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AP Calculus AB Practice Exam 1: printable

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

AP Calculus AB Practice Exam 1

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-1/

Section I, part A: no calculator

  1. What is the limit of x2 − 9x − 3 as x approaches 3?
    1. A) 6
    2. B) 3
    3. C) 9
    4. D) 0
  2. What is the limit of 5x3 − 2x2x3 + 7 as x increases without bound?
    1. A) 0
    2. B) infinity
    3. C) 57
    4. D) 52
  3. What is the limit of √x − 2x − 4 as x approaches 4?
    1. A) 0
    2. B) 4
    3. C) 14
    4. D) 12
  4. The function f is defined by f(x) = x2 for x < 2 and f(x) = 3x − 2 for x ≥ 2. What is the limit of f(x) as x approaches 2?
    1. A) 2
    2. B) 4
    3. C) 8
    4. D) 6
  5. What is the limit of 3x2 + 1x3 − 4 as x increases without bound?
    1. A) 0
    2. B) 3
    3. C) −14
    4. D) infinity
  6. If f(x) = 2x2 − 3x, what is the limit of (f(x + h) − f(x))/h as h approaches 0?
    1. A) 2x − 3
    2. B) 4x
    3. C) 0
    4. D) 4x − 3
  7. If f(x) = 6x2 − 4x + 9, what is f′(x)?
    1. A) 6x − 4
    2. B) 12x + 5
    3. C) 12x − 4
    4. D) 2x3 − 2x2 + 9x
  8. If f(x) = (x2 + 1)(3x − 2), what is f′(x)?
    1. A) 6x
    2. B) 9x2 − 4x + 3
    3. C) 6x2 − 4x + 3
    4. D) 9x2 + 3
  9. If f(x) = 3x − 1x + 2, what is f′(x)?
    1. A) 3
    2. B) −7/(x + 2)2
    3. C) 5/(x + 2)2
    4. D) 7/(x + 2)2
  10. If f(x) = 2sin x − 4cos x, what is f′(x)?
    1. A) 2cos x + 4sin x
    2. B) 2cos x − 4sin x
    3. C) −2cos x + 4sin x
    4. D) −2cos x − 4sin x
  11. If f(x) = (2x + 1)5, what is f′(x)?
    1. A) 5(2x + 1)4
    2. B) 10(2x + 1)4
    3. C) 10(2x + 1)5
    4. D) 2(2x + 1)4
  12. For the curve x2 + y2 = 25, what is dy/dx at the point (3, 4)?
    1. A) 34
    2. B) −43
    3. C) −34
    4. D) 43
  13. If f(x) = ln(x2 + 1), what is f′(x)?
    1. A) 1/(x2 + 1)
    2. B) 2x/(x2 + 1)
    3. C) 2x
    4. D) 1/(2x)
  14. For the curve x2 − y2 = 9, what is the slope of the tangent line at the point (5, 4)?
    1. A) 45
    2. B) −54
    3. C) 54
    4. D) 1
  15. The tangent line to f(x) = √x at x = 9 is used to estimate √9.2. What is the estimate?
    1. A) 3.2
    2. B) 3.1
    3. C) 3
    4. D) 3.033
  16. What is the limit of sin(3x)/x as x approaches 0?
    1. A) 3
    2. B) 0
    3. C) 1
    4. D) 13
  17. The function f(x) = x3 − x satisfies the Mean Value Theorem on [0, 2]. What value of c in (0, 2) does the theorem give?
    1. A) 1
    2. B) 43
    3. C) 2
    4. D) 2/√3
  18. At what point does f(x) = −x2 + 4x + 1 have a relative maximum?
    1. A) (−2, −11)
    2. B) (2, 5)
    3. C) (2, −3)
    4. D) (4, 1)
  19. At what point does the graph of f(x) = x3 − 9x2 + 24x have a point of inflection?
    1. A) (3, 18)
    2. B) (2, 20)
    3. C) (4, 16)
    4. D) (0, 0)
  20. Two positive numbers have a product of 49. What is the smallest possible value of their sum?
    1. A) 7
    2. B) 50
    3. C) 14
    4. D) 49
  21. Which is ∫ (6x2 − 4x + 1) dx?
    1. A) 12x − 4 + C
    2. B) 6x3 − 4x2 + x + C
    3. C) 2x3 − 4x2 + x + C
    4. D) 2x3 − 2x2 + x + C
  22. What is the value of ∫02(3x2 + 1) dx?
    1. A) 8
    2. B) 10
    3. C) 13
    4. D) 12
  23. Which is ∫ 2x(x2 + 1)3 dx?
    1. A) (x2 + 1)4 + C
    2. B) x2(x2 + 1)4/4 + C
    3. C) (x2 + 1)4/4 + C
    4. D) 3(x2 + 1)2 + C
  24. What is the derivative with respect to x of ∫1x(t3 + 1) dt?
    1. A) x3 + 1
    2. B) x44 + x
    3. C) 3x2
    4. D) x3
  25. The area under y = 9 − x2 from x = 0 to x = 3 is estimated with 3 rectangles of equal width, using right endpoints. What is the estimate?
    1. A) 22
    2. B) 8
    3. C) 18
    4. D) 13
  26. Which function y solves dy/dx = 2xy with y(0) = 3?
    1. A) ex2 + 2
    2. B) 3ex2
    3. C) 3e2x
    4. D) 3x2
  27. Which is the general solution of dy/dx = 3x2 − 2?
    1. A) 6x + C
    2. B) x3 − 2 + C
    3. C) x3 − 2x + C
    4. D) 3x3 − 2x + C
  28. What is the average value of f(x) = x2 on the interval [0, 3]?
    1. A) 3
    2. B) 4.5
    3. C) 9
    4. D) 27
  29. What is the area of the region between y = x and y = x2?
    1. A) 12
    2. B) 16
    3. C) 56
    4. D) 13

