AB Practice Test 1

Section I

Part A

TIME: 60 MINUTES

The use of calculators is not permitted for this part of the examination.
There are 30 questions in Part A, for which 60 minutes are allowed. Because there is no deduction for wrong answers, you should answer every question, even if you need to guess.

DIRECTIONS: Choose the best answer for each question.

 

1. 

(A) –5
(B) 0
(C) 1
(D) ∞

2. 

3. If  , then y″(0) equals

(A) 2
(B) 1
(C) 0
(D) –2

Questions 4 and 5. Use the following table, which shows the values of the differentiable functions f and g.

4. The average rate of change of function f on [1,4] is

(A) 7/6
(B) 4/3
(C) 15/8
(D) 15/4

5. If h(x) = g (f(x)) then h′(3) =

(A) 1/2
(B) 1
(C) 4
(D) 6

6. The derivative of a function f is given for all x by

The set of x-values for which f is a relative minimum is

(A) {0,–1,4}
(B) {–1}
(C) {0,4}
(D) {0,–1}

7. If , then  equals

8. The maximum value of the function f(x) = xe^–x is

(A) 
(B) 1
(C) –1
(D) –e

9. Which equation has the slope field shown below?

Questions 10–12. The graph below shows the velocity of an object moving along a line, for 0 ≤ t ≤ 9.

10. At what time does the object attain its maximum acceleration?

(A) 2 < t < 5
(B) t = 6
(C) t = 8
(D) 8 < t < 9

11. The object is farthest from the starting point at t =

(A) 5
(B) 6
(C) 8
(D) 9

12. At t = 8, the object was at position x = 10. At t = 5, the object’s position was x =

(A) 5
(B) 7
(C) 13
(D) 15

13. 

14. You are given two twice-differentiable functions, f(x) and g(x). The table above gives values for f(x) and g(x) and their first and second derivatives at x = 1. Find .

(A) –3
(B) 1
(C) 2
(D) nonexistent

15. A differentiable function has the values shown in this table:

Estimate f′(2.1).

(A) 0.34
(B) 1.56
(C) 1.70
(D) 1.91

16. If  is approximated using various sums with the same number of subdivisions, and if L, R, and T denote, respectively, left Riemann Sum, right Riemann Sum, and trapezoidal sum, then it follows that

(A) R ≤ A ≤ T ≤ L
(B) R ≤ T ≤ A ≤ L
(C) L ≤ T ≤ A ≤ R
(D) L ≤ A ≤ T ≤ R

17. The number of vertical tangents to the graph of y² = x – x³ is

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

18. 

19. The equation of the curve shown below is . What does the area of the shaded region equal?

(A) 8 – π
(B) 8 – 2π
(C) 8 – 4π
(D) 8 – 4 ln 2

20. Over the interval 0 ≤ x ≤ 10, the average value of the function f shown below

(A) is 6.10
(B) is 6.25
(C) does not exist, because f is not continuous
(D) does not exist, because f is not integrable

21. If f′(x) = 2 f(x) and f(2) = 1, then f(x) =

22. If , then f′(t) equals

23. The curve x³ + x tan y = 27 passes through (3,0). Use the tangent line there to estimate the value of y at x = 3.1. The value is

(A) –2.7
(B) –0.9
(C) 0
(D) 0.1

24. 

25. The graph of a function y = f(x) is shown above. Which is true?

26. A function f(x) equals  for all x except x = 1. For the function to be continuous at x = 1, the value of f(1) must be

(A) 0
(B) 1
(C) 2
(D) ∞

27. 

28. Suppose . It follows that

(A) f increases for all x
(B) f has a critical point at x = 0
(C) f has a local min at x = –4
(D) f has a local max at x = –4

29. The graph of f(x) consists of two line segments as shown above. If g(x) = f^–1(x), the inverse function of f(x), find g′(4).

30. Choose the integral that is the limit of the Riemann Sum:

 

Part B

TIME: 45 MINUTES

Some questions in this part of the examination require the use of a graphing calculator. There are 15 questions in Part B, for which 45 minutes are allowed. Because there is no deduction for wrong answers, you should answer every question, even if you need to guess.

DIRECTIONS: Choose the best answer for each question. If the exact numerical value of the correct answer is not listed as a choice, select the choice that is closest to the exact numerical answer.

 

31. An object moving along a line has velocity v(t) = t cos t – ln (t + 2), where 0 ≤ t ≤ 10. The object achieves its maximum speed when t is approximately

(A) 5.107
(B) 6.419
(C) 7.550
(D) 9.538

32. The graph of f′, which consists of a quarter-circle and two line segments, is shown above. At x = 2, which of the following statements is true?

(A) f is not continuous.
(B) f is continuous but not differentiable.
(C) f has a local maximum.
(D) The graph of f has a point of inflection.

33. Let , where f is the function whose graph appears below.

The local linearization of H(x) near x = 3 is H(x) ≈

(A) –2x + 8
(B) 2x – 4
(C) –2x + 4
(D) 2x – 8

34. The table shows the speed of an object, in feet per second, at various times during a 12-second interval.

Estimate the distance the object travels, using the midpoint method with 3 subintervals.

(A) 100 feet
(B) 110 feet
(C) 112 feet
(D) 114 feet

35. In a marathon, when the winner crosses the finish line many runners are still on the course, some quite far behind. If the density of runners x miles from the finish line is given by R(x) = 20[1 – cos(1 + 0.03x²)] runners per mile, how many are within 8 miles of the finish line?

