 + 9x2 + 33x − 8 = 0 has exactly one real root.

 

Section 4.3

 

1–2. Find the critical points and the local extreme values.
1.  f (x) = x3 + 3x − 2.

2.  f (x) = 2x4 − 4x2 + 6.

29. The graph of f ¢ is given. (a) Find the intervals on which f  increases and the intervals on which f  decreases. (b) Find the local maximum(s) and the local minimum(s) of f. Sketch the graph of f given that  f (0) = 1.

38. Suppose that  f (x) = Ax2 + Bx + C has a local minimum at x = 2 and the graph passes through the points (−1, 3) and (3, −1). Find A, B, C.

39. Find a and b given that  f (x) = ax/(x2 + b2 ) has a local minimum at x = −2 and  f ¢ (0) = 1.

43. Show that  f (x) = x4 − 7x2 − 8x − 3 has exactly one critical point c in the interval (2, 3).

 

Section 4.4


1. Find the critical points. Then find and classify all the extreme values.f(x) = √(x + 2)

35. Show that the cubic p(x) = x3 + ax2 + bx + c has extreme values iff a2 > 3b.

47. If the angle of elevation of a cannon is θ and a projectile is fired with muzzle velocity v ft/sec, then the range of the projectile is given by the formula
R = v2 sin2θ/32. What angle of elevation maximizes the range?

48. A piece of wire of length L is to be cut into two pieces, one piece to form a square and the other piece to form an equilateral triangle. How should the wire be cut so as to (a)  maximize the sum of the areas of the square and the triangle? (b)  minimize the sum of the areas of the square and the triangle?

 

Section 4.5

 

1. Find the greatest possible value of xy given that x and y are both positive and x + y = 40.

2. Find the dimensions of the rectangle of perimeter 24 that has the largest area.

3. A rectangular garden 200 square feet in area is to be fenced off against rabbits. Find the dimensions that will require the least amount of fencing given that one side of the garden is already protected by a barn.

4. Find the largest possible area for a rectangle with base on the x-axis and upper vertices on the curve y = 4 − x2.

5. Find the largest possible area for a rectangle inscribed in a circle of radius 4.

6. Find the dimensions of the rectangle of area A that has the smallest perimeter.

7. How much fencing is needed to define two adjacent rectangular playgrounds of the same width and total area 15,000 square feet?

8. A rectangular warehouse will have 5000 square feet of floor space and will be separated into two rectangular rooms by an interior wall. The cost of the exterior walls is $150 per linear foot and the cost of the interior wall is $100 per linear foot. Find the dimensions that will minimize the cost of building the warehouse.

9. Rework Example 3; this time assume that the semicircular portion of the window admits only one-third as much light per square foot as does the rectangular portion.

10. A rectangular plot of land is to be defined on one side by a straight river and on three sides by post-and-rail fencing. Eight hundred feet of fencing are available. How should the fencing be deployed so as to maximize the area of the plot?

11. Find the coordinates of P that maximize the area of the rectangle shown in the figure.

12. A triangle is to be formed as follows: the base of the triangle is to lie on the x-axis, one side is to lie on the line y = 3x, and the third side is to pass through the point (1, 1). Assign a slope to the third side that maximizes the area of the triangle.

13. A triangle is to be formed as follows: two sides are to lie on the coordinate axes and the third side is to pass through the point (2, 5). Assign a slope to the third side that minimizes the area of the triangle.

14. Show that, for the triangle of Exercise 13, it is impossible to assign a slope to the third side that maximizes the area of the triangle.

15. What are the dimensions of the base of the rectangular box of greatest volume that can be constructed from 100 square inches of cardboard if the base is to be twice as long as it is wide? Assume that the box has a top.

