Practice Exercises

In each of Questions A1–A20, a function is given. Choose the alternative that is the derivative, , of the function.

A1. y = x⁵ tan x
(A) 5x⁴ tan x
(B) 5x⁴ sec²x
(C) 5x⁴ + sec²x
(D) 5x⁴ tan x + x⁵ sec²x

A2. 

A3. 

A4. 

A5. y = 3x^2/3 – 4x^1/2 – 2
(A) 2x^1/3 – 2x^–1/2
(B) 3x^–1/3 – 2x^–1/2
(C) 
(D) 2x^–1/3 – 2x^–1/2

A6. 

A7. 

A8. 

A9. 

A10. 

A11. y = ln (sec x + tan x)
(A) sec x
(B) 
(C) 
(D) 

A12. 

(A) 0
(B) 1
(C) 
(D) 

A13. 

A14. 

A15. 

(A) –csc 2x cot 2x
(B) 
(C) –4 csc 2x cot 2x
(D) –csc² 2x

A16. y = e^–x cos 2x

(A) –e^–x(cos 2x + 2 sin 2x)
(B) e^–x(sin 2x – cos 2x)
(C) 2e^–x sin 2x
(D) –e^–x(cos 2x + sin 2x)

A17. y = sec²(x)

(A) 2 sec x
(B) 2 sec x tan x
(C) 2 sec² x tan x
(D) tan x

A18. y = x (ln x)³

(A) 
(B) 3(ln x)²
(C) –3x(ln x)² + (ln x)³
(D) (ln x)³ + 3(ln x)²

A19. 

A20. 

In each of Questions A21–A24, y is a differentiable function of x. Choose the alternative that is the derivative .

A21. x³ – y³ = 1

(A) x
(B) 3x²
(C) 
(D) 

A22. x + cos(x + y) = 0

(A) csc(x + y) – 1
(B) csc(x + y)
(C) 
(D) 

A23. sin x – cos y – 2 = 0

(A) –cot x
(B) 
(C) –csc y cos x
(D) 

A24. 3x² – 2xy + 5y² = 1

(A) 
(B) 
(C) 3x + 5y
(D) 

*A25. If x = t² + 1 and y = 2t³, then 

(A) 3t
(B) 6t²
(C) 
(D) 

A26. If f(x) = x⁴ – 4x³ + 4x² – 1, then the set of values of x for which the derivative equals zero is

(A) {1,2}
(B) {0,–1,–2}
(C) {–1,2}
(D) {0,1,2}

A27. If , then f″(4) is equal to

(A) –16
(B) –4
(C) –2
(D) 

A28. If f(x) = ln (x³), then f″(3) is

(A) 
(B) –1
(C) –3
(D) 1

A29. If a point moves on the curve x² + y² = 25, then, at  is

(A) 0
(B) 
(C) –5
(D) 

*A30. If x = t² – 1 and y = t⁴ – 2t³, then, when  is

(A) 1
(B) –1
(C) 3
(D) 

A31. If f(x) = 5^x and 5^1.002 ≈ 5.016, which is closest to f′(1)?

(A) 0.016
(B) 5.0
(C) 8.0
(D) 32.0

A32. If y = e^x(x – 1), then y″(0) equals

(A) –2
(B) –1
(C) 1
(D) e

*A33. If x = e^θ cos θ and y = e^θ sin θ, then, when  is

(A) 1
(B) 0
(C) e^π/2
(D) –1

A34. Given x = cos t and y = cos 2t, if sin(t) ≠ 0, then  is

(A) 4 cos t
(B) 4
(C) –4
(D) –4 cot t

A35.  is

(A) 0
(B) 1
(C) 6
(D) nonexistent

A36.  is

(A) 0
(B) 
(C) 1
(D) 192

A37.  is

(A) 0
(B) 
(C) 1
(D) e

A38.  is

(A) –1
(B) 0
(C) 1
(D) ∞ 

A39. If , which of the following statements is(are) true?

I.  exists
II. f is continuous at x = 1
III. f is differentiable at x = 1

(A) none
(B) I only
(C) I and II only
(D) I, II, and III

A40. If , which of the following statements is (are) true?

I.  exists
II. g is continuous at x = 3
III. g is differentiable at x = 3

(A) I only
(B) II only
(C) I and II only
(D) I, II, and III

A41. The function f(x) = x^2/3 on [–8,8] does not satisfy the conditions of the Mean Value Theorem because

(A) f(0) is not defined
(B) f(x) is not continuous on [–8,8]
(C) f(x) is not defined for x < 0
(D) f′(0) does not exist

A42. If f(x) = 2x³ – 6x, at what point on the interval , if any, is the tangent to the curve parallel to the secant line on that interval?

