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AP Calculus BC Full-length practice exam 2 — Free with Answer Explanations | Test Practice Hub

AP Calculus BC Full-length practice exam 2

45 questions

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Full-length practice exam 2

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Question 1

Find the arc length of the curve $y = ln (cos x)$ on $[0, \frac{π}{4}]$ .

ln (2 + 1)

\frac{π}{4}

ln 2

sec (π /4) - sec (0)

Question 2

The region bounded by $x = y^{2}$ and $x = 4$ is revolved around the $y$ -axis. Find the volume.

\frac{128 π}{5}

\frac{256 π}{5}

\frac{64 π}{3}

32 π

Question 3

A curve is given parametrically by $x (t) = t^{2} - 1$ and $y (t) = 2 t$ . Find $\frac{d y}{d x}$ in terms of $t$ .

\frac{1}{t}

2 t

4 t

t

Question 4

Let $F (x) = f (g (x))$ where $f (u) = u^{3} + u$ and $g (x) = cos x$ . Find $F^{'} (π /2)$ .

0

- 1

1

3

Question 5

A particle's position is $s (t) = 2 sin (π t)$ for $t \geq 0$ . What is the particle's acceleration at $t = 1$ ?

0

2 π^{2}

- 2 π^{2}

- 2 π

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Question 6

For the cardioid $r = 1 + cos θ$ , find the total arc length.

4

8

2 π

4 π

Question 7

Evaluate $lim_{x \to 0} \frac{sin ( 3 x )}{5 x}$ .

\frac{1}{5}

\frac{3}{5}

\frac{5}{3}

0

Question 8

A particle's position is $r (t) = ⟨ e^{t}, t^{2} ⟩$ . Find the total distance traveled from $t = 0$ to $t = 1$ .

e - 1

\int_{0}^{1} e^{2 t} + 4 t^{2} d t

\int_{0}^{1} (e^{t} + 2 t) d t

(e - 1)^{2} + 1

Question 9

A particle has acceleration $a (t) = ⟨ 2, 6 t ⟩$ with $v (0) = ⟨ 1, 0 ⟩$ and $r (0) = ⟨ 0, 0 ⟩$ . Find $r (1)$ .

⟨ 2, 1 ⟩

⟨ 1, 1 ⟩

⟨ 3, 3 ⟩

⟨ 2, 3 ⟩

Question 10

For the parametric curve $x = t^{3} - 3 t$ , $y = t^{2}$ , find $\frac{d ^{2} y}{d x ^{2}}$ at $t = 2$ .

\frac{2}{9}

\frac{1}{27}

\frac{1}{9}

\frac{2}{27}

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Question 11

The logistic growth model is $\frac{d P}{d t} = 0.3 P (1 - \frac{P}{1000})$ . As $t \to \infty$ , what does $P$ approach?

300

\infty

1000

500

Question 12

Find the Taylor series for $ln (1 + x)$ centered at $x = 0$ (Maclaurin series).

n = 1 \sum \infty \frac{x ^{n}}{n}

n = 1 \sum \infty \frac{( - 1 ) ^{n + 1} x ^{n}}{n}

n = 0 \sum \infty \frac{( - 1 ) ^{n} x ^{n}}{n !}

n = 1 \sum \infty \frac{( - 1 ) ^{n} x ^{n}}{n}

Question 13

Find all inflection points of $h (x) = x^{4} - 6 x^{2} + 5$ .

x = 0

onlyB.

x = \pm 1

x = \pm 3

x = \pm 2

Question 14

Solve the separable ODE $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 3$ .

y = x^{2} + 9

y = x^{2} + 3

y = x + 3

y^{2} = x^{2} + 3

Question 15

Evaluate $\int \frac{x}{4 - x ^{2}} d x$ .

arcsin (\frac{x}{2}) + C

- 4 - x^{2} + C

4 - x^{2} + C

- \frac{1}{2} 4 - x^{2} + C

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Question 16

Two cars approach an intersection; one travels north at $30$ mph and the other travels east at $40$ mph. How fast is the distance between them decreasing when they are $5$ mi and $12$ mi from the intersection, respectively?

50

mphB.

