AP Calculus BC Practice Exam 1: printable

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

AP Calculus BC 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-bc/practice-exam-1/

Section I, part A: no calculator

What is the limit of x² + x − 6x + 3 as x approaches −3?
1. A) 0
2. B) 5
3. C) 1
4. D) −5
What is the limit of ln xx − 1 as x approaches 1?
1. A) 0
2. B) 1
3. C) −1
4. D) infinity
If f(x) = 3x, what is f′(x)?
1. A) −3x²
2. B) −3x
3. C) 3ln x
4. D) 3x²
If f(x) = e^−x, what is f′(x)?
1. A) e^−x
2. B) −xe^{−x − 1}
3. C) −e^−x
4. D) −e^x
If f(x) = 2tan⁻¹(x), what is f′(x)?
1. A) 2/(1 + x²)
2. B) 2/√(1 − x²)
3. C) 1/(1 + x²)
4. D) −2/(1 + x²)
If y = x^3x for x > 0, what is dy/dx?
1. A) 3x × x^{3x − 1}
2. B) x^3x(3ln x + 3)
3. C) x^3xln x
4. D) 3x^3xln x
For y = x² + 3x, what is the differential dy when x = 1 and dx = 0.01?
1. A) 0.0501
2. B) 5
3. C) 0.05
4. D) 0.02
The area of a circle grows at 20 square cm per second. How fast is the radius growing, in cm per second, when the radius is 5 cm?
1. A) 10/π
2. B) 4/π
3. C) 200π
4. D) 2/π
For f(x) = x² + 2x on [−2, 0], what value of c does Rolle's Theorem give?
1. A) 1
2. B) 0
3. C) −1
4. D) −2
At what point does the graph of f(x) = x³ − 6x² + 4 have a point of inflection?
1. A) (2, −12)
2. B) (4, −28)
3. C) (−2, −28)
4. D) (0, 4)
At what point does f(x) = x² + 2x − 3 have a relative minimum?
1. A) (1, 0)
2. B) (−1, −4)
3. C) (−1, −1)
4. D) (−2, −3)
Which is ∫ x e^x dx?
1. A) xe^x + e^x + C
2. B) e^x + C
3. C) x^2e^x/2 + C
4. D) xe^x − e^x + C
What is the value of ∫₁^∞1x² dx?
1. A) infinity
2. B) 1
3. C) 0
4. D) −1
What is the value of ∫₀^∞e^−2x dx?
1. A) 12
2. B) −12
3. C) infinity
4. D) 2
Which is ∫ e^5x dx?
1. A) 5e^5x + C
2. B) e^6x6 + C
3. C) e^5x5 + C
4. D) e^5x + C
Which is ∫ 3x²(x³ + 4)⁵ dx?
1. A) x³(x³ + 4)⁶/6 + C
2. B) (x³ + 4)⁶ + C
3. C) 5(x³ + 4)⁴ + C
4. D) (x³ + 4)⁶/6 + C
Which function y solves dy/dx = 4y with y(0) = −2?
1. A) −2e^x
2. B) e^4x − 3
3. C) −2e^4x
4. D) 2e^4x
Euler's method with step 0.5 is used on dy/dx = −y, starting at y(0) = 4. What is the estimate of y(1)?
1. A) 2
2. B) 0
3. C) 9
4. D) 1
What is the length of the curve y = (43)x from x = 0 to x = 3?
1. A) 5
2. B) 3
3. C) 4
4. D) 7
What is the area of the region between y = 1 and y = x²?
1. A) 23
2. B) 43
3. C) 2
4. D) 83
A curve is given by x = 3t and y = t² − 4. What is dy/dx at t = 1?
1. A) 32
2. B) 2
3. C) 23
4. D) −3
What is the area of the region inside the polar curve r = 3?
1. A) 9π
2. B) 3π
3. C) 6π
4. D) 9π/2
What is the area of one petal of the polar curve r = cos(3θ)?
1. A) π/6
2. B) π/12
3. C) π/3
4. D) π/24
Which is true about the series Σ from n = 1 to ∞ of (n + 4)/n?
1. A) converges
2. B) converges to 4
3. C) converges to 1
4. D) diverges
Which is true about the series Σ from n = 1 to ∞ of (−1)ⁿ⁺¹/√n?
1. A) converges absolutely
2. B) converges conditionally
3. C) diverges
4. D) converges to 0
What is the interval of convergence of the power series Σ from n = 0 to ∞ of (x − 2)ⁿ/3ⁿ?
1. A) [−1, 5]
2. B) (−3, 3)
3. C) (−1, 5)
4. D) (1, 3)
What is the Maclaurin polynomial of degree 5 for sin x?
1. A) x − x³6 + x⁵120
2. B) x + x³6 + x⁵120
3. C) 1 − x²2 + x⁴24
4. D) x − x³3 + x⁵5
Which is true about the series Σ from n = 1 to ∞ of n/(n² + 1)?
1. A) converges
2. B) converges to 12
3. C) converges absolutely
4. D) diverges
Which is true about the series Σ from n = 1 to ∞ of 2ⁿn!?
1. A) diverges
2. B) diverges to infinity
3. C) converges
4. D) cannot be decided by the ratio test