16. Exercise 15 under the assumption that the box has no top.

17. Find the dimensions of the isosceles triangle of largest area with perimeter 12.

18. Find the point(s) on the parabola y = 1/8x2 closest to the point (0, 6).

19. Find the point(s) on the parabola x = y2 closest to the point (0, 3).

20. Find A and B given that the function y = Ax−1/2 + Bx1/2 has a minimum of 6 at x = 9.

21. Find the maximal possible area for a rectangle inscribed in the ellipse 16x2 + 9y2 = 144.

22. Find the maximal possible area for a rectangle inscribed in the ellipse b2 x2 + a2y2 = a2b2.

23. A pentagon with a perimeter of 30 inches is to be constructed by adjoining an equilateral triangle to a rectangle. Find the dimensions of the rectangle and triangle that will maximize the area of the pentagon.

24. A 10-foot section of gutter is made from a 12-inch-wide strip of sheet metal by folding up 4-inch strips on each side so that they make the same angle with the bottom of the gutter. Determine the depth of the gutter that has the greatest carrying capacity.

25. From a 15 × 8 rectangular piece of cardboard four congruent squares are to be cut out, one at each corner. (See the figure.) The remaining crosslike piece is then to be folded into an open box. What size squares should be cut out so as to maximize the volume of the resulting box?

26. A page is to contain 81 square centimeters of print. The margins at the top and bottom are to be 3 centimeters each and, at the sides, 2 centimeters each. Find the most economical dimensions given that the cost of a page varies directly with the perimeter of the page.

27. Let ABC be a triangle with vertices A = (−3, 0), B = (0, 6), C = (3, 0). Let P be a point on the line segment that joins B to the origin. Find the position of P that minimizes the sum of the distances between P and the vertices.

28. Solve Exercise 27 with A = (−6, 0), B = (0, 3), C = (6, 0).

29. An 8-foot-high fence is located 1 foot from a building. Determine the length of the shortest ladder that can be leaned against the building and touch the top of the fence.

30. Two hallways, one 8 feet wide and the other 6 feet wide, meet at right angles. Determine the length of the longest ladder that can be carried horizontally from one hallway into the other.

31. A rectangular banner is to have a red border and a rectangular white center. The width of the border at top and bottom is to be 8 inches, and along the sides 6 inches. The total area is to be 27 square feet. Find the dimensions of the banner that maximize the area of the white center.

32. Conical paper cups are usually made so that the depth is √2 times the radius of the rim. Show that this design requires the least amount of paper per unit volume.

33. A string 28 inches long is to be cut into two pieces, one piece to form a square and the other to form a circle. How should the string be cut so as to
(a) maximize the sum of the two areas? (b) minimize the sum of the two areas?

34. What is the maximum volume for a rectangular box (square base, no top) made from 12 square feet of cardboard?

35. The figure shows a cylinder inscribed in a right circular cone of height 8 and base radius 5. Find the dimensions of the cylinder that maximize its volume.

36. As a variant of Exercise 35, find the dimensions of the cylinder that maximize the area of its curved surface.

37. A rectangular box with square base and top is to be made to contain 1250 cubic feet. The material for the base costs 35 cents per square foot, for the top 15 cents per square foot, and for the sides 20 cents per square foot. Find the dimensions that will minimize the cost of the box.

38. What is the largest possible area for a parallelogram inscribed in a triangle ABC in the manner of the figure?

39. Find the dimensions of the isosceles triangle of least area that circumscribes a circle of radius r.

40. What is the maximum possible area for a triangle inscribed in a circle of radius r?

41. The figure shows a right circular cylinder inscribed in a sphere of radius r. Find the dimensions of the cylinder that maximize the volume of the cylinder.

42. As a variant of Exercise 41, find the dimensions of the right circular cylinder that maximize the lateral area of the cylinder.

43. A right circular cone is inscribed in a sphere of radius r as in the figure. Find the dimensions of the cone that maximize the volume of the cone.

44. What is the largest possible volume for a right circular cone of slant height a?

45. A power line is needed to connect a power station on the shore of a river to an island 4 kilometers downstream and 1 kilometer offshore. Find the minimum cost for such a line given that it costs $50,000 per kilometer to lay wire under water and $30,000 per kilometer to lay wire under ground.

46. A tapestry 7 feet high hangs on a wall. The lower edge is 9 feet above an observer’s eye. How far from the wall should the observer stand to obtain the most favorable view? Namely, what distance from the wall maximizes the visual angle of the observer?

