Practice Exercises

A1. The slope of the curve y³ – xy² = 4 at the point where y = 2 is

(A) –2
(B) 
(C) 
(D) 2

A2. The slope of the curve y² – xy – 3x = 1 at the point (0,–1) is

(A) –1
(B) –2
(C) 1
(D) 2

A3. An equation of the tangent to the curve y = x sin x at the point  is

(A) y = x – π
(B) 
(C) y = π – x
(D) y = x

A4. The tangent to the curve of y = xe^–x is horizontal when x is equal to

(A) 0
(B) 1
(C) –1
(D) 

A5. The minimum value of the slope of the curve y = x⁵ + x³ – 2x is

(A) 2
(B) 6
(C) –2
(D) –6

A6. An equation of the tangent to the hyperbola x² – y² = 12 at the point (4,2) on the curve is

(A) x – 2y + 6 = 0
(B) y = 2x
(C) y = 2x – 6
(D) 

A7. A tangent to the curve y² – xy + 9 = 0 is vertical when

(A) y = 0
(B) 
(C) 
(D) y = ±3

A8. The volume of a sphere is given by . Use a tangent line to approximate the increase in volume, in cubic inches, when the radius of a sphere is increased from 3 to 3.1 inches.

(A) 
(B) 0.04π
(C) 1.2π
(D) 3.6π

A9. When x = 3, the equation 2x² – y³ = 10 has the solution y = 2. Using a tangent line to the graph of the curve, approximate y when x = 3.04.

(A) 1.6
(B) 1.96
(C) 2.04
(D) 2.4

CHALLENGE A10. If the side e of a square is increased by 1%, then the area is increased approximately

(A) 0.02e
(B) 0.02e²
(C) 0.01e²
(D) 0.01e

CHALLENGE A11. The edge of a cube has length 10 in., with a possible error of 1%. The possible error, in cubic inches, in the volume of the cube is

(A) 1
(B) 3
(C) 10
(D) 30

A12. The function f(x) = x⁴ – 4x² has

(A) one local minimum and two local maxima
(B) one local minimum and no local maximum
(C) no local minimum and one local maximum
(D) two local minima and one local maximum

A13. The number of inflection points on the graph of f(x) = x⁴ – 4x² is

(A) 0
(B) 1
(C) 2
(D) 3

A14. The maximum value of the function  is

(A) 0
(B) –4
(C) –2
(D) 2

A15. The total number of local maximum and minimum points of the function whose derivative, for all x, is given by f′(x) = x(x – 3)²(x + 1)⁴ is

(A) 0
(B) 1
(C) 2
(D) 3

A16. For which curve shown below are both f′ and f ″ negative?

(A)

 

(B)

 

(C)

(D)

A17. For which curve shown in Question A16 is f″ positive but f′ negative?

(A) Curve (A)
(B) Curve (B)
(C) Curve (C)
(D) Curve (D)

In Questions A18–A21, the position of a particle moving along a horizontal line is given by s = t³ – 6t² + 12t – 8.

A18. The object is moving to the right for

(A) t < 2
(B) all t except t = 2
(C) all t
(D) t > 2

A19. The minimum value of the speed is

(A) 0
(B) 1
(C) 2
(D) 3

A20. The acceleration is positive

(A) when t > 2
(B) for all t, t ≠ 2
(C) when t < 2
(D) for 1 < t < 2

A21. The speed of the particle is decreasing for

(A) t < 2
(B) all t
(C) t < 1 or t > 2
(D) t > 2

In Questions A22–A24, a particle moves along a horizontal line and its position at time t is s = t⁴ – 6t³ + 12t² + 3.