(A) 1
(B) 
(C) 0
(D) nowhere

A43. If h is the inverse function of f and if , then h′(3) =

A44.  equals

(A) 0
(B) 1
(C) 
(D) ∞ 

A45. If sin(xy) = x, then 

A46.  is

(A) 1
(B) 2
(C) 
(D) 0

A47. is

A48. is

(A) 0
(B) 1
(C) 2
(D) ∞

A49.  is

(A) 0
(B) 1
(C) π
(D) ∞

CHALLENGE A50. 

(A) is 1
(B) is 0
(C) is ∞
(D) oscillates between –1 and 1

*A51. The graph in the xy-plane represented by x = 3 + 2 sin t and y = 2 cos t – 1, for –π ≤ t ≤ π, is

(A) a semicircle
(B) a circle
(C) an ellipse
(D) half of an ellipse

A52. 

(A) = 0
(B) 
(C) = 1
(D) = 2

In each of Questions A53–A56, a pair of equations that represent a curve parametrically is given. Choose the alternative that is the derivative .

*A53. x = t – sin t and y = 1 – cos t

*A54. x = cos³θ and y = sin³θ

(A) –cot θ
(B) cot θ
(C) –tan θ
(D) –tan²θ

*A55. x = 1 – e^–t and y = t + e^–t

(A) 
(B) e^t + 1
(C) e^t – e^–2t
(D) e^t– 1

*A56.  and y = 1 – ln(1 – t) (t < 1)

A57. The “left half” of the parabola defined by y = x² – 8x + 10 for x ≤ 4 is a one-to-one function; therefore, its inverse is also a function. Call that inverse g. Find g′(3).

A58. If f(u) = sin u and u = g (x) = x² – 9, then (f ∘ g)′(3) equals

(A) 0
(B) 1
(C) 3
(D) 6

A59. If , then the set of x’s for which f′(x) exists is

(A) all reals
(B) all reals except x = 1 and. x = –1
(C) all reals except x = –1
(D) all reals except. x = 1

A60. If  and , then the derivative of f(g(x)) is

A61. Suppose y = f(x) = 2x³ – 3x. If h(x) is the inverse function of f, then h′(–1) =

(A) –1
(B) 
(C) 1
(D) 3

A62. If f(x) = x³ – 3x² + 8x + 5 and g(x) = f^–1(x), then g′(5) =

(A) 8
(B) 
(C) 
(D) 53

 

In Questions B1–B8, differentiable functions f and g have the values shown in the table.

B1. If A(x) = f(x) + 2g(x), then A′(3) =

(A) –2
(B) 2
(C) 7
(D) 8

B2. If B(x) = f(x) · g(x), then B′(2) =

(A) –20
(B) –7
(C) –6
(D) 13

B3. If , then D′(1) =

B4. If , then H′(3) =

B5. If , then K′(0) =

B6. If M(x) = f(g(x)), then M′(1) =

(A) –12
(B) –6
(C) 6
(D) 12

B7. If P(x) = f(x³), then P′(1) =

(A) 2
(B) 6
(C) 8
(D) 12

B8. If S(x) = f^–1(x), then S′(3) =

(A) –2
(B) 
(C) 
(D) 2

B9. The graph of g′ is shown here. Which of the following statements is (are) true of g?

I. g is continuous at x = a
II. g is differentiable at x = a
III. g is increasing in an interval containing x = a

(A) I only
(B) III only
(C) I and III only
(D) I, II, and III

B10. A function f has the derivative shown. Which of the following statements must be false?

(A) f is continuous at x = a
(B) f has a vertical asymptote at x = a
(C) f has a jump discontinuity at x = a
(D) f has a removable discontinuity at x = a

B11. The function f whose graph is shown has f′ = 0 at x =

(A) 2 only
(B) 2 and 5
(C) 4 and 7
(D) 2, 4, and 7

B12. A differentiable function f has the values shown. Estimate f′(1.5).

(A) 8
(B) 18
(C) 40
(D) 80

B13. Water is poured into a conical reservoir at a constant rate. If h(t) is the rate of change of the depth of the water, then h is

(A) linear and increasing
(B) linear and decreasing
(C) nonlinear and increasing
(D) nonlinear and decreasing

Use the figure to answer Questions B14–B16. The graph of f consists of two line segments and a semicircle of radius 2.

B14. f′(x) = 0 for x =

(A) 1 only
(B) 2 only
(C) 4 only
(D) 1 and 4

B15. For 0 < x < 6, f′(x) does not exist for x =

(A) 1 only
(B) 2 only
(C) 1 and 2
(D) 2 and 6

B16. f′(5) =

B17. At how many points on the interval [–5,5] is a tangent to y = x + cos x parallel to the secant line on [–5,5]?