47. An object of weight W is dragged along a horizontal plane by means of a force P whose line of action makes an angle θ with the plane. The magnitude of the force is given by the formula

where μ denotes the coefficient of friction. Find the value of θ that minimizes P.

48. The range of a projectile fired with elevation angle θ at an inclined plane is given by the formula

where α is the inclination of the target plane, and v and g are constants. Calculate θ for maximum range.

49. Two sources of heat are placed s meters apart—a source of intensity a at A and a source of intensity b at B. The intensity of heat at a point P on the line segment between A and B is given by the formula

where x is the distance between P and A measured in meters. At what point between A and B will the temperature be lowest?

50. The distance from a point to a line is the distance from that point to the closest point of the line. What point of the line Ax + By + C = 0 is closest to the point (x1 , y1 )? What is the distance from (x1 , y1) to the line?

51. Let f be a differentiable function defined on an open interval I. Let P(a, b) be a point not on the graph of f. Show that if PQ is the longest or shortest line segment that joins P to the graph of f, then PQ is perpendicular to the graph of f.

52. Draw the parabola y = x2. On the parabola mark a point P ≠ O. Through P draw the normal line. The normal line intersects the parabola at another point Q. Show that the distance between P and Q is minimized by setting .

53. For each integer n, set f(n) = 6n4 − 16n3 + 9n2 . Find the integer n that minimizes  f (n).

54. A local bus company offers charter trips to Blue Mountain Museum at a fare of $37 per person if 16 to 35 passengers sign up for the trip. The company does not charter trips for fewer than 16 passengers. The bus has 48 seats. If more than 35 passengers sign up, then the fare for every passenger is reduced by 50 cents for each passenger in excess of 35 that signs up. Determine the number of passengers that generates the greatest revenue for the bus company.

55. The Hotwheels Rent-A-Car Company derives an average net profit of $12 per customer if it services 50 customers or fewer. If it services more than 50 customers, then the average net profit is decreased by 6 cents for each customer over 50. What number of customers produces the greatest total net profit for the company?

56. A steel plant has the capacity to produce x tons per day of low-grade steel and y tons per day of high-grade steel where
Given that the market price of low-grade steel is half that of high-grade steel, show that about tons of low-grade steel should be produced per day for maximum revenue.

57. The path of a ball is the curve  . Here the origin is taken as the point from which the ball is thrown and m is the initial slope of the trajectory. At a distance which depends on m, the ball returns to the height from which it was thrown. What value of m maximizes this distance?

58. Given the trajectory of Exercise 57, what value of m maximizes the height at which the ball strikes a vertical wall 300 feet away?

59. A truck is to be driven 300 miles on a freeway at a constant speed of ν miles per hour. Speed laws require that 35 ≤ ν ≤ 70. Assume that the fuel costs $2.60 per gallon and is consumed at the rate of  gallons per hour. Given that the driver’s wages are $20 per hour, at what speed should the truck be driven to minimize the truck owner’s expenses?

60. A tour boat heads out on a 100-kilometer sight-seeing trip. Given that the fixed costs of operating the boat total $2500 per hour, that the cost of fuel varies directly with the square of the speed of the boat, and at 10 kilometers per hour the cost of the fuel is $400 per hour, find the speed that minimizes the boat owner’s expenses. Is the speed that minimizes the owner’s expenses dependent on the length of the trip?

61. An oil drum is to be made in the form of a right circular cylinder to contain 16π cubic feet. The upright drum is to be taller than it is wide, but not more than 6 feet tall. Determine the dimensions of the drum that minimize surface area.

62. The cost of erecting a small office building is $1,000,000 for the first story, $1,100,000 for the second, $1,200,000 for the third, and so on. Other expenses (lot, basement, etc.) are $5,000,000. Assume that the annual rent is $200,000 per story. How many stories will provide the greatest return on investment?

