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AP Calculus BC Full-length practice exam 3

45 questions

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Full-length practice exam 3

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Question 1

A population grows according to $P^{'} = 0.04 P$ . If $P (0) = 500$ , find $P (10)$ .

500 e^{0.4}

500 (1.04)^{10}

500 + 0.04 (10) (500)

500 e^{0.04}

Question 2

Find the volume of the solid generated by revolving the region bounded by $y = x^{2}$ and $y = x$ around the line $y = - 1$ using the washer method.

\frac{π}{2}

\frac{7 π}{15}

\frac{11 π}{15}

\frac{π}{15}

Question 3

Differentiate $f (x) = e^{s i n^{2} x}$ .

e^{s i n^{2} x} \cdot cos^{2} x

e^{s i n^{2} x} \cdot sin (2 x)

e^{2 s i n x} cos x

2 sin x cos x

Question 4

Find the equation of the normal line to $y = x^{2} - 3 x$ at the point $(1, - 2)$ .

y = x - 3

y = x - 3

y = - x - 1

y = - x + 1

Question 5

Using the MVT on $f (x) = x$ over $[4, 9]$ , find $c$ such that $f^{'} (c)$ equals the average rate of change.

c = \frac{25}{4}

c = \frac{13}{2}

c = 6

c = \frac{25}{4}

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Question 6

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 7

Let $f (x) = x^{4/3} - 2 x^{1/3}$ . Find all $x$ where $f^{'} (x) = 0$ .

x = 0

onlyB.

x = \frac{1}{2}

onlyC.

x = 0

and

x = \frac{1}{2}

x = 2

only

Question 8

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

Question 9

The derivative of $f (x) = \frac{x ^{2} + 1}{x - 3}$ at $x = 4$ is:

7

17

- 7

3

Question 10

A particle moves with position $r (t) = ⟨ t^{2} - 1, 2 t ⟩$ . Find the speed at $t = 2$ .

25

20

(2 t)^{2} + 4

evaluated naively

= 20

6

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Question 11

Find $\frac{d}{d x} [arcsin (x)]$ .

\frac{1}{x ( 1 - x )}

\frac{1}{2 x 1 - x}

\frac{1}{1 - x}

\frac{- 1}{2 x ( 1 - x )}

Question 12

Evaluate $lim_{x \to 0} \frac{e ^{x} - 1 - x}{x ^{2}}$ using L'Hôpital's Rule.

0

1

\frac{1}{2}

2

Question 13

Use the Maclaurin series $sin x = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots$ to find $x \to 0 lim \frac{sin x - x}{x ^{3}}$ .

- \frac{1}{6}

0

\frac{1}{6}

1

Question 14

A point moves along the curve $y = x^{2}$ . When $x = 3$ the $x$ -coordinate increases at $2$ units/sec. How fast is the distance from the origin increasing at that moment?

\frac{78}{85}

units/secB.

122

units/secC.

2

units/secD.

\frac{12}{10}

units/sec

Question 15

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

0

1

\infty

- 1

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Question 16

Find the Lagrange error bound for approximating $sin (0.1)$ using the third-degree Taylor polynomial about $x = 0$ .

\frac{( 0.1 ) ^{4}}{4 !}

\frac{( 0.1 ) ^{3}}{3 !}

\frac{( 0.1 ) ^{5}}{5 !}

\frac{( 0.1 ) ^{4}}{4}

Question 17

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 18

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 19

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 20

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

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Question 21

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 22

Find $\frac{d}{d x} [\frac{x ^{2} + 1}{x - 3}]$ at $x = 4$ .

- 9

8

7

17

Question 23

A vector-valued function is $r (t) = ⟨ t^{2}, sin t ⟩$ . Find $r^{'} (t)$ .

⟨ 2 t, cos t ⟩

⟨ 2 t, - cos t ⟩

⟨ t^{2}, sin t ⟩

⟨ 2, cos t ⟩

Question 24

The function $f (x) = \frac{x ^{2} - 4}{x - 2}$ has what type of discontinuity at $x = 2$ ?

A.Vertical asymptoteB.Jump discontinuityC.Removable discontinuityD.Essential discontinuity

Question 25

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}

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Question 26

For the polar curve $r = 2 cos θ$ , find the area enclosed.

π

2 π

4 π

\frac{π}{2}

Question 27

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 28

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

\frac{1}{5}

\frac{3}{5}

\frac{5}{3}

0

Question 29

A slope field for $\frac{d y}{d x} = x - y$ shows slopes of $0$ along the line:

y = x

y = - x

y = 0

x = 0

Question 30

The solution to a logistic equation passes through $(0, 100)$ and approaches $L = 800$ as $t \to \infty$ . Using $P = \frac{L}{1 + A e ^{- r t}}$ with $r = 0.5$ , find $A$ .

A = 7

A = 8

A = 700

A = 0.125

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Question 31

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 32

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 33

Verify that $y = C e^{2 x} + 3$ is the general solution to $\frac{d y}{d x} = 2 (y - 3)$ . What is the particular solution satisfying $y (0) = 7$ ?

y = 4 e^{2 x} + 3

y = 7 e^{2 x}

y = 4 e^{2 x}

y = C e^{2 x}

with

C = 7

Question 34

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 35

Evaluate $\int x cos x d x$ using integration by parts.

x sin x + cos x + C

x sin x - cos x + C

cos x - x sin x + C

x cos x + sin x + C

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Question 36

Find the area of the region inside $r = 2$ and outside $r = 2 - 2 cos θ$ .

π + 4

2 π - 4

π

4 - π

Question 37

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 38

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 39

On what interval is $f (x) = x e^{- x}$ concave down?

(2, \infty)

(- \infty, 1)

(- \infty, 2)

(0, \infty)

Question 40

For the logistic model $d P / d t = 0.2 P (1 - P /500)$ with $P (0) = 50$ , at what population is the growth rate $d P / d t$ maximized?

P = 50

P = 500

P = 250

P = 100

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Question 41

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

4

8

2 π

4 π

Question 42

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}

Question 43

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 44

If $f (x) = e^{2 x} cos x$ , find $f^{'} (x)$ .

e^{2 x} (2 cos x - sin x)

e^{2 x} (2 cos x + sin x)

2 e^{2 x} cos x

e^{2 x} (- sin x)

Question 45

Use implicit differentiation to find $\frac{d y}{d x}$ for $x^{3} + y^{3} = 6 x y$ .

\frac{6 y - 3 x ^{2}}{3 y ^{2} - 6 x}

\frac{3 x ^{2} - 6 y}{6 x - 3 y ^{2}}

\frac{6 y - 3 x ^{2}}{6 x}

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

AP Calculus BC Full-length practice exam 3 — Free with Answer Explanations | Test Practice Hub