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AP Calculus AB Practice Exam 2: printable

▶ Section I: multiple choice ✎ Section II: free response Answers and scoring guide

AP Calculus AB Practice Exam 2

Name: ______________________   Date: __________

Part A: multiple choice: 29 questions, 62 minutes, no calculator. Part B: multiple choice: 13 questions, 38 minutes, graphing calculator. Part A: free response: 2 questions, 30 minutes, graphing calculator. Part B: free response: 4 questions, 60 minutes, no calculator. steplyclass.com/ap-calculus-ab/practice-exam-2/

Section I, part A: no calculator

  1. What is the limit of x2 + x − 6x − 2 as x approaches 2?
    1. A) 0
    2. B) 5
    3. C) 4
    4. D) 3
  2. What is the limit of 4x2 − x2x2 + 3 as x increases without bound?
    1. A) 0
    2. B) −13
    3. C) 2
    4. D) infinity
  3. What is the limit of √x − 3x − 9 as x approaches 9?
    1. A) 16
    2. B) 0
    3. C) 6
    4. D) 13
  4. The function f is defined by f(x) = x2 + 1 for x < 1 and f(x) = 3 − x for x ≥ 1. What is the limit of f(x) as x approaches 1?
    1. A) 1
    2. B) 4
    3. C) 3
    4. D) 2
  5. Using the definition of the derivative, f(x) = x2 + 5x has f′(x) equal to which of these?
    1. A) 2x
    2. B) 2x + 5
    3. C) x + 5
    4. D) 0
  6. If f(x) = 4x3 − 5x2 + 7, what is f′(x)?
    1. A) 12x2 − 10x
    2. B) 12x2 − 10x + 7
    3. C) 4x2 − 5x
    4. D) 12x3 − 10x2
  7. If f(x) = (2x + 1)(x − 4), what is f′(x)?
    1. A) 2
    2. B) 2x − 8
    3. C) 4x − 7
    4. D) 4x − 8
  8. If f(x) = x + 2x − 1, what is f′(x)?
    1. A) 1
    2. B) 3/(x − 1)2
    3. C) 1/(x − 1)2
    4. D) −3/(x − 1)2
  9. Which is an equation of the line tangent to the graph of y = x3 − 3x at x = 2?
    1. A) y = 9x
    2. B) y = 9x − 16
    3. C) y = 3x − 4
    4. D) y = 9x + 20
  10. If f(x) = 6ex + 2ln x, what is f′(x)?
    1. A) 6ex + 2x
    2. B) 6ex + 2x
    3. C) 6xex − 1 + 2x
    4. D) 6ex + 1/(2x)
  11. If f(x) = (x2 − 5)3, what is f′(x)?
    1. A) 3(x2 − 5)2
    2. B) 6x(x2 − 5)3
    3. C) 6x(x2 − 5)2
    4. D) 2x(x2 − 5)2
  12. If f(x) = cos(x2), what is f′(x)?
    1. A) −sin(x2)
    2. B) 2x sin(x2)
    3. C) −2x sin(x2)
    4. D) −sin(2x)
  13. The curve x2 + xy + y2 = 7 goes through (1, 2). What is the slope of the curve at that point?
    1. A) 45
    2. B) −45
    3. C) −54
    4. D) −12
  14. If f(x) = tan⁻¹(3x), what is f′(x)?
    1. A) 1/(1 + 9x2)
    2. B) 3/(1 + 3x2)
    3. C) 3/(1 + 9x2)
    4. D) 3/(1 + x2)
  15. What is the limit of (e3x − 1)/x as x approaches 0?
    1. A) 0
    2. B) 1
    3. C) 13
    4. D) 3
  16. What is the limit of (1 − cos(2x))/x2 as x approaches 0?
    1. A) 0
    2. B) 12
    3. C) 2
    4. D) 4
  17. The line tangent to y = √x at x = 16 is used to estimate √16.4. What is the estimate?
    1. A) 4.05
    2. B) 4.4
    3. C) 4.2
    4. D) 4
  18. The radius of a circle grows at 3 cm per second. How fast is the area growing, in square cm per second, when the radius is 10 cm?
    1. A) 60π
    2. B) 30π
    3. C) 100π
    4. D) 6π
  19. The function f(x) = x2 − 6x + 2 has a relative minimum. At which point is it?
    1. A) (3, −7)
    2. B) (−3, 29)
    3. C) (3, 2)
    4. D) (6, 2)
  20. At what point does the graph of f(x) = x3 − 6x2 + 5 have its point of inflection?
    1. A) (0, 5)
    2. B) (4, −27)
    3. C) (2, −11)
    4. D) (−2, −27)
  21. The function f(x) = x2 + 2x satisfies the Mean Value Theorem on [0, 4]. What value of c does the theorem give?
    1. A) 6
    2. B) 2
    3. C) 4
    4. D) 3
  22. Two numbers add up to 30. What is the largest possible value of their product?
    1. A) 225
    2. B) 15
    3. C) 900
    4. D) 200
  23. Two positive numbers have a product of 64. What is the smallest possible value of their sum?
    1. A) 8
    2. B) 65
    3. C) 16
    4. D) 20
  24. What is ∫ (8x3 − 6x + 5) dx?
    1. A) 24x2 − 6 + C
    2. B) 2x4 − 3x2 + 5x + C
    3. C) 8x4 − 6x2 + 5x + C
    4. D) 2x4 − 3x2 + C
  25. What is the value of ∫12(4x − 1) dx?
    1. A) 7
    2. B) 6
    3. C) 5
    4. D) 2
  26. What is ∫ 3x2(x3 − 2)4 dx?
    1. A) (x3 − 2)5/5 + C
    2. B) (x3 − 2)5 + C
    3. C) 4(x3 − 2)3 + C
    4. D) x3(x3 − 2)5/5 + C
  27. What is the derivative, with respect to x, of ∫0x√(t2 + 4) dt?
    1. A) √(x2 + 4) − 2
    2. B) x/√(x2 + 4)
    3. C) √(x2 + 4)
    4. D) √(t2 + 4)
  28. A left Riemann sum with 4 equal parts is used to estimate the area under y = x2 from x = 0 to x = 4. What is the estimate?
    1. A) 30
    2. B) 14
    3. C) 643
    4. D) 13
  29. The function y satisfies dy/dx = −2y and y(0) = 5. Which is y?
    1. A) y = 5e2x
    2. B) y = e−2x + 4
    3. C) y = 5e−2x
    4. D) y = 5 − 2x

