AB Practice Test 3

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 cylindrical tank, shown in the figure above, is partially full of water at time t = 0, when more water begins flowing in at a constant rate. The tank becomes half full when t = 4 and is completely full when t = 12. Let h represent the height of the water at time t. During which interval is  increasing?

(A) 0 < t < 4
(B) 0 < t < 8
(C) 0 < t < 12
(D) 4 < t < 12

2. 

(A) 1
(B) –1
(C) 0
(D) nonexistent

3. If f(x) = x ln x, then f′″(e) equals

4. An equation of the tangent to the curve 2x² – y⁴ = 1 at the point (–1,1) is

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

5. On which interval(s) does the function f(x) = x⁴ – 4x³ + 4x² + 6 increase?

(A) x < 0 and 1 < x < 2
(B) x > 2 only
(C) 0 < x < 1 and x > 2
(D) 0 < x < 1 only

6. equals

(A) ln |4 + 2 sin x| + C
(B) –2 ln |4 + 2 sin x| + C
(C) 
(D) 2 ln |4 + 2 sin x| + C

7. Suppose the function f is defined as

Which of the following is true?

(A) The function f is both continuous and differentiable at x = 2.
(B) The function f is continuous but not differentiable at x = 2.
(C). The function f is differentiable but not continuous at x = 2.
(D) The function f is neither continuous nor differentiable at x = 2.

8. If a particle moves on a line according to the law s = t⁵ + 2t³, then the number of times it reverses direction is

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

9. A particular solution of the differential equation whose slope field is shown above contains point P. This solution may also contain which other point?

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

10. Let . Which of the following statements is (are) true?

I. The domain of F is x ≠ ±1.
II. F(2) > 0.
III. The graph of F is concave upward.

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

11. As the tides change, the water level in a bay varies sinusoidally. At high tide today at 8 A.M., the water level was 15 feet; at low tide, 6 hours later at 2 P.M., it was 3 feet. How fast, in feet per hour, was the water level dropping at noon today?

12. A smooth curve with equation y = f(x) is such that its slope at each x equals x². If the curve goes through the point (–1,2), then its equation is

(A) y = x³ + 3
(B) 
(C) 
(D) y = 3x³ + 5

13. is equal to

14. 

15. If G(2) = 5 and , then an estimate of G(2.2) using a tangent-line approximation is

(A) 4.4
(B) 5.8
(C) 6
(D) 11.6

16. The area bounded by the parabola y = x² and the lines y = 1 and y = 9 equals

17. Suppose  if x ≠ 0 and f(0) = 1. Which of the following statements is (are) true of f?

I. f is defined at x = 0.
II.  exists.
III. f is continuous at x = 0.

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

18. Which function could have the graph shown below?

 

19. Suppose the graph of f is both increasing and concave up on a ≤ x ≤ b. If  is approximated using various sums with the same number of subintervals, and if L, R, M, and T denote, respectively, left Riemann Sum, right Riemann Sum, midpoint Riemann Sum, and trapezoidal sum, then it follows that

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

20.  is

(A) + ∞
(B) 0
(C) 
(D) nonexistent

21. The only function that does not satisfy the Mean Value Theorem on the interval specified is

22. Suppose f’(x) = x(x — 2)²(x + 3). Which of the following is (are) true?

I. f has a local maximum at x = –3.
II. f has a local minimum at x = 0.
III. f has neither a local maximum nor a local minimum at x = 2.

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

23. If , then  is

24. The graph of function f shown above consists of three quarter-circles. Which of the following is (are) equivalent to ?

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

25. The base of a solid is the first-quadrant region bounded by , and each cross section perpendicular to the x-axis is a semicircle with a diameter in the xy-plane. The volume of the solid is

26. The average value of f(x) = 3 + |x| on the interval [–2,4] is

27. is

28. The area of the region in the xy-plane bounded by the curves y = e^x, y = e–x, and x = 1 is equal to

29. If , then f′(x) =

30. Choose the Riemann Sum whose limit is the integral 

 

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. A particle moves on a line according to the law s = f(t) so that its velocity is v = ks, where k is a nonzero constant. The acceleration of the particle is

(A) k²v
(B) k²s
(C) k
(D) 0

32. A cup of coffee placed on a table cools at a rate of  per minute, where H represents the temperature of the coffee and t is time in minutes. If the coffee was at 120°F initially, what will its temperature be, to the nearest degree, 10 minutes later?

