Master AP Calculus AB & BC
Part III. FOUR PRACTICE TESTS
ANSWER SHEET PRACTICE TEST 2
Section I, Part A
Section I, Part B
Practice Test 2: AP Calculus AB
SECTION I, PART A
55 Minutes • 28 Questions
A CALCULATOR MAY NOT BE USED FOR THIS PART OF THE EXAMINATION.
Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.
1. What is the instantaneous rate of change for f(x) =
The rate at which cars cross a bridge in cars per minute is given by the preceding graph. A good approximation for the total number of cars that crossed the bridge by 12:00 noon is
The function f is continuous on the closed interval [0,2] and has values as defined by the table above. Which of the following statements must be true?
(A) f must be increasing on [0,2],
(B) f must be concave up on (0,2).
(D) The average rate of increase of f over [0,2] is 3.
(E) f has no points of inflection on [0,2].
6. What is the slope of the curve defined by 3x2 + 2xy + 6y2 — 3x — 8y = 0 at the point (1,1)?
(E) It is undefined.
8. The radius of a sphere is increasing at a rate of 2 inches per minute. At what rate (in cubic inches per minute) is the volume increasing when the surface area of the sphere is 9π square inches?
The area of the shaded region in the preceding diagram is equivalent to
10. What is the average rate of change of f(x) = x3 — 3x2 + x — 1 over [—1,4]?
11. If the graph of the second derivative of some function, f, is a line of slope 6, then f could be which type of function?
12. Let f be defined as
What is the average value of f over [-4,4]?
f is a twice differentiable function with a horizontal tangent line at x = 1, as shown in the diagram above. Which of these statements must be true?
14. Let f be a continuous function on [—4,12]. If f(—4) = —2 and f(12) = 6, then the mean value theorem guarantees that
(A) f(4) = 2
(B) f’(4) = 1/2
(C) f'(c) = 1/2 for at least one c between — 4 and 12
(D) f(c) = 0 for at least one c between — 4 and 12
(E) f(4) = 0
16. Let f(x) = ex. If the rate of change of f at x = c is e3 times its rate of change at x = 2, then c =
Let f, g, and their derivatives be defined by the table above. If h(x) = f(g(x)), then for what value, c, is h(c) = h’(c)?
(E) None of the above
18. Let f be a differentiable function over [0,10] such that f(0) = 0 and f(10) = 3. If there are exactly two solutions to f(x) = 4 over (0,10), then which of these statements must be true?
(A) f'(c) = 0 for some c on (0,10).
(B) f has a local maximum at x = 5.
(C) f"(c) = 0 for some c on (0,10).
(D) 0 is the absolute minimum of f.
(E) f is strictly monotonic.
19. The normal line to the curve at the point (2,2) has slope
20. What are all the values for k such that
(C) -2 and 2
(D) -2, 0, and 2
(E) 0 and 2
21. If the rate of change of y is directly proportional to y, then it’s possible that
22. The graph of y = 3x3 — 2x2 + 6x — 2 is decreasing for which interval(s)?
(E) None of the above
23. Determine the value for c on [2,5] that satisfies the mean value theorem for
24. Below is the slope field graph of some differential equation (Note: Each dot on the axes marks one unit.)
Which of the following equations is the easiest possible differential equation for the characteristics shown in the graph?
The area of the shaded region in the preceding diagram is
26. The function f is continuous on the closed interval [0,2]. It is given that f(0) = -1 and f(2) = 2. If f’(x) > 0 for all x on [0,2] and f"(x) < 0 for all x on (0,2), then f(1) could be
27. The water level in a cylindrical barrel is falling at a rate of one inch per minute. If the radius of the barrel is ten inches, what is the rate that water is leaving the barrel (in cubic inches per minute) when the volume is 500π cubic inches?
28. If f(x) = arctan(x2), then f'(√3) =
STOP. END OF SECTION I, FART A. IF YOU HAVE ANY TIME LEFT, GO OVER YOUR WORK IN THIS PART ONLY. DO NOT WORK IN ANY OTHER PART OF THE TEST.
SECTION I, PART B
50 Minutes • 17 Questions
A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS IN THIS PART OF THE EXAMINATION.
Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.
In this test: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.
29. A particle starts at the origin and moves along the x-axis with decreasing positive velocity. Which of these could be the graph of the distance, s(t), of the particle from the origin at time t?
30. Let f be the function given by f(x) = 3 ln 2x, and let g be the function given by g(x) = x3 + 2x. At what value of x do the graphs off and g have parallel tangent lines?
(A) - 0.782
31. Let f be some function such that the rate of increase of the derivative of f is 2 for all x. If f'(2) = 4 and f(1) = 2, find f(3).
(E) It is nonexistent.
Let f be a continuous function with values as represented in the table above. Approximate using a right-hand Riemann sum with three subintervals of equal length.
The graph of f', the derivative of f is shown above. Which of the following describes all relative extrema of f on (a,b)?