(A) 30
(B) 40
(C) 157
(D) 166

36. Which best describes the behavior of the function  at x = 1?

(A) It has a jump discontinuity.
(B) It has an infinite discontinuity.
(C) It has a removable discontinuity.
(D) It is continuous.

37. Let G(x) = [f(x)]². In an interval around x = a, the graph of f is increasing and concave downward, while G is decreasing. Which describes the graph of G there?

(A) concave downward
(B) concave upward
(C) point of inflection
(D) quadratic

38. The value of c for which  has a local minimum at x = 3 is

(A) –9
(B) 0
(C) 6
(D) 9

39. The function g is a differentiable function. It is known that g′(x) ≤ 4 for 3 ≤ x ≤ 10 and that g(7) = 8. Which of the following could be true?

I. g(5) = 0
II. g(8) = –4
III. g(9) = 17

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

40. At which point on the graph of y = f(x) shown above is f′(x) < 0 and f″(x) > 0?

(A) A
(B) B
(C) C
(D) D 

41. Let f(x) = x⁵ + 1, and let g be the inverse function of f. What is the value of g′(0)?

42. The hypotenuse AB of a right triangle ABC is 5 feet, and one leg, AC, is decreasing at the rate of 2 feet per second. The rate, in square feet per second, at which the area is changing when AC = 3 is

43. At how many points on the interval [0,π] does f(x) = 2 sin x + sin 4x satisfy the Mean Value Theorem?

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

44. If the radius r of a sphere is increasing at a constant rate, then the rate of increase of the volume of the sphere is

(A) constant
(B) increasing
(C) decreasing
(D) decreasing for r < 1 and increasing for r > 1

45. The rate at which a purification process can remove contaminants from a tank of water is proportional to the amount of contaminant remaining. If 20% of the contaminant can be removed during the first minute of the process and 98% must be removed to make the water safe, approximately how long will the decontamination process take?

(A) 2 minutes
(B) 5 minutes
(C) 7 minutes
(D) 18 minutes

 

Section II

Part A

TIME: 30 MINUTES
2 PROBLEMS

A graphing calculator is required for some of these problems. See instructions on page 8.

 

1. A function f is defined on the interval [0,4] with f(2) = 3, f′(x) = e^sinx – 2 cos(3x), and f″(x) = (cosx) · e^sinx + 6 sin(3x).

(a) Find all values of x in the interval where f has a critical point. Classify each critical point as a local maximum, local minimum, or neither. Justify your answers.
(b) On what subinterval(s) of (0,4), if any, is the graph of f concave down? Justify your answer.
(c) Write an equation for the line tangent to the graph of f at x = 1.5.

 

2. The rate of sales of a new software product is given by the differentiable and increasing function S(t), where S is measured in units per month and t is measured in months from the initial release date. The software company recorded these sales data:

(a) Use the data in the table to estimate S′(7). Show the work that leads to your answer. Indicate units of measure.
(b) Use a left Riemann Sum with the four subintervals indicated in the table to estimate the total number of units of software sold during the first 9 months of sales. Is this an underestimate or an overestimate of the total number of software units sold? Give a reason for your answer.
(c) For 1 ≤ t ≤ 9, must there be a time t when the rate of sales is increasing at  per month? Justify your answer.
(d) For 9 ≤ t ≤ 12, the rate of sales is modeled by the function R(t) = 120 · (2)^x/3. Given that the actual sales in the first 9 months is 3636 software units, use this model to find the number of units of the software sold during the first 12 months of sales. Round your answer to the nearest whole unit.

 

Part B

TIME: 60 MINUTES
4 PROBLEMS

No calculator is allowed for any of these problems.

If you finish Part B before time has expired, you may return to work on Part A, but you may not use a calculator.

3. The graph of function y = f(x) passes through the point (1,1) and satisfies the differential equation .

(a) Sketch the slope field for the differential equation at the 12 indicated points on the axes provided.

(b) Find an equation of the line tangent to f(x) at the point (1,1) and use the linear equation to estimate f(1.2).
(c) Solve the differential equation, and find the particular solution for y = f(x) that passes through the point (1,1).

 

4. Let R represent the first-quadrant region bounded by the y-axis and the curves y = 2^x and , as shown in the graph.

(a) Find the area of region R.
(b) Set up, but do not evaluate, an integral expression for the volume of the solid formed when R is rotated around the x-axis.
(c) Set up, but do not evaluate, an integral expression for the volume of the solid whose base is R and all cross sections in planes perpendicular to the x-axis are squares.

 

5. The graph of the function f is given above. f is twice differentiable and is defined on the interval – 8 ≤ x ≤ 6. The function g is also twice differentiable and is defined as .

(a) Write an equation for the line tangent to the graph of g(x) at x = – 8. Show the work that leads to your answer.
(b) Using your tangent line from part (a), approximate g(– 7). Is your approximation greater than or less than g(– 7)? Give a reason for your answer.
(c) Evaluate: . Show the work that leads to your answer.
(d) Find the absolute minimum value of g on the interval – 8 ≤ x ≤ 6. Justify your answer.

 

6. A particle moves along the x-axis from t = 0 to t = 12. The velocity function of the particle is . The initial position of the particle is x = 4 when t = 0.

(a) When is the particle moving to the right from t = 0 to t = 12?
(b) What is the speed of the particle at t = 9
(c) Find the acceleration function of the particle. Is the speed of the particle increasing or decreasing at time t = 6? Explain your reasoning.
(d) Find the position of the particle at time t = 6.