Section I, part B: calculator

  1. The radius of a circle grows at 2 cm per second. How fast is the area growing when the radius is 5 cm, in square cm per second, to the nearest hundredth?
    1. A) 31.42
    2. B) 62.83
    3. C) 78.54
    4. D) 12.57
  2. A 10-foot ladder leans against a wall. The top slides down the wall at 2 feet per second. How fast, in feet per second, is the bottom moving away from the wall when the top is 8 feet above the floor?
    1. A) 2
    2. B) 1.5
    3. C) 1.6
    4. D) 2.67
  3. Air flows into a balloon at a rate of r(t) = 2t + 10 litres per second. How many litres flow in from t = 2 to t = 6 seconds?
    1. A) 22
    2. B) 40
    3. C) 72
    4. D) 76
  4. The function f(x) = −x2 + 3x satisfies the Mean Value Theorem on [−1, 2]. What value of c does the theorem give?
    1. A) 12
    2. B) 32
    3. C) 0
    4. D) −12
  5. A rectangular pen with an area of 72 square feet is fenced on three sides; a barn wall is the fourth side. What is the least length of fence needed, in feet?
    1. A) 30
    2. B) 36
    3. C) 24
    4. D) 18
  6. At what point does the graph of f(x) = x3 + 3x2 − 2 have a point of inflection?
    1. A) (0, −2)
    2. B) (−2, 2)
    3. C) (1, 2)
    4. D) (−1, 0)
  7. What is the value of ∫01(x2 + 1) dx?
    1. A) 43
    2. B) 13
    3. C) 1
    4. D) 2
  8. The area under y = 2x + 1 from x = 0 to x = 4 is estimated with 4 rectangles of width 1, using left endpoints. What is the estimate?
    1. A) 24
    2. B) 16
    3. C) 20
    4. D) 12
  9. A population P grows by dP/dt = 0.05P, with P(0) = 200. What is P(10), to the nearest hundredth?
    1. A) 300
    2. B) 210
    3. C) 329.74
    4. D) 543.66
  10. What is the average value of f(x) = 3x2 − 2x on the interval [1, 3]?
    1. A) 18
    2. B) 9
    3. C) 6
    4. D) 11
  11. The region between y = x and y = x2 is turned around the x-axis. What is the volume of the solid, to the nearest hundredth?
    1. A) 0.42
    2. B) 0.10
    3. C) 1.05
    4. D) 0.13
  12. What is the area of the region between y = x2 + 1 and y = 2x + 4?
    1. A) 9
    2. B) 24
    3. C) −323
    4. D) 323
  13. What is the area of the region between y = x2 − 1 and the x-axis from x = 0 to x = 3?
    1. A) 6
    2. B) 203
    3. C) 223
    4. D) 8

Section II, part A: calculator

Question 1: Water in a tank (9 points)

Water is pumped into a tank at the rate r(t) = 3t2 + 2t litres per minute, for 0 ≤ t ≤ 3. At the same time water drains out of the tank at a constant rate of 10 litres per minute. At t = 0 the tank holds 50 litres.

(a) How many litres of water are pumped into the tank from t = 0 to t = 3? Show the integral you use.

(b) Is the amount of water in the tank increasing or decreasing at t = 1? Give a reason.

(c) Write an expression for A(t), the litres of water in the tank at time t. Use it to find A(3).

(d) At what time t, for 0 < t < 3, is the amount of water in the tank least? Justify your answer.

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

A particle moves along the x-axis. Its velocity is v(t) = t2 − 4t + 3 feet per second, for 0 ≤ t ≤ 5. At t = 0 the particle is at x = 2.

(a) At what times in 0 < t < 5 is the particle at rest? Show how you find them.