A22. The particle is at rest when t is equal to

(A) 1 or 2
(B) 0
(C) 
(D) 0, 1, or 2

A23. The velocity, v, is increasing when

(A) t > 1
(B) 1 < t < 2
(C) t < 2
(D) t < 1 or t > 2

A24. The speed of the particle is increasing for

(A) 0 < t < 1 or t > 2
(B) 1 < t < 2
(C) t < 2
(D) t < 0 or t > 2

A25. The displacement from the origin of a particle moving on a line is given by s = t⁴ – 4t³. The maximum displacement during the time interval –2 ≤ t ≤ 4 is

(A) 27
(B) 3
(C) 48
(D) 16

A26. If a particle moves along a line according to the law s = t⁵ + 5t⁴, then the number of times it reverses direction is

(A) 0
(B) 1
(C) 2
(D) 3

*In Questions A27–A30,  is the (position) vector 〈x,y〉 from the origin to a moving point P(x,y) at time t.

*A27. A single equation in x and y for the path of the point is

(A) x² + y² = 13
(B) 9x² + 4y² = 36
(C) 2x² + 3y² = 13
(D) 4x² + 9y² = 36

*A28. When t = 3, the speed of the particle is
(A) 
(B) 2
(C) 3
(D) 

*A29. The magnitude of the acceleration when t = 3 is

*A30. At the point where , the slope of the curve along which the particle moves is

A31. A balloon is being filled with helium at the rate of 4 ft³/min. The rate, in square feet per minute, at which the surface area is increasing when the volume is  is

(A) 4π
(B) 2
(C) 4
(D) 1

A32. A circular conical reservoir, vertex down, has depth 20 ft and radius at the top 10 ft. Water is leaking out so that the surface is falling at the rate of . The rate, in cubic feet per hour, at which the water is leaving the reservoir when the water is 8 ft deep is

(A) 4π
(B) 8π
(C) 
(D) 

A33. A local minimum value of the function  is
(A) 
(B) 1
(C) –1
(D) e

CHALLENGE A34. The area of the largest rectangle that can be drawn with one side along the x-axis and two vertices on the curve of  is

CHALLENGE A35. A line with a negative slope is drawn through the point (1,2) forming a right triangle with the positive x- and y-axes. The slope of the line forming the triangle of least area is

(A) –1
(B) –2
(C) –3
(D) –4

CHALLENGE A36. The point(s) on the curve x² – y² = 4 closest to the point (6,0) is (are)

A37. The sum of the squares of two positive numbers is 200; their minimum product is

(A) 100
(B) 20
(C) 0
(D) there is no minimum

CHALLENGE A38. The first-quadrant point on the curve y²x = 18 that is closest to the point (2,0) is

 

A39. If h is a small negative number, then the local linear approximation for 

A40. If f(x) = xe^–x, then at x = 0

(A) f is increasing
(B) f is decreasing
(C) f has a relative maximum
(D) f has a relative minimum

A41. A function f has a derivative for each x such that |x| < 2 and has a local minimum at (2,–5). Which statement below must be true?

(A) f′(2) = 0
(B) f′ exists at x = 2
(C) f′(x) < 0 if x < 2, f′(x) > 0 if x > 2
(D) none of the preceding is necessarily true

A42. The height of a rectangular box is 10 in. Its length increases at the rate of 2 in./sec; its width decreases at the rate of 4 in./sec. When the length is 8 in. and the width is 6 in., the rate, in cubic inches per second, at which the volume of the box is changing is

(A) 200
(B) 80
(C) –80
(D) –200

A43. The tangent to the curve x³ + x^2y + 4y = 1 at the point (3,–2) has slope

A44. If f(x) = ax⁴ + bx2 and ab > 0, then

(A) the curve has no horizontal tangents
(B) the curve is concave up for all x
(C) the curve has no inflection point
(D) none of the preceding is necessarily true

A45. A function f is continuous and differentiable on the interval [0,4], where f′ is positive but f″ is negative. Which table could represent points on f?

*A46. An equation of the tangent to the curve with parametric equations x = 2t + 1, y = 3 – t³ at the point where t = 1 is

(A) 2x + 3y = 12
(B) 3x + 2y = 13
(C) 6x + y = 20
(D) 3x – 2y = 5

A47. Given the function , it is known that f(64) = 4. Using the tangent line to the graph of f(x), approximately how much less than 4 is ?