 

Section 4.6

 

27. Find: (a) the intervals on which f  increases and the intervals on which f decreases; (b) the local maxima and
the local minima; (c) the intervals on which the graph is concave up and the intervals on which the graph is concave down; (d) the points of inflection. Use this information to sketch the graph
f (x) = x3 − 9x.

39. Find d given that (d, f (d)) is a point of inflection of the graph of f (x) = (x − a)(x − b)(x − c).

40. Find c given that the graph of f (x) = cx2 + x−2 has a point of inflection at (1, f (1)).

41. Find  a  and  b  given  that  the  graph  of  f (x) = ax3 + bx2 passes through the point (−1, 1) and has a point of inflection where .

42. Determine A and B so that the curve y = Ax1/2 + Bx−1/2 has a point of inflection at (1, 4).

43. Determine A and B so that the curve y = A cos 2x + B sin 3x has a point of inflection at (π/6, 5).

44. Find necessary and sufficient conditions on A and B for f (x) = Ax2 + Bx + C
(a)  to decrease between A and B with graph concave up.
(b)  to increase between A and B with graph concave down.

45. Find a function f  with  f ¢(x) = 3x2 − 6x + 3 for all real x and (1, −2) a point of inflection. How many such functions are there?

46. Set f (x) = sin x. Show that the graph of f is concave down above the x-axis and concave up below the x-axis. Does g(x) = cos x have the same property?

 

Section 4.9

 

1. An object moves along a coordinate line, its position at each time t ≥ 0 given by x(t). Find the position, velocity, and acceleration at time t0 . What is the speed at time t0?
1. x(t) = 4 + 3t − t2 ;    t0 = 5

7. An object moves along the x-axis, its position at each time t ≥ 0 given by x(t). Determine the times, if any, at
which (a) the velocity is zero, (b) the acceleration is zero.
x(t) = 5t + 1.

Exercises 11-20. Objects A, B, C move along the x-axis. Their positions x(t) from time t = 0 to time t = t3 have been graphed in the figure as functions of t.
11. Which object begins farthest to the right?

12. Which object finishes farthest to the right?

13. Which object has the greatest speed at time t1 ?

14. Which object maintains the same direction during the time interval [t1 , t3 ]?

15. Which object begins moving left?

16. Which object finishes moving left?

17. Which object changes direction at time t2 ?

18. Which object speeds up throughout the time interval [0, t1 ]?

19. Which objects slow down during the time interval [t1 , t2 ]?

20. Which object  changes  direction  during  the  time  interval [t2 , t3]?

Exercises 21–28. An object moves along the x-axis, its position at each time t ≥ 0 given by x(t). Determine the time interval(s), if any, during which the object satisfies the given condition.
21. x(t) = t4 − 12t3 + 28t2 ; moves right.

22. x(t) = t3 − 12t2 + 21t; moves left.

23. x(t) = 5t4 − t5 ; speeds up.

24. x(t) = 6t2 − t4 ; slows down.
25. x(t) = t3 − 6t2 − 15t; moves left slowing down.
26. x(t) = t3 − 6t2 − 15t; moves right slowing down.
27. x(t) = t4 − 8t3 − 16t2 ; moves right speeding up.
28. x(t) = t4 − 8t3 − 16t2 ; moves left speeding up.

Exercises 29–32. An object moves along a coordinate line, its position at each time t ≥ 0 being given by x(t). Find the times t at which the object changes direction.
29. x(t) = (t + 1)2 (t − 9)3.

39. An object is dropped and hits the ground 6 seconds later. From what height, in feet, was it dropped?

40. Supplies are dropped from a stationary helicopter and seconds later hit the ground at 98 meters per second. How high was the helicopter?

41. An object is projected vertically upward from ground level with velocity v. Find the height in meters attained by the object.

42. An object projected vertically upward from ground level returns to earth in 8 seconds. Give the initial velocity in feet per second.

43. An object projected vertically upward passes every height less than the maximum twice, once on the way up and once on the way down. Show that the speed is the same in each direction. Measure height in feet.

44. An object is projected vertically upward from the ground. Show that it takes the object the same amount of time to reach its maximum height as it takes for it to drop from that height back to the ground. Measure height in meters.