(b) Is the particle speeding up or slowing down at t = 4? Give a reason.

(c) Find the displacement of the particle from t = 0 to t = 5. Show the integral you use.

(d) Find the position of the particle at t = 5 and the total distance it travels from t = 0 to t = 5.

Section II, part B: no calculator

Question 3: Analysing a cubic (9 points)

Let f(x) = x3 − 3x2 − 9x + 5.

(a) Find f′(x).

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

(c) On what interval is the graph of f concave up? Give a reason.

(d) Find the absolute minimum value of f on the closed interval [0, 4]. Justify your answer.

Question 4: A differential equation (9 points)

Consider the differential equation dy/dx = 6x2 y.

(a) Write an equation of the tangent line to the solution curve through the point (1, 2). Use it to estimate y(1.1).

(b) Find d²y/dx² at the point (1, 2). Is the estimate in part (a) too big or too small? Give a reason.

(c) Find the particular solution y = f(x) with f(0) = 2. Show your separation of variables.

Question 5: Area and volume (9 points)

Let R be the region between the graph of y = 4x − 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.

(c) The vertical line x = k divides R into two regions of equal area. Write, but do not solve, an equation for k.

Question 6: An accumulation function (9 points)

Let g(x) = ∫1x(t2 − 4) dt.

(a) Find g′(x).

(b) For 0 < x < 5, at what x-value does g have a relative minimum? Justify your answer.

(c) Find g(3).

(d) Find the limit of g(x)/⁠(x − 1) as x approaches 1.

Answer key

Section I, part A: no calculator

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

Section I, part B: calculator

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

Section II

Question 1: Water in a tank (9 points)

(a) How many litres of water are pumped into the tank from t = 0 to t = 3? Show the integral you use.

amount = 36 litres

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 amount of water in the tank increasing or decreasing at t = 1? Give a reason.

At t = 1 water comes in at r(1) = 3 + 2 = 5 litres per minute.

It drains out at 10 litres per minute, which is more than 5.

So the amount of water is decreasing at t = 1.

Scoring guide (2 points):

  • 1 point: finds r(1) = 5
  • 1 point: compares 5 with the drain rate 10 and says decreasing

(c) Write an expression for A(t), the litres of water in the tank at time t. Use it to find A(3).

A(t) = 50 + ∫0t(r(s) − 10) ds.

At t = 3: ∫03r(s) ds = 36 (part (a)), and 10 × 3 = 30 litres drain out.

A(3) = 50 + 36 − 30 = 56 litres.

Scoring guide (3 points):

  • 1 point: the integral of r(s) − 10 (in minus out)
  • 1 point: the starting 50 litres added
  • 1 point: A(3) = 56 litres

(d) At what time t, for 0 < t < 3, is the amount of water in the tank least? Justify your answer.

A′(t) = r(t) − 10 = 3t2 + 2t − 10. A′(t) = 0 when t = (−2 + √124)/6 ≈ 1.523 (the other answer is negative).

A′(t) < 0 before 1.523 and A′(t) > 0 after it: the water goes down, then up.

So the amount is least at t ≈ 1.523 minutes.

Scoring guide (2 points):

  • 1 point: sets A′(t) = r(t) − 10 = 0 and finds t ≈ 1.523
  • 1 point: justifies: A′ changes from negative to positive there

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

(a) At what times in 0 < t < 5 is the particle at rest? Show how you find them.

At rest means v(t) = 0: t2 − 4t + 3 = 0.

(t − 1)(t − 3) = 0, so t = 1 and t = 3 seconds.

Scoring guide (2 points):

  • 1 point: sets v(t) = 0
  • 1 point: t = 1 and t = 3

(b) Is the particle speeding up or slowing down at t = 4? Give a reason.

v(4) = 16 − 16 + 3 = 3, which is positive.

a(t) = v′(t) = 2t − 4, so a(4) = 4, which is positive.

The velocity and the acceleration have the same sign, so the particle is speeding up.

Scoring guide (2 points):

  • 1 point: finds v(4) = 3 and a(4) = 4
  • 1 point: speeding up, because v and a have the same sign

(c) Find the displacement of the particle from t = 0 to t = 5. Show the integral you use.

displacement = 203 ft

Scoring guide (3 points):

  • 1 point: the integral of v(t) from 0 to 5
  • 1 point: the antiderivative t33 − 2t2 + 3t
  • 1 point: displacement 203 feet

(d) Find the position of the particle at t = 5 and the total distance it travels from t = 0 to t = 5.

Position: x(5) = 2 + 203 = 263 feet.

Distance: the particle turns at t = 1 and t = 3. ∫01v dt = 43, ∫13v dt = −43, ∫35v dt = 20/3.

Total distance = 43 + 43 + 203 = 283 feet.