Section I, part B: calculator

  1. A 13-foot ladder leans against a wall. The bottom slides away from the wall at 2 feet per second. How fast, in feet per second, is the top sliding down the wall when the bottom is 5 feet from the wall?
    1. A) 0.83
    2. B) 2
    3. C) 4.8
    4. D) 0.77
  2. Air is pumped into a round balloon at 100 cubic cm per second. How fast, in cm per second, is the radius growing when the radius is 5 cm?
    1. A) 1.27
    2. B) 0.32
    3. C) 3.18
    4. D) 0.19
  3. The area of a circle grows at 30 square cm per second. How fast, in cm per second, is the radius growing when the radius is 3 cm?
    1. A) 3.18
    2. B) 5
    3. C) 1.59
    4. D) 1.06
  4. A population P grows so that dP/dt = 0.03P, where t is in years. At t = 0 there are 800. To the nearest hundredth, what is P after 5 years?
    1. A) 920
    2. B) 824.36
    3. C) 927.42
    4. D) 929.47
  5. A cup of tea cools in a 20° room so that dT/dt = −0.1(T − 20), where t is in minutes. At t = 0 the tea is 90°. To the nearest hundredth, what is its temperature after 10 minutes?
    1. A) 33.11
    2. B) 45.75
    3. C) 20
    4. D) 53.11
  6. What is the value of ∫132x dx, to the nearest thousandth?
    1. A) 1.099
    2. B) 1.333
    3. C) 4
    4. D) 2.197
  7. The region under y = x2 from x = 0 to x = 2 is turned around the x-axis. To the nearest hundredth, what is the volume of the solid?
    1. A) 20.11
    2. B) 8.38
    3. C) 6.4
    4. D) 2.67
  8. The function f(x) = x3 satisfies the Mean Value Theorem on [0, 3]. To the nearest hundredth, what value of c does the theorem give?
    1. A) 1.5
    2. B) 3
    3. C) 1.73
    4. D) 9
  9. What is the area of the region between y = x2 and y = x + 2?
    1. A) 7.5
    2. B) 3
    3. C) 10.5
    4. D) 4.5
  10. What is the value of ∫016e3x dx, to the nearest thousandth?
    1. A) 38.171
    2. B) 114.513
    3. C) 40.171
    4. D) 3.437
  11. Water flows into a tank at a rate of r(t) = 3t2 + 2 litres per minute. How many litres flow in from t = 1 to t = 3 minutes?
    1. A) 29
    2. B) 26
    3. C) 33
    4. D) 30
  12. What is the limit of x2 + 2x − 3x2 − 1 as x approaches 1?
    1. A) 0
    2. B) 2
    3. C) 1
    4. D) 4
  13. What is the average value of f(x) = 3x2 + 1 on the interval [0, 2]?
    1. A) 10
    2. B) 5
    3. C) 7
    4. D) 4