(A) One relative maximum and one relative minimum
(B) Two relative maximums and one relative minimum
(C) One relative maximum and no relative minimum
(D) No relative maximum and two relative minimums
(E) One relative maximum and two relative minimums
35. Let What value on [0,4] satisfies the mean value theorem for f?
36. The position for a particle moving on the x-axis is given by At what time, t, on [0,3] is the particle’s instantaneous velocity equal to its average velocity over [0,3]?
37. Let f be defined as
and g be defined as Which of the following statements about f and g is false?
38. Let f(x) = x2 + 3. Using the trapezoidal rule, with n = 5, approximate
39. Population y grows according to the equation dy/dt = ky, where k is a constant and t is measured in years. If the population triples every five years, then k —
40. The circumference of a circle is increasing at a rate of 2π/5 inches per minute. When the circumference is 10π inches, how fast is the area of the circle increasing in square inches per minute?
The base of a solid is the region in the first quadrant bounded by the x-axis and the parabola y = —x2 + 6x, as shown in the figure above. If cross sections perpendicular to the x-axis are equilateral triangles, what is the volume of the solid?
42. Let f be the function given by f(x) = x2 + 4x — 8. The tangent line to the graph at x = 2 is used to approximate values of f. For what value(s) of x is the tangent line approximation twice that of f?
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III
43. The first derivative of a function, f, is given by How many critical values does f have on the open interval (0,10)?
45. Let f be defined as for a constant, k. For what value of k will
(A) - 2
(E) None of the above
STOP. END OF SECTION I, PART B. IF YOU HAVE ANY TIME LEFT, GO OVER YOUR WORK IN THIS PART ONLY. DO NOT WORK IN ANY OTHER PART OF THE TEST.
SECTION II, PART A
45 Minutes • 3 Questions
A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION.
SHOW ALL YOUR WORK. It is important to show your setups for these problems because partial credit will be awarded. If you use decimal approximations, they should be accurate to three decimal places.
1. Examine the function, f, defined as for 0 ≤ x ≤ 10.
(a) Use a Riemann sum with five equal subintervals evaluated at the midpoint to approximate the area under f from x = 0 to x = 10.
(b) Again using five equal subintervals, use the trapezoidal rule to approximate the area under f from x = 0 to x = 10.
(c) Using your result from part B, approximate the average value of the function, f, from x = 0 to x = 10.
(d) Determine the actual average value of the function, f, from x = 0 to x = 10.
2. A man is observing a horserace. He is standing at some point, O, 100 feet from the track. The line of sight from the observer to some point P located on the track forms a 30° angle with the track, as shown in the diagram below. Horse H is galloping at a constant rate of 45 feet per second.
(a) At what rate is the distance from the horse to the observer changing 4 seconds after the horse passes point P?
(b) At what rate is the area of the triangle formed by P, H, and O changing 4 seconds after the horse passes point P?
(c) At the instant the horse gallops past him, the observer begins running at a constant rate of 10 feet per second on a line perpendicular to and toward the track. At what rate is the distance between the observer and the horse changing when the observer is 50 feet from the track?
3. Let v(t) be the velocity, in feet per second, of a race car at time t seconds, t ≥ 0. At time t = 0, while traveling at 197.28 feet per second, the driver applies the brakes such that the car’s velocity satisfies the differential equation
(a) Find an expression for v in terms of t where t is measured in seconds.
(b) Flow far does the car travel before coming to a stop?
(c) Write an equation for the tangent line to the velocity curve at t = 9 seconds.
(d) Find the car’s average velocity from t = 0 until it stops.
STOP. END OF SECTION II, PART A. IF YOU HAVE ANY TIME LEFT, GO OVER YOUR WORK IN THIS PART ONLY. DO NOT WORK IN ANY OTHER PART OF THE TEST.
SECTION II, PART B
45 Minutes • 3 Questions
A CALCULATOR IS A/OT PERMITTED FOR THIS PART OF THE EXAMINATION.
4. Let R be defined as the region in the first quadrant bounded by the curves y = x2 and y = 8 — x2.
(a) Sketch and label the region on the axes provided.
(b) Determine the area of R.
(c) Determine the volume of the solid formed when R is rotated about the x-axis.
(d) Determine the volume of the solid whose base is R and whose cross sections perpendicular to the x-axis are semicircles.
5. The graph below represents the derivative, f', of some function f.
(a) At what value of x does f achieve a local maximum? Explain your reasoning.
(b) Put these values in order from least to greatest: f(4), f(5), and f(7). Explain your reasoning.
(c) Does/have any points of inflection? If so, what are they? Explain your reasoning.
6. Examine the curve defined by 2exy — y = 0.
(b) Find for the family of curves bexy - y = 0.
(c) Determine the y-intercept(s) of bexy - y = 0.
(d) Write the equation for the tangent line at the y-intercept.
STOP. END OF SECTION II, PART B. IF YOU HAVE ANY TIME LEFT, GO OVER YOUR WORK IN THIS PART ONLY. DO NOT WORK IN ANY OTHER FART OF THE TEST.