Scoring guide (2 points):

  • 1 point: x(5) = 263
  • 1 point: total distance 283 (the part from 1 to 3 counted as positive)

Question 3: Analysing a cubic (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.

f′(x) = 3x2 − 6x − 9 = 3(x − 3)(x + 1) = 0, so x = −1 or x = 3.

f′ changes from positive to negative at x = −1: a relative maximum.

f′ changes from negative to positive at x = 3: a relative minimum.

Scoring guide (3 points):

  • 1 point: x = −1 and x = 3 from f′(x) = 0
  • 1 point: relative maximum at x = −1, with the sign change of f′
  • 1 point: relative minimum at x = 3, with the sign change of f′

(c) On what interval is the graph of f concave up? Give a reason.

f″(x) = 6x − 6.

f″(x) > 0 when x > 1, so the graph is concave up for x > 1.

Scoring guide (2 points):

  • 1 point: f″(x) = 6x − 6
  • 1 point: concave up for x > 1, because f″ > 0 there

(d) Find the absolute minimum value of f on the closed interval [0, 4]. Justify your answer.

Check the ends and the critical point inside: f(0) = 5, f(3) = 27 − 27 − 27 + 5 = −22, f(4) = 64 − 48 − 36 + 5 = −15.

The smallest is −22, so the absolute minimum is −22, at x = 3.

Scoring guide (2 points):

  • 1 point: compares f(0), f(3) and f(4)
  • 1 point: absolute minimum −22

Question 4: A differential equation (9 points)

(a) Write an equation of the tangent line to the solution curve through the point (1, 2). Use it to estimate y(1.1).

At (1, 2): dy/dx = 6 × 12 × 2 = 12.

Tangent line: y − 2 = 12(x − 1).

At x = 1.1: y ≈ 2 + 12 × 0.1 = 3.2.

Scoring guide (2 points):

  • 1 point: slope 12 and the tangent line
  • 1 point: estimate 3.2

(b) Find d²y/dx² at the point (1, 2). Is the estimate in part (a) too big or too small? Give a reason.

d²y/dx² = 12xy + 6x2 (dy/dx) = 12xy + 6x2 (6x2 y).

At (1, 2): 12 × 2 + 6 × 12 = 24 + 72 = 96, which is positive.

The curve is concave up there, so the tangent line lies below it: the estimate is too small.

Scoring guide (2 points):

  • 1 point: d²y/dx² = 96 at (1, 2)
  • 1 point: too small, because the curve is concave up

(c) Find the particular solution y = f(x) with f(0) = 2. Show your separation of variables.

y = 2e^(2x³)

Scoring guide (5 points):

  • 1 point: separates the variables: dy/y = 6x2 dx
  • 1 point: ln|y| on the left
  • 1 point: 2x3 + C on the right
  • 1 point: uses y(0) = 2 to find the constant
  • 1 point: y = 2e2x3

Question 5: Area and volume (9 points)

(a) Find the area of R.

area = 323 square units

Scoring guide (3 points):

  • 1 point: the curve meets the axis at x = 0 and x = 4
  • 1 point: the integral of 4x − x2 from 0 to 4
  • 1 point: area 323

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

V = 512π15 ≈ 107.23

Scoring guide (3 points):

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

(c) The vertical line x = k divides R into two regions of equal area. Write, but do not solve, an equation for k.

The whole area is 323, so each part has area 16/3.

∫0k(4x − x2) dx = 16/3.

Scoring guide (3 points):

  • 1 point: the integral of 4x − x2 from 0 to k (or from k to 4)
  • 1 point: uses half the area, 163
  • 1 point: a complete equation

Question 6: An accumulation function (9 points)

(a) Find g′(x).

ddx∫ from 1 to x of (t2 − 4) dt = x2 − 4

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 < 5, at what x-value does g have a relative minimum? Justify your answer.

g′(x) = x2 − 4 = 0 when x = 2 (x = −2 is not in the interval).

g′ changes from negative to positive at x = 2, so g has a relative minimum at x = 2.

Scoring guide (2 points):

  • 1 point: g′(x) = 0 at x = 2
  • 1 point: the sign change of g′ from negative to positive

(c) Find g(3).

= 23

Scoring guide (2 points):

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

(d) Find the limit of g(x)/⁠(x − 1) as x approaches 1.

g(1) = ∫ from 1 to 1 = 0, so the limit has the form 0/0.

L'Hôpital's rule: the limit = g′(1)/1 = (1 − 4)/1.

The limit is −3.

Scoring guide (3 points):

  • 1 point: notes the form 00 (g(1) = 0)
  • 1 point: uses L'Hôpital's rule with g′(x)
  • 1 point: the limit −3

Every answer solved step by step: steplyclass.com/ap-calculus-ab/practice-exam-1/answers/

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