Section II, part A: calculator

Question 1: Sand on a pile (9 points)

Sand is added to a pile at the rate r(t) = 16 − t2 cubic metres per hour, for 0 ≤ t ≤ 4. At t = 0 the pile holds 20 cubic metres of sand. No sand is taken away.

(a) How many cubic metres of sand are added to the pile from t = 0 to t = 3? Show the integral you use.

(b) Find r′(2). Using correct units, explain what r′(2) means in this story.

(c) What is the average rate, in cubic metres per hour, at which sand is added over 0 ≤ t ≤ 4?

(d) How many cubic metres of sand are in the pile at t = 4?

Question 2: A particle on a line (9 points)

A particle moves along the x-axis. Its velocity is v(t) = t2 − 5t + 4 metres per second, for 0 ≤ t ≤ 5. At t = 0 the particle is at x = 3.

(a) At what times t is the particle at rest? Show how you find them.

(b) Is the particle speeding up or slowing down at t = 3? Give a reason.

(c) Find the displacement and the total distance the particle travels from t = 0 to t = 5.

(d) Where is the particle at t = 5?

Section II, part B: no calculator

Question 3: A cubic function (9 points)

The function f is defined by f(x) = x3 + 3x2 − 9x − 4.

(a) Find f′(x).

(b) Find the x-values where f has a relative maximum and a relative minimum. Justify your answers.

(c) Find the x-value of the point of inflection of the graph of f. Give a reason.

(d) Find the absolute maximum and the absolute minimum values of f on the closed interval [−4, 2].

Question 4: A differential equation (9 points)

Consider the differential equation dy/dx = 4x3 y.

(a) Find d²y/dx² in terms of x and y.

(b) A solution curve goes through the point (1, 2). Write an equation of its tangent line at x = 1, and use it to estimate y at x = 1.1.

(c) Find y = f(x), the particular solution of the differential equation with f(0) = 3.

Question 5: A region and its solids (9 points)

R is the region between the graph of y = 3x − x2 and the x-axis.

(a) Find the area of R.

(b) R is turned around the x-axis. Find the volume of the solid made.

(c) R is the base of another solid. Its cross sections at right angles to the x-axis are squares. Write, but do not work out, an integral for its volume.

Question 6: An accumulation function (9 points)

The function g is defined by g(x) = ∫2x(t2 − 9) dt.

(a) Find g′(x).

(b) For 0 < x < 6, at what x-value does g have a relative minimum? Justify your answer.

(c) Find g(5), that is, ∫25(x2 − 9) dx.

(d) Write an equation of the line tangent to the graph of g at x = 5.

