Loading test…

AP Calculus AB Full-length practice exam 1

45 questions

Header banner

Full-length practice exam 1

0 of 45 answered

~26 min left

Question 1

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 2

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

Question 3

Solve $\frac{d y}{d x} = - 2 x y^{2}$ with $y (0) = 1$ .

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

y = \frac{1}{x ^{2} + 1}

y = \frac{1}{x ^{2} - 1}

y = - 2 x

Question 4

The graph of $f^{'}$ is shown (not provided here — described): $f^{'} (x) > 0$ for $x < 2$ , $f^{'} (2) = 0$ , and $f^{'} (x) < 0$ for $x > 2$ . What can be concluded about $f$ at $x = 2$ ?

f

has a local minimum at

x = 2

.B.

f

has a local maximum at

x = 2

.C.

f

has an inflection point at

x = 2

.D.

f

is not differentiable at

x = 2

Question 5

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.

In-content ad

Question 6

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 7

For the initial value problem $\frac{d y}{d x} = x - y$ , $y (0) = 1$ , use Euler's method with step size $h = 1$ to estimate $y (2)$ .

0

1

- 1

2

Question 8

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

0

1

\infty

D.Does not exist.

Question 9

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 10

A slope field for $\frac{d y}{d x} = \frac{x}{y}$ is sketched. Which statement best describes a solution curve passing through $(0, 3)$ ?

A.A line with slope

0

through

(0, 3)

.B.The upper semicircle of radius

3

centered at the origin.C.The upper branch of the hyperbola

y^{2} - x^{2} = 9

.D.The parabola

y = x^{2} + 3

In-content ad

Question 11

Use the washer method to find the volume when the region between $y = x$ and $y = x^{2}$ (on $[0, 1]$ ) is rotated about the $x$ -axis.

\frac{π}{6}

\frac{2 π}{15}

\frac{π}{5}

\frac{π}{15}

Question 12

Find the absolute maximum of $f (x) = - x^{2} + 4 x + 1$ on the interval $[0, 5]$ .

1

5

4

6

Question 13

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 14

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 15

A ladder 10 ft long leans against a vertical wall. The base slides away from the wall at $2$ ft/s. How fast is the top of the ladder sliding down when the base is $6$ ft from the wall?

- \frac{3}{2}

ft/sB.

- 2

ft/sC.

- \frac{6}{4}

ft/sD.

\frac{3}{2}

ft/s

In-content ad

Question 16

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 17

The radius of a circle is increasing at a rate of $3$ cm/s. At what rate is the area increasing when the radius is $5$ cm?

15 π

cm²/sB.

30 π

cm²/sC.

10 π

cm²/sD.

75 π

cm²/s

Question 18

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

0

\frac{1}{2}

1

D.Does not exist.

Question 19

A 5-foot tall person walks away from a 20-foot lamppost at $3$ ft/s. How fast is their shadow lengthening?

1

ft/sB.

3

ft/sC.

4

ft/sD.

\frac{3}{4}

ft/s

Question 20

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

\frac{8}{3}

\frac{14}{3}

5

10

In-content ad

Question 21

Differentiate $y = ln (x^{4} + 7)$ with respect to $x$ .

\frac{1}{x ^{4} + 7}

\frac{4}{x ^{4} + 7}

\frac{4 x ^{3}}{x ^{4} + 7}

4 ln (x^{3})

Question 22

A particle moves along a number line so that its position is $s (t) = t^{3} - 6 t + 2$ (feet, $t$ in seconds). What is the velocity at $t = 2$ ?

6

ft/sB.

2

ft/sC.

12

ft/sD.

- 6

ft/s

Question 23

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 24

Find the volume of the solid formed when the region bounded by $y = e^{x}$ , the $x$ -axis, $x = 0$ , and $x = 1$ is rotated about the $x$ -axis.

π (e - 1)

\frac{π ( e ^{2} - 1 )}{2}

π e^{2}

e^{2} - 1

Question 25

If $y = arctan (2 x)$ , what is $\frac{d y}{d x}$ ?

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

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

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

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

In-content ad

Question 26

A function $f$ has $f^{'} (c) = 0$ and $f^{''} (c) = 0$ . What can be concluded?

f

has a local maximum at

x = c

.B.

f

has a local minimum at

x = c

.C.

f

has an inflection point at

x = c

.D.The Second Derivative Test is inconclusive; further analysis is needed.

Question 27

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 28

Solve the separable ODE $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 4$ . (Give the solution for $y > 0$ .)

y = x^{2} + 4

y = x^{2} + 16

y = x^{2} + 4

y = \frac{x ^{2}}{2} + 4

Question 29

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 30

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}

In-content ad

Question 31

The function $f (x) = {x^{2} a x + b x < 1 x \geq 1$ is continuous and differentiable at $x = 1$ . What are $a$ and $b$ ?

a = 1

b = 0

a = 2

b = - 1

a = 2

b = 1

a = - 1

b = 2

Question 32

Find the area enclosed between $y = x$ and $y = x^{2}$ on $[0, 1]$ .

\frac{1}{3}

\frac{1}{6}

\frac{1}{2}

1

Question 33

If a particle has velocity $v (t) = 3 t^{2} - 6 t$ (m/s), what is its displacement from $t = 0$ to $t = 4$ ?

8 m

16 m

24 m

32 m

Question 34

A spherical balloon is being inflated so that its volume increases at $100$ cm³/s. How fast is the radius increasing when the radius is $5$ cm? (Recall $V = \frac{4}{3} π r^{3}$ .)

\frac{1}{π}

cm/sB.

4 π

cm/sC.

\frac{1}{4 π}

cm/sD.

\frac{100}{4 π}

cm/s

Question 35

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

In-content ad

Question 36

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

0

2

4

5

Question 37

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 38

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 39

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 40

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

In-content ad

Question 41

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 42

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 43

If $f^{'} (x) > 0$ on $(a, b)$ , then $f$ is:

A.Concave up on

(a, b)

.B.Increasing on

(a, b)

.C.Decreasing on

(a, b)

.D.Constant on

(a, b)

Question 44

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 45

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}

AP Calculus AB Full-length practice exam 1 — Free with Answer Explanations | Test Practice Hub