45. A rubber ball is thrown straight down from a height of 224 feet at a speed of 80 feet per second. If the ball always rebounds with one-fourth of its impact speed, what will be the speed of the ball the third time it hits the ground?

46. A ball is thrown straight up from ground level. How high will the ball go if it reaches a height of 64 feet in 2 seconds?

47. A stone is thrown upward from ground level. The initial speed is 32 feet per second. (a) In how many seconds will the stone hit the ground? (b) How high will it go? (c) With what minimum speed should the stone be thrown so as to reach a height of at least 36 feet?

48. To estimate the height of a bridge, a man drops a stone into the water below. How high is the bridge (a) if the stone hits the water 3 seconds later? (b) if the man hears the splash 3 seconds later? (Use 1080 feet per second as the speed of sound.)

49. A falling stone is at a certain instant 100 feet above the ground. Two seconds later it is only 16 feet above the ground. (a) From what height was it dropped? (b) If it was thrown down with an initial speed of 5 feet per second, from what height was it thrown? (c) If it was thrown upward with an initial speed of 10 feet per second, from what height was it thrown?

50. A rubber ball is thrown straight down from a height of 4 feet. If the ball rebounds with one-half of its impact speed and returns exactly to its original height before falling again, how fast was it thrown originally?

51. Ballast dropped from a balloon that was rising at the rate of 5 feet per second reached the ground in 8 seconds. How high was the balloon when the ballast was dropped?

52. Had the balloon of Exercise 51 been falling at the rate of 5 feet per second, how long would it have taken for the ballast to reach the ground?

53. Two race horses start a race at the same time and finish in a tie. Prove that there must have been at least one time t during the race at which the two horses had exactly the same speed.

54. Suppose that the two horses of Exercise 53 cross the finish line together at the same speed. Show that they had the same acceleration at some instant during the race.

55. A certain tollroad is 120 miles long and the speed limit is 65 miles per hour. If a driver’s entry ticket at one end of the tollroad is stamped 12 noon and she exits at the other end at 1:40 p.m., should she be given a speeding ticket? Explain.

56. At 1:00 p.m. a car’s speedometer reads 30 miles per hour and at 1:15 p.m. it reads 60 miles per hour. Prove that the car’s acceleration was exactly 120 miles per hour per hour at least once between 1:00 and 1:15.

57. A car is stationary at a toll booth. Twenty minutes later, at a point 20 miles down the road, the car is clocked at 60 mph. Explain how you know that the car must have exceeded the 60-mph speed limit some time before being clocked at 60 mph.

58. The results of an investigation of a car accident showed that the driver applied his brakes and skidded 280 feet in 6 seconds. If the speed limit on the street where the accident occurred was 30 miles per hour, was the driver exceeding the speed limit at the instant he applied his brakes? Explain.

59. (Simple harmonic motion) A bob suspended from a spring oscillates up and down about an equilibrium point, its vertical position at time t given by
y(t) = A sin (ωt + ϕ0 )
where A, ω, ϕ0 are positive constants.
(a)  Show that at all times t the acceleration of the bob y ²(t) is related to the position of the bob by the equation
y ²(t) + ω2 = 0.
(b)  It is clear that the bob oscillates from −A to A, and the speed of the bob is zero at these points. At what position does the bob attain maximum speed? What is this maximum speed?
(c)  What are the extreme values of the acceleration function? Where does the bob attain these extreme values?

 

Section 4.10

 

1. A point moves along the line x + 2y = 2. Find (a) the rate of change of the y-coordinate, given that the x-coordinate is increasing at the rate of 4 units per second; (b) the rate of change of the x-coordinate, given that the y-coordinate is decreasing at the rate of 2 units per second.

2. A particle is moving in the circular orbit x2 + y2 = 25. As it passes through the point (3, 4), its y-coordinate is decreasing at the rate of 2 units per second. At what rate is the x-coordinate changing?

3. A particle is moving along the parabola y2 = 4(x + 2). As it passes through the point (7, 6), its y-coordinate is increasing at the rate of 3 units per second. How fast is the x-coordinate changing at this instant?