(A) 73°F
(B) 95°F
(C) 100°F
(D) 105°F

33. An investment of $4000 grows at the rate of 320e^0.08t dollars per year after t years. Its value after 10 years is approximately

(A) $12902
(B) $8902
(C) $7122
(D) $4902

34. The density of the population of a city is 10,000 people per square mile at the beginning of a certain year. We can model the time, in years (t), it will take until the population reaches a certain density, in people per square mile (D), by using the function t = g(D). What are the units of g′(D)?

(A) years
(B) years per people per square mile
(C) people per square mile per year
(D) people per square mile

Questions 35 and 36. The graph below shows the velocity of an object during the interval 0 ≤ t ≤ 9.

35. The object attains its greatest speed at t =

(A) 3
(B) 5
(C) 6
(D) 8

36. The object was at the origin at t = 3. It returned to the origin

(A) during 6 < t < 7
(B) at t = 7
(C) during 7 < t < 8
(D) at t = 8

37. When the region bounded by the y-axis, y = e^x, and y = 2 is rotated around the y-axis, it forms a solid with volume

(A) 0.188
(B) 0.386
(C) 0.592
(D) 1.214

38. If  is replaced by u, then  is equivalent to

39. The line tangent to the graph of function f at the point (8,1) intersects the y-axis at y = 3. Find f′(8).

40. How many points of inflection does the function f have on the interval 0 ≤ x ≤ 6 if ?

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

41. The graph above shows the rate at which tickets were sold at a movie theater during the last hour before showtime. Using the right-rectangle method, estimate the size of the audience.

(A) 270
(B) 300
(C) 330
(D) 360

42. Find the x-coordinate where f(x) = 4^{sin x} and g(x) = ln (x²) intersect and their derivatives have the same sign.

(A) –5.240
(B) –3.961
(C) –1.151
(D) 2.642

43. Which statement is true?

(A) If f′(c) = 0, then f has a local maximum or minimum at (c, f(c)).
(B) If f″(c) = 0, then the graph of f has an inflection point at (c, f(c)).
(C) If f is differentiable at x = c, then f is continuous at x = c.
(D) If f is continuous on (a,b), then f attains a maximum value on (a,b).

44. The graph of f′ is shown above. Which statement(s) about f must be true for a < x < b?

I. f is increasing.
II. f is continuous.
III. f is differentiable.

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

45. Given  and g′(x) < 0 for all x, select the table that could represent g on the interval [–1,4].

(A)

 

(B)

(C) 

(D)

 

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 curve is defined by x²y – 3y² = 48.

(a) Verify that .
(b) Write an equation of the line tangent to this curve at (5,3).
(c) Using your equation from part (b), estimate the y-coordinate of the point on the curve where x = 4.93.
(d) Show that this curve has no horizontal tangent lines.

 

2. The table shows the depth of water, W in a river, as measured at 4-hour intervals during a daylong flood. Assume that W is a differentiable function of time t.

(a) Find the approximate value of W′(16). Indicate units of measure.
(b) Estimate the average depth of the water, in feet, over the time interval 0 ≤ t ≤ 24 hours by using a trapezoidal approximation with subintervals of length Δt = 4 hours.
(c) Scientists studying the flooding believe they can model the depth of the water with the function , where F(t) represents the depth of the water, in feet, after t hours. Find F′(16) and explain the meaning of your answer, with appropriate units, in terms of the river depth.
(d) Use the function F to find the average depth of the water, in feet, over the time interval 0 ≤ t ≤ 24 hours.

 

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 region R is bounded by the curves f(x) = cos(πx) – 1 and g(x) = x(2 – x), as shown in the figure.

(a) Find the area of R.
(b) A solid has base R, and each cross section perpendicular to the x-axis is an isosceles right triangle whose hypotenuse lies in R. Set up, but do not evaluate, an integral for the volume of this solid.
(c) Set up, but do not evaluate, an integral for the volume of the solid formed when R is rotated around the line y = 3.

 

4. Two autos, P and Q, start from the same point and race along a straight road for 10 seconds. The velocity of P is given by  feet per second. The velocity of Q is shown in the graph.

(a) At what time is P’s actual acceleration (in ft/sec²) equal to its average acceleration for the entire race?
(b) What is Q’s acceleration (in ft/sec²) then?
(c) At the end of the race, which auto was ahead? Explain.

5. Given the differential equation  

(a) Sketch the slope field for this differential equation at the points shown in the figure.

(b) Let f be the particular solution to the differential equation whose graph passes through (0,1). Express f as a function of x, and state its domain.

 

6. The graph shown is for .

(a) What is ?
(b) What is ?
(c) At what value of x does f(x) = 0?
(d) Over what interval is f′(x) negative?
(e) Let . Sketch the graph of G on the same axes.