\frac{630}{13}

mph decreasingC.

\frac{70}{13}

mph decreasingD.

\frac{1200}{13}

mph decreasing

Question 17

For what value of $c$ is $f (x) = {x^{2} + c 3 x - 1 x < 2 x \geq 2$ continuous at $x = 2$ ?

c = 1

c = - 1

c = 3

c = 5

Question 18

Evaluate $lim_{x \to \infty} \frac{4 x ^{3} - 2 x + 7}{6 x ^{3} + 5 x ^{2} - 1}$ .

0

\frac{4}{5}

\frac{2}{3}

\infty

Question 19

Use Euler's method with step size $h = 0.5$ to approximate $y (1)$ for $\frac{d y}{d x} = x + y$ with $y (0) = 1$ .

2.75

2.5

3.0

1.5

Question 20

Evaluate $x \to \infty lim (1 + \frac{3}{x})^{x}$ .

1

e^{3}

3 e

\infty

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Question 21

If $G (x) = \int_{1}^{x^{3}} ln t d t$ , find $G^{'} (x)$ .

ln (x^{3})

3 x^{2} ln (x^{3})

\frac{3 x ^{2}}{x ^{3}}

3 ln x

Question 22

Find $\frac{d}{d x} [x^{3} ln x]$ for $x > 0$ .

3 x^{2} ln x

x^{2}

x^{2} (3 ln x + 1)

3 x^{2} ln x + x^{2}

Question 23

Find the volume of the solid formed by revolving the region between $y = x$ and $y = x$ on $[0, 1]$ around the $x$ -axis using the washer method.

\frac{π}{6}

\frac{π}{3}

\frac{π}{2}

\frac{π}{12}

Question 24

Use L'Hôpital's Rule to evaluate $lim_{x \to 1} \frac{x ^{3} - 3 x + 2}{x ^{2} - 2 x + 1}$ .

0

\frac{3}{2}

3

\infty

Question 25

Use the Ratio Test to determine whether $n = 1 \sum \infty \frac{n !}{2 ^{n}}$ converges or diverges.

A.Converges by Ratio TestB.Diverges by Ratio Test (limit

= \infty > 1

)C.Inconclusive (limit

= 1

)D.Converges by comparison with

\sum 1/ 2^{n}

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Question 26

Which of the following differential equations is NOT separable?

\frac{d y}{d x} = x y + 3 x

\frac{d y}{d x} = sin x cos y

\frac{d y}{d x} = x + y

\frac{d y}{d x} = \frac{x ^{2}}{y}

Question 27

Solve the separable ODE $\frac{d y}{d x} = 3 x^{2} y$ with initial condition $y (0) = 2$ .

y = 2 e^{x^{3}}

y = e^{x^{3}} + 1

y = 2 e^{3 x}

y = 6 x^{2}

Question 28

Evaluate $\int x e^{x^{2}} d x$ .

\frac{1}{2} e^{x^{2}} + C

e^{x^{2}} + C

x^{2} e^{x^{2}} /2 + C

2 x e^{x^{2}} + C

Question 29

Evaluate $\int_{1}^{4} x d x$ .

\frac{14}{3}

\frac{4}{3}

\frac{2}{3} (8 - 1) = \frac{14}{3}

... wait:

3

\frac{7}{3}

Question 30

Find the interval of convergence of the power series $n = 0 \sum \infty \frac{x ^{n}}{n + 1}$ .

(- 1, 1)

[- 1, 1]

[- 1, 1)

(- \infty, \infty)

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Question 31

For the ODE $\frac{d y}{d x} = \frac{y ^{2}}{x}$ , find the general solution.

y = \frac{- 1}{ln ∣ x ∣ + C}

y = \frac{1}{ln ∣ x ∣ + C}

y = C e^{l n x}

y^{3} = 3 ln ∣ x ∣ + C

Question 32

For the function $f (x) = \frac{∣ x ^{2} - 4∣}{x ^{2} - 4}$ , evaluate $lim_{x \to 2^{-}} f (x)$ and $lim_{x \to 2^{+}} f (x)$ , then classify the discontinuity at $x = 2$ .