4. A particle is moving along the parabola 4y = (x + 2)2 in such a way that its x-coordinate is increasing at the constant rate of 2 units per second. How fast is the particle’s distance from the point (−2, 0) changing as it passes through the point (2, 4)?

5. A particle is moving along the ellipse x2/16 + y2/4 = 1. At each time t its x- and y-coordinates are given by x = 4 cos t, y = 2 sin t. At what rate is the particle’s distance from the origin changing at time t? At what rate is this distance from the origin changing when t = π/4?

6. A particle is moving along the curve y = x√x, x ≥ 0. Find  the points on the curve, if any, at which both coordinates are changing at the same rate.

7. A heap of rubbish in the shape of a cube is being compacted into a smaller cube. Given that the volume decreases at the rate of 2 cubic meters per minute, find the rate of change of an edge of the cube when the volume is exactly 27 cubic meters. What is the rate of change of the surface area of the cube at that instant?

8. The volume of a spherical balloon is increasing at the constant rate of 8 cubic feet per minute. How fast is the radius increasing when the radius is exactly 10 feet? How fast is the surface area increasing at that time?

9. At a certain instant the side of an equilateral triangle is α centimeters long and increasing at the rate of k centimeters per minute. How fast is the area increasing?

10. The dimensions of a rectangle are changing in such a way that the perimeter remains 24 inches. Show that when the area is 32 square inches, the area is either increasing or decreasing 4 times as fast as the length is increasing.

11. A rectangle is inscribed in a circle of radius 5 inches. If the length of the rectangle is decreasing at the rate of 2 inches per second, how fast is the area changing when the length is 6 inches?

12. A boat is held by a bow line that is wound about a bollard 6 feet higher than the bow of the boat. If the boat is drifting away at the rate of 8 feet per minute, how fast is the line unwinding when the bow is 30 feet from the bollard?

13. Two boats are racing with constant speed toward a finish marker, boat A sailing from the south at 13 mph and boat B approaching from the east. When equidistant from the marker, the boats are 16 miles apart and the distance between them is decreasing at the rate of 17 mph. Which boat will win the race?

14. A spherical snowball is melting in such a manner that its radius is changing at a constant rate, decreasing from 16 cm to 10 cm in 30 minutes. How fast is the volume of the snowball changing when the radius is 12 cm?

15. A 13-foot ladder is leaning against a vertical wall. If the bottom of the ladder is being pulled away from the wall at the rate of 2 feet per second, how fast is the area of the triangle formed by the wall, the ground, and the ladder changing when the bottom of the ladder is 12 feet from the wall?

16. A ladder 13 feet long is leaning against a wall. If the foot of the ladder is pulled away from the wall at the rate of 0.5 feet per second, how fast will the top of the ladder be dropping when the base is 5 feet from the wall?

17. A tank contains 1000 cubic feet of natural gas at a pressure of 5 pounds per square inch. Find the rate of change of the volume if the pressure decreases at a rate of 0.05 pounds per square inch per hour. (Assume  Boyle’s  law: pressure × volume = constant.)

18. The adiabatic law for the expansion of air is PV1.4 = C. At a given instant the volume is 10 cubic feet and the pressure is 50 pounds per square inch. At what rate is the pressure changing if the volume is decreasing at a rate of 1 cubic foot per second?

19. A man standing 3 feet from the base of a lamppost casts a shadow 4 feet long. If the man is 6 feet tall and walks away from the lamppost at a speed of 400 feet per minute, at what rate will his shadow lengthen? How fast is the tip of his shadow moving?

20. A light is attached to the wall of a building 64 feet above the ground. A ball is dropped from that height, but 20 feet away from the side of the building. The height y of the ball at time t is given by y(t) = 64 − 16t2 . Here we are measuring y in feet and t in seconds. How fast is the shadow of the ball moving along the ground 1 second after the ball is dropped?