A48. The best linear approximation for f(x) = tan x near  is y =

A49. Given f(x) = e^kx, approximate f(h), where h is near zero, using a tangent-line approximation. f(h) ≈

(A) k
(B) kh
(C) 1 + k
(D) 1 + kh

A50. If f(x) = cx² + dx + e for the function shown in the graph, then

(A) c, d, and e are all positive
(B) c > 0, d < 0, e > 0
(C) c < 0, d > 0, e > 0
(D) c < 0, d < 0, e > 0

A51. Given f(x) = log₁₀x and log₁₀(102) ≈ 2.0086, which is closest to f′(100)?

(A) 0.0043
(B) 0.0086
(C) 0.01
(D) 1.0043

B1. The point on the curve  at which the tangent is parallel to the line x – 3y = 6 is

(A) (4,3)
(B) (0,1)
(C) 
(D) 

B2. An equation of the tangent to the curve x² = 4y at the point on the curve where x = –2 is

(A) x + y – 3 = 0
(B) x – y + 3 = 0
(C) x + y – 1 = 0
(D) x + y + 1 = 0

B3. The table shows the velocity at time t of an object moving along a line. Estimate the acceleration (in ft/sec²) at t = 6 sec.

(A) –6
(B) –1.5
(C) 1.5
(D) 6

Use the graph shown, sketched on [0,7], for Questions B4–B6.

B4. From the graph it follows that

(A) f is discontinuous at x = 4
(B) f is decreasing for 4 < x < 7
(C) f(5) < f(0)
(D) f(2) < f(3)

B5. Which statement best describes f at x = 5?

(A) f has a root.
(B) f has a maximum.
(C) f has a minimum.
(D) The graph of f has a point of inflection.

B6. For which interval is the graph of f concave downward?

(A) (0,4)
(B) (4,5)
(C) (5,7)
(D) (4,7)

Use the graph shown for Questions B7–B13. It shows the velocity of an object moving along a straight line during the time interval 0 ≤ t ≤ 5.

B7. The object attains its maximum speed when t =

(A) 0
(B) 1
(C) 2
(D) 3

B8. The speed of the object is increasing during the time interval

(A) (1,2)
(B) (0,2)
(C) (2,3)
(D) (3,5)

B9. The acceleration of the object is positive during the time interval

(A) (1,2)
(B) (0,2)
(C) (2,3)
(D) (3,5)

B10. How many times on 0 < t < 5 is the object’s acceleration undefined?

(A) none
(B) 1
(C) 2
(D) 3

B11. During 2 < t < 3 the object’s acceleration (in ft/sec²) is
(A) –10
(B) –5
(C) 5
(D) 10

B12. The object is farthest to the right when t =

(A) 0
(B) 1
(C) 2
(D) 5

B13. The object’s average acceleration (in ft/sec²) for the interval 0 ≤ t ≤ 3 is

(A) –15
(B) –5
(C) –3
(D) –1

B14. The line y = 3x + k is tangent to the curve y = x³ when k is equal to

(A) 1 or –1
(B) 2 or –2
(C) 3 or –3
(D) 4 or –4

B15. The two tangents that can be drawn from the point (3,5) to the parabola y = x² have slopes

(A) 1 and 5
(B) 0 and 4
(C) 2 and 10
(D) 2 and 4

B16. The table shows the velocity at various times of an object moving along a line. An estimate of its acceleration (in ft/sec²) at t = 1 is

(A) 0.8
(B) 1.0
(C) 1.2
(D) 1.6

For Questions B17 and B18, f′(x) = x sin x – cos x for 0 < x < 4.

B17. f has a local maximum when x is approximately

(A) 1.192
(B) 2.289
(C) 3.426
(D) 3.809

B18. The graph of f has a point of inflection when x is approximately

(A) 1.192
(B) 2.289
(C) 3.426
(D) 3.809

*In Questions B19–B22, the motion of a particle in a plane is given by the pair of equations x = 2t and y = 4t – t².