A.Both limits equal

1

; removable discontinuity.B.

lim_{x \to 2^{-}} f (x) = - 1

lim_{x \to 2^{+}} f (x) = 1

; jump discontinuity.C.The limit does not exist and

f

has an infinite discontinuity.D.

lim_{x \to 2^{-}} f (x) = 1

lim_{x \to 2^{+}} f (x) = - 1

; jump discontinuity.

Question 33

For $f (x) = x^{3} - 6 x^{2} + 9 x$ , on what intervals is $f$ increasing?

(1, 3)

(- \infty, 1)

and

(3, \infty)

(0, 3)

(- \infty, \infty)

Question 34

Find $\frac{d}{d x} [arctan (\frac{y}{x})]$ treating $y$ as a differentiable function of $x$ .

\frac{1}{1 + ( y / x ) ^{2}}

\frac{y ^{'} x - y}{x ^{2} + y ^{2}}

\frac{y ^{'}}{x ^{2} + y ^{2}}

\frac{x y ^{'} - y}{x ^{2} ( 1 + y ^{2} / x ^{2} )}

Question 35

Find $\frac{d}{d x} [x \cdot e^{x}]$ .

e^{x} (x + \frac{1}{x})

e^{x} (x + \frac{1}{2 x})

\frac{e ^{x}}{2 x}

x e^{x}

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Question 36

The Maclaurin series for $\frac{1}{1 - x}$ is $\sum_{n = 0}^{\infty} x^{n}$ . Use this to find a series for $\frac{x}{( 1 - x ) ^{2}}$ .

n = 1 \sum \infty n x^{n}

n = 0 \sum \infty x^{n + 1}

n = 0 \sum \infty (n + 1) x^{n}

n = 0 \sum \infty n x^{n - 1}

Question 37

Evaluate $\int_{0}^{π} sin^{2} x d x$ .

0

\frac{π}{2}

π

1

Question 38

Find the area of the region bounded by $y = x^{2}$ and $y = 2 x$ .

\frac{4}{3}

\frac{8}{3}

2

\frac{2}{3}

Question 39

For $\frac{d y}{d x} = k y$ with $k < 0$ , the general solution is $y = C e^{k x}$ . This represents exponential decay. If $y (0) = 10$ and $y (3) = 5$ , find $k$ .

k = - \frac{ln 2}{3}

k = - \frac{1}{3}

k = \frac{ln 2}{3}

k = - \frac{1}{2}

Question 40

Find the Maclaurin series for $f (x) = e^{- x}$ up to the $x^{3}$ term.

1 - x + \frac{x ^{2}}{2} - \frac{x ^{3}}{6} + \dots

1 + x + \frac{x ^{2}}{2} + \frac{x ^{3}}{6} + \dots

- 1 + x - \frac{x ^{2}}{2} + \frac{x ^{3}}{6} + \dots

x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots

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Question 41

Evaluate $\int x^{2} e^{x} d x$ .

e^{x} (x^{2} - 2 x + 2) + C

e^{x} (x^{2} + 2 x + 2) + C

e^{x} (x^{2} - 2 x) + C

x^{2} e^{x} - e^{x} + C

Question 42

For $p (x) = \frac{sin x}{x ( x - π )}$ , identify and classify all discontinuities on $(- \infty, \infty)$ .

A.Vertical asymptote at

x = 0

; vertical asymptote at

x = π

.B.Removable at

x = 0

; removable at

x = π

.C.Removable at

x = 0

; vertical asymptote at

x = π

.D.Jump discontinuity at

x = 0

; removable at

x = π

Question 43

Let $f$ be continuous on $R$ with $f (x) = \frac{sin ( x ^{2} )}{x}$ for $x \neq = 0$ . What must $f (0)$ equal?

0

1

- 1

f (0)

can be any real number.

Question 44

Find $\frac{d y}{d x}$ by implicit differentiation if $x^{2} + y^{2} = 25$ .

\frac{x}{y}

- \frac{y}{x}

- \frac{x}{y}

\frac{y}{x}

Question 45

Find the slope of the tangent line to the polar curve $r = 1 + sin θ$ at $θ = π /2$ .

0

1

\infty

- 1