Answer key

Section I, part A: no calculator

  1. B
  2. C
  3. A
  4. D
  5. B
  6. A
  7. C
  8. D
  9. B
  10. B
  11. C
  12. C
  13. B
  14. C
  15. D
  16. C
  17. A
  18. A
  19. A
  20. C
  21. B
  22. A
  23. C
  24. B
  25. C
  26. A
  27. C
  28. B
  29. C

Section I, part B: calculator

  1. A
  2. B
  3. C
  4. D
  5. B
  6. D
  7. A
  8. C
  9. D
  10. A
  11. D
  12. B
  13. B

Section II

Question 1: Sand on a pile (9 points)

(a) How many cubic metres of sand are added to the pile from t = 0 to t = 3? Show the integral you use.

amount = 39 cubic metres

Scoring guide (2 points):

  • 1 point: working: one mark for each right line (Check my steps)
  • 1 point: the right answer, finished

(b) Find r′(2). Using correct units, explain what r′(2) means in this story.

r′(t) = −2t, so r′(2) = −4 cubic metres per hour per hour.

At t = 2 hours, the rate at which sand is added is going down by 4 cubic metres per hour, each hour.

Scoring guide (2 points):

  • 1 point: r′(2) = −4
  • 1 point: the meaning: the rate is going down by 4 cubic metres per hour per hour at t = 2 (with units)

(c) What is the average rate, in cubic metres per hour, at which sand is added over 0 ≤ t ≤ 4?

average = 12834 = 323

Scoring guide (3 points):

  • 1 point: the average value formula: 14 × ∫04r(t) dt
  • 1 point: the integral: ∫04(16 − t2) dt = 1283
  • 1 point: the answer: 323 ≈ 10.667 cubic metres per hour

(d) How many cubic metres of sand are in the pile at t = 4?

Sand at t = 4 = 20 + ∫04r(t) dt.

The integral is 1283 (part (c): 4 hours × the average rate 323).

So the pile holds 20 + 1283 = 1883 ≈ 62.667 cubic metres.

Scoring guide (2 points):

  • 1 point: 20 + ∫04r(t) dt
  • 1 point: the answer: 1883 ≈ 62.667 cubic metres

Question 2: A particle on a line (9 points)

(a) At what times t is the particle at rest? Show how you find them.

t = 1 or t = 4

Scoring guide (2 points):

  • 1 point: working: one mark for each right line (Check my steps)
  • 1 point: the right answer, finished

(b) Is the particle speeding up or slowing down at t = 3? Give a reason.

v(3) = 9 − 15 + 4 = −2, which is negative.

a(t) = v′(t) = 2t − 5, so a(3) = 1, which is positive.

The velocity and the acceleration have different signs, so the particle is slowing down.

Scoring guide (2 points):

  • 1 point: finds v(3) = −2 and a(3) = 1
  • 1 point: slowing down, because v and a have different signs

(c) Find the displacement and the total distance the particle travels from t = 0 to t = 5.

displacement = −56 m

Scoring guide (3 points):

  • 1 point: the displacement: ∫05v(t) dt = −56 metres
  • 1 point: splits the distance where v changes sign: t = 1 and t = 4
  • 1 point: the total distance: 496 ≈ 8.167 metres

(d) Where is the particle at t = 5?

x(5) = x(0) + displacement = 3 + (−56) (part (c)).

x(5) = 136 ≈ 2.167, so the particle is at x = 13/6.

Scoring guide (2 points):

  • 1 point: x(0) + ∫05v(t) dt
  • 1 point: the answer: x = 136 ≈ 2.167

Question 3: A cubic function (9 points)

(a) Find f′(x).

f′(x) = 3x2 + 6x − 9

Scoring guide (2 points):

  • 1 point: working: one mark for each right line (Check my steps)
  • 1 point: the right answer, finished

(b) Find the x-values where f has a relative maximum and a relative minimum. Justify your answers.

max (−3, 23), min (1, −9)

Scoring guide (3 points):

  • 1 point: f′(x) = 3x2 + 6x − 9 = 0 at x = −3 and x = 1
  • 1 point: relative maximum at x = −3, because f′ changes from positive to negative there
  • 1 point: relative minimum at x = 1, because f′ changes from negative to positive there

(c) Find the x-value of the point of inflection of the graph of f. Give a reason.

inflection : (−1, 7)

Scoring guide (2 points):

  • 1 point: f″(x) = 6x + 6 = 0 at x = −1
  • 1 point: inflection at x = −1, because f″ changes sign there

(d) Find the absolute maximum and the absolute minimum values of f on the closed interval [−4, 2].

min : −9 at x = 1

Scoring guide (2 points):

  • 1 point: tests the ends and the critical points: f(−4) = 16, f(−3) = 23, f(1) = −9, f(2) = −2
  • 1 point: absolute maximum 23 (at x = −3) and absolute minimum −9 (at x = 1)

Question 4: A differential equation (9 points)

(a) Find d²y/dx² in terms of x and y.

d²y/dx² = 12x2 y + 4x3 × dy/dx (the product rule).