Section I, part B: calculator

What is the limit of 8x³ − 12x³ + x as x increases without bound?
1. A) 0
2. B) 4
3. C) −1
4. D) 8
If f(x) = 5^x, what is f′(x)?
1. A) x × 5^{x − 1}
2. B) 5^x
3. C) 5^x ln 5
4. D) 5^x/ln 5
For the curve x² + y² = 100, what is the slope of the tangent line at the point (6, −8)?
1. A) 34
2. B) 43
3. C) −34
4. D) −43
What is the limit of (1 − cos x)/x² as x approaches 0?
1. A) infinity
2. B) 0
3. C) 1
4. D) 12
At what point does f(x) = −x² − 6x have a relative maximum?
1. A) (−3, 9)
2. B) (3, −27)
3. C) (−3, −9)
4. D) (0, 0)
Two positive numbers have a sum of 20. What is the largest possible value of their product?
1. A) 20
2. B) 100
3. C) 99
4. D) 400
The area under y = x² + 1 from x = 0 to x = 2 is estimated with 2 rectangles of width 1, using right endpoints. What is the estimate?
1. A) 3
2. B) 143
3. C) 7
4. D) 5
Which is ∫ x² ln x dx, for x > 0?
1. A) x³3 ln x + C
2. B) x³3 ln x + x³9 + C
3. C) 2x ln x + x + C
4. D) x³3 ln x − x³9 + C
A substance decays by dm/dt = −km. It starts at 80 grams and its half-life is 5 years. How many grams are left after 12 years, to the nearest hundredth?
1. A) 40
2. B) 15.16
3. C) 20
4. D) 10
The region under y = 3 from x = 0 to x = 2 is turned around the x-axis. What is the volume of the solid, to the nearest hundredth?
1. A) 56.55
2. B) 18.85
3. C) 18
4. D) 226.19
A curve is given by x = t³ and y = 6t^2. What is dy/dx at t = 2?
1. A) 12
2. B) 24
3. C) 4
4. D) 2
What is the length of the curve y = (23)x^3/2 from x = 0 to x = 8?
1. A) 32√23
2. B) 8
3. C) 523
4. D) 40
Which is true about the series Σ from n = 1 to ∞ of n2ⁿ?
1. A) diverges
2. B) diverges to infinity
3. C) cannot be decided by the ratio test
4. D) converges

Section II, part A: calculator

Question 1: A particle in the plane (9 points)

A particle moves in the xy-plane. Its position at time t is x(t) = t² + 1 and y(t) = t³ − 3t, for 0 ≤ t ≤ 2.

(a) Find dy/dx at t = 2.

(b) Find the speed of the particle at t = 2.

(c) Find the total distance the particle travels from t = 0 to t = 2.

(d) At what point (x, y) is the particle moving straight up or straight down? Give a reason.

Question 2: A polar curve (9 points)

The curve C is given by the polar equation r = 2 + 2cos θ, for 0 ≤ θ ≤ 2π.

(a) Find the area of the region inside C.

(b) Find dr/dθ at θ = π/2. What does it say about the point on C as θ grows past π/2?

(c) Find the slope dy/dx of C at θ = π/2.

(d) Find the area of the region inside C and outside the circle r = 2.

Section II, part B: no calculator

Question 3: A Taylor polynomial (9 points)

Let f(x) = e^−x.

(a) Find the Maclaurin polynomial of degree 3 for f.

(b) Use your polynomial to estimate f(0.1).

(c) The Maclaurin series of f at x = 0.1 is alternating, with terms that shrink to 0. Show that the estimate in (b) is within 0.00001 of f(0.1).

(d) Write the first four nonzero terms of the Maclaurin series of x e^−x.

Question 4: Euler's method (9 points)

Let y = f(x) be the solution of dy/dx = x + y with f(0) = 1.

