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

AP Calculus AB Full-length practice exam 3

45 questions

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Full-length practice exam 3

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Question 1

Find $x \to \infty lim \frac{2 x ^{2} - 5 x + 1}{3 x ^{2} + x - 7}$ .

0

\frac{2}{3}

\frac{3}{2}

\infty

Question 2

The temperature $T$ (°F) of a cooling object satisfies $T^{'} (t) = - 0.1 (T - 70)$ . At time $t = 0$ , $T = 200$ °F. Using the differential equation, what is $T^{'} (0)$ ?

- 20

- 13

- 0.1

130

Question 3

Evaluate $\int_{0}^{2} (x^{2} + 1) d x$ .

\frac{8}{3}

\frac{14}{3}

5

10

Question 4

If $f (x) = e^{s i n x}$ , find $f^{'} (x)$ .

e^{c o s x}

e^{s i n x}

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

sin x \cdot e^{c o s x}

Question 5

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

\frac{1}{sin x}

cot x

tan x

\frac{sin x}{cos x}

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Question 6

Newton's law of cooling gives $\frac{d T}{d t} = - k (T - T_{s})$ , where $T_{s} = 20$ °C is room temperature. If an object cools from $100$ °C to $60$ °C in $10$ minutes, what is the temperature at $t = 20$ minutes?

20

°CB.

40

°CC.

36

°CD.

28

°C

Question 7

Find all inflection points of $f (x) = x^{4} - 4 x^{3}$ .

x = 0

onlyB.

x = 3

onlyC.

x = 2

onlyD.

x = 0

and

x = 2

Question 8

A bacteria culture doubles every 3 hours. If the initial count is $N_{0}$ , what is the count after $t$ hours?

N (t) = N_{0} e^{3 t}

N (t) = N_{0} \cdot 2^{t /3}

N (t) = N_{0} \cdot 2^{3 t}

N (t) = N_{0} (1 + t /3)

Question 9

If $f (x) = \frac{x ^{2} - 4}{x - 2}$ for $x \neq = 2$ and $f (2) = 5$ , then $f$ is:

A.Continuous everywhere.B.Continuous everywhere except at

x = 2

.C.Discontinuous everywhere.D.Has a vertical asymptote at

x = 2

Question 10

Water drains from a conical tank (vertex down) of height 10 m and radius 4 m at $2$ m³/min. How fast is the water level dropping when $h = 5$ m? (Recall $V = \frac{1}{3} π r^{2} h$ .)

- \frac{1}{2 π}

m/minB.

- \frac{2}{25 π}

m/minC.

- \frac{1}{4 π}

m/minD.

- \frac{5}{2 π}

m/min

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Question 11

If the position of a particle is $s (t) = t^{3} - 6 t^{2} + 9 t$ (in meters, $t$ in seconds), what is its velocity at $t = 2$ ?

- 3 m/s

- 1 m/s

2 m/s

9 m/s

Question 12

On a slope field for $\frac{d y}{d x} = y - x$ , the slopes along the line $y = x$ are:

1

everywhere along

y = x

.B.

0

everywhere along

y = x

.C.Equal to

2 x

along

y = x

.D.Undefined along

y = x

Question 13

If $F (x) = \int_{0}^{x} t^{3} d t$ , what is $F^{'} (x)$ ?

3 x^{2}

x^{3}

\frac{x ^{4}}{4}

0

Question 14

Find $\int x cos (x^{2}) d x$ .

sin (x^{2}) + C

\frac{1}{2} sin (x^{2}) + C

x sin (x^{2}) + C

2 sin (x^{2}) + C

Question 15

A tank contains $100$ gallons of water. Water drains at rate $r (t) = 5 t$ gal/min. How much water drains out in the first $4$ minutes?

20

gallonsB.

40

gallonsC.

80

gallonsD.

10

gallons

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Question 16

Apply L'Hôpital's rule to evaluate $x \to 0 lim \frac{e ^{x} - 1 - x}{x ^{2}}$ .

0

1

\frac{1}{2}

D.Does not exist.

Question 17

A particle moves so that $x (t) = t^{3} - 6 t^{2} + 9 t$ for $t \geq 0$ . On which interval(s) is the particle's speed increasing?

(0, 1)

onlyB.

(1, 2)

and

(3, \infty)

(3, \infty)

onlyD.

(0, 3)

Question 18

Use the limit definition of the derivative to find $f^{'} (2)$ for $f (x) = x^{2} + 1$ .

0

2

4

5

Question 19

A particle's velocity is $v (t) = t^{2} - 4 t + 3$ for $0 \leq t \leq 4$ . What is the total distance traveled?

\frac{4}{3}

4

\frac{16}{3}

\frac{8}{3}

Question 20

What is the average value of $f (x) = 4 x$ on $[1, 3]$ ?

4

8

16

12

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Question 21

A slope field for a differential equation shows short line segments with slope $x + y$ at each point $(x, y)$ . Which differential equation corresponds to this slope field?

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

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

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

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

Question 22

If $f^{''} (x) < 0$ on an interval, then the graph of $f$ is:

A.Concave up.B.Decreasing.C.Concave down.D.At a local maximum.