21. An object that weighs 150 pounds on the surface of the earth will weigh 150(1 + r/4000)-2 pounds when it is r miles above the earth. Given that the altitude of the object is increasing at the rate of 10 miles per minute, how fast is the weight decreasing when the object is 400 miles above the surface?

22. In the special theory of relativity the mass of a particle moving at speed ν is given by the expression m/√(1 – v2/c2) where m is the mass at rest and c is the speed of light. At what rate is the mass of the particle changing when the speed of the particle is ½ c and is increasing at the rate of 0.01c per second?

23. Water is dripping through the bottom of a conical cup 4 inches across and 6 inches deep. Given that the cup loses half a cubic inch of water per minute, how fast is the water level dropping when the water is 3 inches deep?

24. Water is poured into a conical container, vertex down, at the rate of 2 cubic feet per minute. The container is 6 feet deep and the open end is 8 feet across. How fast is the level of the water rising when the container is half full?

25. At what rate is the volume of a sphere changing at the instant when the surface area is increasing at the rate of 4 square centimeters per minute and the radius is increasing at the rate of 0.1 centimeter per minute?

26. Water flows from a faucet into a hemispherical basin 14 inches in diameter at the rate of 2 cubic inches per second. How fast does the water rise (a) when the water is exactly halfway to the top? (b) just as it runs over?

27. The base of an isosceles triangle is 6 feet. Given that the altitude is 4 feet and increasing at the rate of 2 inches per minute, at what rate is the vertex angle changing?

28. As a boy winds up the cord, his kite is moving horizontally at a height of 60 feet with a speed of 10 feet per minute. How fast is the inclination of the cord changing when the cord is 100 feet long?

29. A revolving searchlight mile from a straight shoreline makes 1 revolution per minute. How fast is the light moving along the shore as it passes over a shore point 1 mile from the shore point nearest to the  searchlight?

30. A revolving searchlight 1 mile from a straight shoreline turns at the rate of 2 revolutions per minute in the counterclockwise direction.
(a)  How fast is the light moving along the shore when it makes an angle of 45o with the shore?
(b)  How fast is the light moving when the angle is 90o?

31. A man starts at a point A and walks 40 feet north. He then turns and walks due east at 4 feet per second. A searchlight placed at A follows him. At what rate is the light turning 15 seconds after the man started walking east?

32. The diameter and height of a right circular cylinder are found at a certain instant to be 10 centimeters and 20 centimeters, respectively. If the diameter is increasing at the rate of 1 centimeter per second, what change in height will keep the volume constant?

33. A horizontal trough 12 feet long has a vertical cross section in the form of a trapezoid. The bottom is 3 feet wide, and the sides are inclined to the vertical at an angle with sine 4/5. Given that water is poured into the trough at the rate of 10 cubic feet per minute, how fast is the water level rising when the water is exactly 2 feet deep?

34. Two cars, car A traveling east at 30 mph and car B traveling north at 22.5 mph, are heading toward an intersection I. At what rate is the angle IAB changing when cars A and B are 300 feet and 400 feet, respectively, from the intersection?

35. A rope 32 feet long is attached to a weight and passed over a pulley 16 feet above the ground. The other end of the rope is pulled away along the ground at the rate of 3 feet per second. At what rate is the angle between the rope and the ground changing when the weight is exactly 4 feet off the ground?

36. A slingshot is made by fastening the two ends of a 10-inch rubber strip 6 inches apart. If the midpoint of the strip is drawn back at the rate of 1 inch per second, at what rate is the angle between the segments of the strip changing 8 seconds later?

37. A balloon is released 500 feet away from an observer. If the balloon rises vertically at the rate of 100 feet per minute and at the same time the wind is carrying it away horizontally at the rate of 75 feet per minute, at what rate is the inclination of the observer’s line of sight changing 6 minutes after the balloon has been released?

38. A searchlight is continually trained on a plane that flies directly above it at an altitude of 2 miles at a speed of 400 miles per hour. How fast does the light turn 2 seconds after the plane passes directly overhead?

39. A baseball diamond is a square 90 feet on a side. A player is running from second base to third base at the rate of 15 feet per second. Find the rate of change of the distance from the player to home plate at the instant the player is 10 feet from third base. (If you are not familiar with baseball, skip this problem.)