(a) Use Euler's method, starting at x = 0 with two steps of equal size, to estimate f(1). Show the work that leads to your answer.

(b) Find d²y/dx² in terms of x and y. Is the solution curve concave up or concave down at (0, 1)?

(c) Show that y = 2e^x − x − 1 is the solution with f(0) = 1.

(d) Find f(1) exactly. Is the Euler estimate too big or too small? Give a reason.

Question 5: Integration by parts and an improper integral (9 points)

This question has three integrals.

(a) Find ∫ x cos x dx.

(b) Find ∫₀^∞e^−3x dx, or show that it diverges.

(c) Does ∫₁^∞1x dx converge or diverge? Give a reason.

Question 6: A power series (9 points)

Consider the power series Σ from n = 1 to ∞ of (x − 1)ⁿ/(n × 2ⁿ).

(a) Find the interval of convergence. Check the endpoints.

(b) At x = −1 the series converges. Does it converge absolutely or conditionally? Give a reason.

(c) At x = 2, find the sum of the first three terms of the series.

Answer key

Section I, part A: no calculator

Section I, part B: calculator

Section II

Question 1: A particle in the plane (9 points)

(a) Find dy/dx at t = 2.

slope = 94

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 speed of the particle at t = 2.

dx/dt = 2t = 4 and dy/dt = 3t² − 3 = 9 at t = 2.

Speed = √(4² + 9²) = √97 ≈ 9.849.

Scoring guide (2 points):

1 point: dx/dt = 4 and dy/dt = 9
1 point: speed √97 ≈ 9.849

(c) Find the total distance the particle travels from t = 0 to t = 2.

Distance = ∫₀²√((2t)² + (3t² − 3)²) dt.

With a calculator: ≈ 7.605.

Scoring guide (3 points):

1 point: the integrand √((dx/dt)² + (dy/dt)²)
1 point: limits 0 and 2
1 point: ≈ 7.605

(d) At what point (x, y) is the particle moving straight up or straight down? Give a reason.

Straight up or down means dx/dt = 0 and dy/dt ≠ 0.

dx/dt = 2t = 0 at t = 0, where dy/dt = −3 ≠ 0.

The point is (x(0), y(0)) = (1, 0); the particle is moving straight down there.

Scoring guide (2 points):

1 point: dx/dt = 0 with dy/dt ≠ 0, at t = 0
1 point: the point (1, 0)

Question 2: A polar curve (9 points)

(a) Find the area of the region inside C.

A = 6π ≈ 18.85

Scoring guide (3 points):

1 point: uses ½∫ r² dθ
1 point: limits 0 and 2π
1 point: area 6π

(b) Find dr/dθ at θ = π/2. What does it say about the point on C as θ grows past π/2?

dr/dθ = −2sin θ, which is −2 at θ = π/2.

dr/dθ < 0, so r is getting smaller: the point is moving closer to the origin.

Scoring guide (2 points):

1 point: dr/dθ = −2
1 point: moving closer to the origin, because r is decreasing

(c) Find the slope dy/dx of C at θ = π/2.

x = r cos θ and y = r sin θ. At θ = π/2: r = 2 and dr/dθ = −2.

dy/dθ = (dr/dθ)sin θ + r cos θ = −2 × 1 + 2 × 0 = −2.

dx/dθ = (dr/dθ)cos θ − r sin θ = 0 − 2 = −2.

dy/dx = −2−2 = 1.

Scoring guide (2 points):

1 point: dy/dθ and dx/dθ at θ = π/2
1 point: slope 1

(d) Find the area of the region inside C and outside the circle r = 2.

C is outside r = 2 when cos θ > 0, for −π/2 < θ < π/2.

Area = ½∫_−π/2^π/2((2 + 2cos θ)² − 2²) dθ = 8 + π ≈ 11.142.

Scoring guide (2 points):

1 point: the integral with limits −π/2 and π/2
1 point: 8 + π ≈ 11.142

Question 3: A Taylor polynomial (9 points)

(a) Find the Maclaurin polynomial of degree 3 for f.

P₃(x) = 1 − x + x²2 − x³6

Scoring guide (3 points):

1 point: f(0) = 1 and f′(0) = −1
1 point: f″(0) = 1 and f‴(0) = −1
1 point: 1 − x + x²2 − x³6

(b) Use your polynomial to estimate f(0.1).

P₃(0.1) = 1 − 0.1 + 0.012 − 0.0016 = 1 − 0.1 + 0.005 − 0.000167 ≈ 0.904833.