Question 23

A population $P$ satisfies $\frac{d P}{d t} = 0.05 P$ . If $P (0) = 200$ , find $P (t)$ .

P (t) = 200 + 0.05 t

P (t) = 200 e^{0.05 t}

P (t) = e^{0.05 t}

P (t) = 200 (0.05)^{t}

Question 24

Apply the squeeze theorem to evaluate $x \to 0 lim x^{2} sin (\frac{1}{x})$ .

0

1

\infty

D.Does not exist.

Question 25

Evaluate $x \to 3 lim \frac{x ^{2} - 9}{x - 3}$ .

0

3

6

D.Does not exist.

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Question 26

Differentiate $y = tan^{- 1} (\frac{x}{3})$ with respect to $x$ .

\frac{3}{9 + x ^{2}}

\frac{1}{9 + x ^{2}}

\frac{1}{1 + x ^{2} /9}

\frac{x}{3 ( 1 + x ^{2} )}

Question 27

If $f$ is differentiable at $x = a$ , then $f$ is:

A.Continuous at

x = a

.B.Equal to its derivative at

x = a

.C.Continuous at every

x

.D.Linear near

x = a

Question 28

Using the Second Derivative Test, classify the critical point $x = 1$ of $f (x) = x^{3} - 3 x + 2$ .

A.Local maximumB.Local minimumC.Inflection pointD.

x = 1

is not a critical point.

Question 29

A particle moves along the $x$ -axis with velocity $v (t) = t^{2} - 4$ for $0 \leq t \leq 3$ . What is the total displacement?

- 3

3

\frac{11}{3}

0

Question 30

If $f (x) = x sin x$ , find $f^{'} (x)$ .

sin x + x cos x

cos x

x cos x

sin x cos x

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Question 31

Use L'Hôpital's rule to find $x \to 0^{+} lim x ln x$ .

1

- 1

0

- \infty

Question 32

Let $f (x) = arcsin (x)$ . Using the derivative of an inverse function, find $f^{'} (1/2)$ .

3

\frac{2}{3}

2

\frac{1}{2}

Question 33

If $x \to 4^{-} lim f (x) = 5$ and $x \to 4^{+} lim f (x) = 7$ , then:

lim_{x \to 4} f (x) = 6

.B.

lim_{x \to 4} f (x) = 12

.C.

lim_{x \to 4} f (x)

does not exist.D.

f

is continuous at

4

Question 34

Let $f (x) = x^{3} - 3 x^{2} - 9 x + 5$ . On what interval(s) is $f$ increasing?

(- \infty, - 1)

and

(3, \infty)

(- 1, 3)

(- \infty, - 1)

and

(3, \infty)

(- \infty, 3)

Question 35

Use implicit differentiation to find $\frac{d y}{d x}$ for $x^{2} + y^{2} = 25$ .

\frac{x}{y}

- \frac{y}{x}

- \frac{x}{y}

\frac{2 x}{2 y}

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Question 36

If $y = cos^{3} (x^{2})$ , find $\frac{d y}{d x}$ .

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

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

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

3 cos^{2} (x^{2}) sin (x^{2}) \cdot 2 x

Question 37

Using the disk method, find the volume of the solid formed by rotating $y = x$ on $[0, 4]$ about the $x$ -axis.

4 π

8 π

16 π

2 π

Question 38

Find $\frac{d}{d x} [\frac{x ^{2}}{x + 1}]$ .

2 x

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

\frac{2 x}{x + 1}

\frac{2 x + 1}{x + 1}

Question 39

The general solution to $\frac{d y}{d x} = k y$ is $y = C e^{k x}$ . If $y (0) = 5$ and $y (2) = 20$ , find $k$ .

k = \frac{ln 2}{2}

k = ln 2

k = 2

k = \frac{ln 4}{4}

Question 40

Find the volume of the solid with square cross-sections perpendicular to the $x$ -axis, where the base is the region bounded by $y = x$ and $y = 0$ on $[0, 4]$ .

4

8

8 π

16

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Question 41

Evaluate $\int (3 x^{2} + 4 x - 5) d x$ .

x^{3} + 2 x^{2} - 5 x + C

3 x^{3} + 4 x^{2} - 5 x + C

x^{3} + 2 x^{2} - 5 x

6 x + 4 + C

Question 42

Find $f^{'} (x)$ if $f (x) = e^{x} cos x$ .

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

e^{x} cos x

- e^{x} sin x

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

Question 43

The position of a particle is $s (t) = t^{3} - 9 t^{2} + 24 t$ for $t \geq 0$ . At what time(s) is the particle at rest?

t = 1

and

t = 8

t = 2

and

t = 4

t = 3

onlyD.

t = 6

only

Question 44

The acceleration of a particle is $a (t) = 6 t$ with $v (0) = - 2$ and $s (0) = 4$ . Find $s (3)$ .

27

25

31

22

Question 45

Evaluate $x \to 0 lim \frac{1 - cos x}{x ^{2}}$ .

0

\frac{1}{2}

1

D.Does not exist.