Put in dy/dx = 4x3 y: d²y/dx² = 12x2 y + 4x3 × 4x3 y = 12x2 y + 16x6 y.

Scoring guide (2 points):

  • 1 point: the product rule: 12x2 y + 4x3 dy/dx
  • 1 point: puts in dy/dx: 12x2 y + 16x6 y

(b) A solution curve goes through the point (1, 2). Write an equation of its tangent line at x = 1, and use it to estimate y at x = 1.1.

The slope at (1, 2) is dy/dx = 4 × 13 × 2 = 8.

The tangent line: y = 2 + 8(x − 1).

At x = 1.1: y ≈ 2 + 8 × 0.1 = 2.8.

Scoring guide (2 points):

  • 1 point: the tangent line y = 2 + 8(x − 1)
  • 1 point: the estimate y(1.1) ≈ 2.8

(c) Find y = f(x), the particular solution of the differential equation with f(0) = 3.

y = 3e^(x⁴)

Scoring guide (5 points):

  • 1 point: separates the letters: dy/y = 4x3 dx
  • 2 points: the antiderivatives: ln|y| = x4 (1 point each side)
  • 1 point: the constant of integration, found with f(0) = 3
  • 1 point: the answer: y = 3ex4

Question 5: A region and its solids (9 points)

(a) Find the area of R.

area = 92 square units

Scoring guide (3 points):

  • 1 point: the limits 0 and 3 (where 3x − x2 = 0)
  • 1 point: the integral ∫03(3x − x2) dx
  • 1 point: the answer: 92

(b) R is turned around the x-axis. Find the volume of the solid made.

V = 81π10 ≈ 25.45

Scoring guide (3 points):

  • 2 points: working: one mark for each right line (Check my steps)
  • 1 point: the right answer, finished

(c) R is the base of another solid. Its cross sections at right angles to the x-axis are squares. Write, but do not work out, an integral for its volume.

Each square has side 3x − x2 (the height of R at x), so its area is (3x − x2)2.

Volume = ∫03(3x − x2)2 dx.

(It is the integral of part (b) without the π: 81/10.)

Scoring guide (3 points):

  • 1 point: the area of one square: (3x − x2)2
  • 1 point: the limits 0 and 3
  • 1 point: the whole integral ∫03(3x − x2)2 dx

Question 6: An accumulation function (9 points)

(a) Find g′(x).

ddx∫ from 2 to x of (t2 − 9) dt = x2 − 9

Scoring guide (2 points):

  • 1 point: working: one mark for each right line (Check my steps)
  • 1 point: the right answer, finished

(b) For 0 < x < 6, at what x-value does g have a relative minimum? Justify your answer.

g′(x) = x2 − 9 = 0 at x = 3 (x = −3 is not in 0 < x < 6).

g′(x) < 0 for 0 < x < 3 and g′(x) > 0 for 3 < x < 6, so g has a relative minimum at x = 3.

Scoring guide (2 points):

  • 1 point: g′(x) = 0 at x = 3
  • 1 point: justifies: g′ changes from negative to positive at x = 3

(c) Find g(5), that is, ∫25(x2 − 9) dx.

= 12

Scoring guide (2 points):

  • 1 point: working: one mark for each right line (Check my steps)
  • 1 point: the right answer, finished

(d) Write an equation of the line tangent to the graph of g at x = 5.

g(5) = 12 (part (c)).

g′(5) = 52 − 9 = 16 (part (a)).

The tangent line: y = 12 + 16(x − 5).

Scoring guide (3 points):

  • 1 point: g(5) = 12
  • 1 point: g′(5) = 16
  • 1 point: the line y = 12 + 16(x − 5)

Every answer solved step by step: steplyclass.com/ap-calculus-ab/practice-exam-2/answers/

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