40. An airplane is flying at constant speed and altitude on a line that will take it directly over a radar station on the ground. At the instant the plane is 12 miles from the station, it is noted that the plane’s angle of elevation is 30o and is increasing at the rate of 0.5 o per second. Give the speed of the plane in miles per hour.

41. An athlete is running around a circular track of radius 50 meters at the rate of 5 meters per second. A spectator is 200 meters from the center of the track. How fast is the distance between the two changing when the runner is approaching the spectator and the distance between them is 200 meters?

 

Section 4.11

 

1. Use a differential to estimate the change in the volume of a cube caused by an increase h in the length of each side. Interpret geometrically the error of your estimate DV − dV .

2. Use a differential to estimate the area of a ring of inner radius r and width h. What is the exact area?

13. Estimate f (2.8) given that f (3) = 2 and f ′(x) = (x3 + 5)1/5.

14. Estimate  f (5.4) given that  f (5) = 1 and f ′(x) = 3√(x2 + 2).

15. Find the approximate volume of a thin cylindrical shell with open ends given that the inner radius is r, the height is h, and the thickness is t.

16. The diameter of a steel ball is measured to be 16 centimeters, with a maximum error of 0.3 centimeters. Estimate by differentials the maximum error (a) in the surface area as calculated from the formula S = 4πr2 ; (b) in the volume as calculated from the formula V = 4/3 πr3.

17. A box is to be constructed in the form of a cube to hold 1000 cubic feet. Use a differential to estimate how accurately the inner edge must be made so that the volume will be correct to within 3 cubic feet.

18. Use differentials to estimate the values of x for which √(x + 1) - √x < 0.01.

19. A hemispherical dome with a 50-foot radius will be given a coat of paint 0.01 inch thick. The contractor for the job wants to estimate the number of gallons of paint that will be needed. Use a differential to obtain an estimate. (There are 231 cubic inches in a gallon.)

20. View the earth as a sphere of radius 4000 miles. The volume of ice that covers the north and south poles is estimated to be 8 million cubic miles. Suppose that this ice melts and the water produced distributes itself uniformly over the surface of the earth. Estimate the depth of this water.

21. The period P of the small oscillations of a simple pendulum is related to the length L of the pendulum by the equation P = 2π√(L/g) where g is the (constant) acceleration of gravity. Show that a small change dL in the length of a pendulum produces a change dP in the period that satisfies the equation dP/P = ½ dL/L.

22. Suppose that the pendulum of a clock is 90 centimeters long. Use the result in Exercise 21 to determine how the length of the pendulum should be adjusted if the clock is losing 15 seconds per hour.

23. A pendulum of length 3.26 feet goes through one complete oscillation in 2 seconds. Use Exercise 21 to find the approximate change in P if the pendulum is lengthened by 0.01 feet.

24. A metal cube is heated and the length of each edge is thereby increased by 0.1%. Use a differential to show that the volume of the cube is then increased by about 0.3%.

25. We want to determine the area of a circle by measuring the diameter x and then applying the formula A = 1/4πr2. Use a differential to estimate how accurately we must measure the diameter for our area formula to yield a result that is accurate within 1%.

26. Estimate by differentials how precisely x must be determined (a) for our calculation of xn to be accurate within 1%; (b) for our estimate of x1/n to be accurate within 1%. (Here n is a positive integer.)

 

Section 4.12

 

1.Use the Newton-Raphson method to estimate a root of the equation f (x) = 0 starting at the indicated value of x:
(a) Express xn+1 in terms of xn .
(b) Give x4 rounded off to five decimal places and evaluate f at that approximation.
f(x) = x2 - 24; x1 = 5

16. Set f (x) = x4− 7x2 − 8x − 3.
(a) Show that f has exactly one critical point c in the interval (2, 3).
(b) Use the Newton-Raphson method to estimate c by calculating x3. Round off your answer to four decimal places. Does f have a local maximum at c, a local minimum, or neither?


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