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

B18. From the values of f shown, estimate f′(2).

(A) –0.10
(B) –0.20
(C) –5
(D) –10

B19. Using the values shown in the table for Question B18, estimate (f^–1)′(4)

(A) –0.2
(B) –0.1
(C) –5
(D) –10

B20. The table below shows some points on a function f that is both continuous and differentiable on the closed interval [2,10].

Which must be true?

(A) f′(6) = 0
(B) f′(8) > 0
(C) The maximum value of f on the interval [2,10] is 30.
(D) For some value of x on the interval [2,10] f′(x) = 0.

B21. If f is differentiable and difference quotients overestimate the slope of f at x = a for all h > 0, which must be true?

(A) f′(x) ≥ 0 on [a,h]
(B) f′(x) ≤ 0 on [a,h]
(C) f″(x) ≥ 0 on [a,h]
(D) f″(x) ≤ 0 on [a,h]

B22. If f(a) = f(b) = 0 and f(x) is continuous on [a,b], then

(A) f(x) must be identically zero
(B) f(x) may be different from zero for all x on [a,b]
(C) there exists at least one number c, a < c < b, such that f′(c) = 0
(D) f(x) must exist for every x on (a,b)

B23. Suppose f′(1) = 2, f′(1) = 3, and f′(2) = 4. Then (f^–1)′(2)

B24. Suppose . It follows necessarily that

(A) g is not defined at x = 0
(B) g is not continuous at x = 0
(C) the limit of g(x) as x approaches 0 equals 1
(D) g′(0) = 1

Use this graph of y = f(x) for Questions B25 and B26.

B25. f′(3) is most closely approximated by

(A) 0.3
(B) 0.8
(C) 1.5
(D) 1.8

B26. The rate of change of f(x) is least at x ≈

(A) –3
(B) –1.3
(C) 0.7
(D) 2.7

Use the following definition of the symmetric difference quotient as an approximation for f′(x₀) for Questions B27–B29. For small values of h,

B27. For f(x) = 5^x, what is the estimate of f′(2) obtained by using the symmetric difference quotient with h = 0.03?

(A) 40.25
(B) 40.252
(C) 41.818
(D) 80.503

B28. To how many places is the symmetric difference quotient accurate when it is used to approximate f′(0) for f(x) = 4^x and h = 0.08?

(A) 1
(B) 2
(C) 3
(D) more than 3

B29. To how many places is f′(x₀) accurate when it is used to approximate f′(0) for f(x) = 4^x and h = 0.001?

(A) 1
(B) 2
(C) 3
(D) more than 3

B30. The value of f′(0) obtained using the symmetric difference quotient with f(x) = |x| and h = 0.001 is

(A) –1
(B) 0
(C) 1
(D) indeterminate

B31. If   and h(x) = sin x, then  equals

(A) g(sin x)
(B) cos x · g(x)
(C) cos x · g (sin x)
(D) sin x · g(sin x)

B32. Let f(x) = 3^x – x³. The tangent to the curve is parallel to the secant through (0,1) and (3,0) for x

(A) 1.244 only
(B) 2.727 only
(C) 1.244 and 2.727 only
(D) no such value of x exists

Questions B33–B37 are based on the following graph of f(x), sketched on –6 ≤ x ≤ 7. Assume the horizontal and vertical grid lines are equally spaced at unit intervals.

B33. On the interval 1 < x < 2, f(x) equals

(A) –x – 2
(B) –x – 3
(C) –x – 4
(D) –x + 2

B34. Over which of the following intervals does f′(x) equal zero?

I. (–6,–3)
II. (–3,–1)
III. (2,5)

(A) II only
(B) I and II only
(C) I and III only
(D) II and III only

B35. How many points of discontinuity does f′(x) have on the interval –6 < x < 7?

(A) 2
(B) 3
(C) 4
(D) 5

B36. For –6 < x < –3, f′(x) equals
(A) 
(B) –1
(C) 1
(D) 

B37. Which of the following statements about the graph of f′(x) is false?
(A) f′(x) consists of six horizontal segments
(B) f′(x) has four jump discontinuities
(C) f′(x) is discontinuous at each x in the set {–3,–1,1,2,5}
(D) On the interval –1 < x < 1, f′(x) = –3

B38. The table gives the values of a function f that is differentiable on the interval [0,1]:

According to this table, the best approximation of f′(0.10) is

(A) 0.12
(B) 1.08
(C) 1.17
(D) 1.77

B39. At how many points on the interval [a,b] does the function graphed satisfy the Mean Value Theorem?

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