Scoring guide (2 points):

1 point: puts x = 0.1 into P₃
1 point: ≈ 0.9048

(c) The Maclaurin series of f at x = 0.1 is alternating, with terms that shrink to 0. Show that the estimate in (b) is within 0.00001 of f(0.1).

The error of an alternating series is at most the first term left out: (0.1)⁴/4! = 0.0001/24.

0.000124 ≈ 0.0000042, which is less than 0.00001.

Scoring guide (2 points):

1 point: the next term (0.1)⁴/4!
1 point: ≈ 0.0000042 < 0.00001

(d) Write the first four nonzero terms of the Maclaurin series of x e^−x.

Multiply each term of the series of e^−x by x.

x − x² + x³2 − x⁴/6.

Scoring guide (2 points):

1 point: multiplies the series by x
1 point: x − x² + x³2 − x⁴6

Question 4: Euler's method (9 points)

(a) Use Euler's method, starting at x = 0 with two steps of equal size, to estimate f(1). Show the work that leads to your answer.

y(1) ≈ 2.5

Scoring guide (3 points):

1 point: first step: y(0.5) ≈ 1 + 0.5 × 1 = 1.5
1 point: second step uses the slope 0.5 + 1.5 = 2
1 point: f(1) ≈ 2.5

(b) Find d²y/dx² in terms of x and y. Is the solution curve concave up or concave down at (0, 1)?

d²y/dx² = 1 + dy/dx = 1 + x + y.

At (0, 1): 1 + 0 + 1 = 2, which is positive, so the curve is concave up there.

Scoring guide (2 points):

1 point: d²y/dx² = 1 + x + y
1 point: concave up, because d²y/dx² = 2 > 0

(c) Show that y = 2e^x − x − 1 is the solution with f(0) = 1.

dy/dx = 2e^x − 1, and x + y = x + 2e^x − x − 1 = 2e^x − 1: the same, so it solves the equation.

At x = 0: y = 2 − 0 − 1 = 1, so f(0) = 1.

Scoring guide (2 points):

1 point: dy/dx = x + y checked
1 point: f(0) = 1 checked

(d) Find f(1) exactly. Is the Euler estimate too big or too small? Give a reason.

f(1) = 2e − 2 ≈ 3.437.

The estimate 2.5 is too small: the curve is concave up, so each Euler step stays under it.

Scoring guide (2 points):

1 point: f(1) = 2e − 2
1 point: too small, with the reason

Question 5: Integration by parts and an improper integral (9 points)

(a) Find ∫ x cos x dx.

= x sin x + cos x + C

Scoring guide (3 points):

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

(b) Find ∫₀^∞e^−3x dx, or show that it diverges.

= 13

Scoring guide (3 points):

1 point: writes it as a limit as b → ∞
1 point: the antiderivative −e^−3x3
1 point: converges to 13

(c) Does ∫₁^∞1x dx converge or diverge? Give a reason.

∫₁^b1x dx = ln b − ln 1 = ln b.

As b grows without bound, ln b grows without bound.

So the integral diverges.

Scoring guide (3 points):

1 point: writes it as a limit as b → ∞
1 point: the antiderivative ln b
1 point: diverges, because ln b → ∞

Question 6: A power series (9 points)

(a) Find the interval of convergence. Check the endpoints.

interval : [−1, 3)

Scoring guide (4 points):

1 point: ratio test: |x − 1|/2 < 1
1 point: radius 2: −1 < x < 3
1 point: x = 3: the harmonic series diverges
1 point: x = −1: converges by the alternating series test; interval [−1, 3)

(b) At x = −1 the series converges. Does it converge absolutely or conditionally? Give a reason.

At x = −1 the terms are (−2)ⁿ/(n × 2ⁿ) = (−1)ⁿ/n.

Σ (−1)ⁿ/n converges by the alternating series test (1n shrinks to 0).

Σ 1n, the harmonic series, diverges: so it converges conditionally.

Scoring guide (3 points):

1 point: the series Σ (−1)ⁿ/n at x = −1
1 point: the alternating series test
1 point: conditionally, because Σ 1n diverges

(c) At x = 2, find the sum of the first three terms of the series.

At x = 2 the terms are 1/(n × 2ⁿ): 12, 18, 1/24.

12 + 18 + 124 = 1224 + 324 + 124 = 1624 = 2/3.

Scoring guide (2 points):

1 point: the three terms 12, 18, 124
1 point: 23

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

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