*B19. The particle moves along

(A) an ellipse
(B) a hyperbola
(C) a line
(D) a parabola

*B20. The speed of the particle at any time t is

*B21. The minimum speed of the particle is

(A) 0
(B) 1
(C) 2
(D) 4

*B22. The acceleration of the particle

(A) depends on t
(B) is always directed upward
(C) is constant both in magnitude and in direction
(C) never exceeds 1 in magnitude

*B23. If a particle moves along a curve with constant speed, then
(A) the magnitude of its acceleration must equal zero
(B) the direction of acceleration must be constant
(C) the curve along which the particle moves must be a straight line
(D) its velocity and acceleration vectors must be perpendicular

*B24. A particle is moving on the curve of y = 2x – ln x so that  at all times t. At the point (1,2),  is

(A) –4
(B) –2
(C) 2
(D) 4

* In Questions B25 and B26, a particle is in motion along the polar curve r = 6 cos 2θ such that  when .

*B25. At , find the rate of change (in units per second) of the particle’s distance from the origin.

*B26. At , what is the horizontal component of the particle’s velocity?

Use the graph of f′ on [0,5], shown below, for Questions B27 and B28.

B27. f has a local minimum at x =

(A) 0
(B) 1
(C) 2
(D) 3

B28. The graph of f has a point of inflection at x =

(A) 1 only
(B) 2 only
(C) 3 only
(D) 2 and 3 only

B29. It follows from the graph of f′, shown below, that

(A) f is not continuous at x = a
(B) f is continuous but not differentiable at x = a
(C) f has a relative maximum at x = a
(D) The graph of f has a point of inflection at x = a

B30. A vertical circular cylinder has radius r ft and height h ft. If the height and radius both increase at the constant rate of 2 ft/sec, then the rate, in square feet per second, at which the lateral surface area increases is

(A) 4πr
(B) 2π(r + h)
(C) 4π(r + h)
(D) 4πh

B31. A tangent drawn to the parabola y = 4 – x² at the point (1,3) forms a right triangle with the coordinate axes. The area of the triangle is

B32. Two cars are traveling along perpendicular roads, car A at 40 mph and car B at 60 mph. At noon, when car A reaches the intersection, car B is 90 miles away, and moving toward the intersection. At 1 P.M. the rate, in miles per hour, at which the distance between the cars is changing is

(A) –68
(B) –4
(C) 4
(D) 68

B33. Two cars are traveling along perpendicular roads, car A at 40 mph and car B at 60 mph. At noon, when car A reaches the intersection, car B is 90 miles away and moving toward the intersection. If t represents the time, in hours, traveled after noon, then the cars are closest together when t is

(A) 1.038
(B) 1.077
(C) 1.5
(D) 1.8

The graph for Questions B34 and B35 shows the velocity of an object moving along a straight line during the time interval 0 ≤ t ≤ 12.

B34. For what t does this object attain its maximum acceleration?

(A) 0 < t < 4
(B) 4 < t < 8
(C) t = 5
(D) t = 8

B35. The object reverses direction at t =

(A) 4 only
(B) 5 only
(C) 8 only
(D) 5 and 8

B36. Given f′ as graphed, which could be the graph of f?

(A)

 

(B)

 

(C)

 

(D)

 

B37. The graph of f′ is shown above. If we know that f(2) = 10, then the local linearization of f at x = 2 is f(x) ≈

Use the following graph for Questions B38–B40.

B38. At which labeled point do both  and  equal zero?

(A) P
(B) Q
(C) R
(D) S 

B39. At which labeled point is  positive and  equal to zero?

(A) P
(B) Q
(C) R
(D) S 

B40. At which labeled point is  equal to zero and  positive?

(A) P
(B) Q
(C) R
(D) S 

B41. If f(6) = 30 and , estimate f(6.02) using the line tangent to f at x = 6.

(A) 29.92
(B) 30.02
(C) 30.08
(D) 30.16

B42. A local linear approximation for  near x = –3 is