Free Printable Calculus AB: Units 1

1.

Evaluate.

(a)

2.

Find the limit shown.

(a)

3.

$\lim_{x\rightarrow\infty}\ \frac{x^3-x^2+1}{5-3x-x^4}=...$

a)

0

b)

-1

c)

-5

d)

$\infty$

e)

$-\infty$

4.

$\lim_{x\rightarrow-\infty}\ \frac{x+7}{\sqrt{4x^2+3x}}=...$

a)

$-\infty$

b)

$\infty$

c)

$\frac{1}{2}$

d)

0

e)

$-\frac{1}{2}$

5.

Determine whether $f\left(x\right)=\frac{x^2+5x-6}{x-1}$ is continuous or discontinuous at $x=1.$

a)

continuous

b)

discontinuous, removable

c)

discontinuous, jump

d)

discontinuous, essential

6.

a)

0

b)

-1

c)

2

d)

DNE

7.

$\lim_{x\rightarrow0}\ \frac{7\ \sin\ x\ \cos\ x}{2x}$

a)

-3.5

b)

$-\ 3,5$

c)

$\frac{7}{2}$

d)

7

e)

2

8.

Hint: $\lim_{x\rightarrow\ 0}\frac{x\ }{\sin x}$ has the same value as $\lim_{x\rightarrow\ 0}\frac{\sin x}{x}$

(a)

9.

Check all the boxes that contain true statements.

a)

$\lim_{x\rightarrow-2^{+\ \ }}g\left(x\right)=-3$

b)

$\lim_{x\rightarrow-2}\ \ g\left(x\right)\ =e^{-2}$

c)

$\lim_{x\rightarrow0^-}\ \ g\left(x\right)=1$

d)

$\lim_{x\rightarrow0}\ \ g\left(x\right)=1$

10.

The function shown

a)

is continuous at x = 0

b)

is differentiable at x = 0

c)

has a limit that exists at x = 0

d)

exists at x = 0

11.

The function shown

a)

is continuous at x = 1.5

b)

is differentiable at x = 1.5

c)

has a limit that exists at x = 1.5

d)

exists at x = 1.5

12.

Which one of the following statements is always true?

a)

When a graph is increasing, its derivative is negative.

b)

When a graph is decreasing, so is its derivative.

c)

When a graph is decreasing, its derivative is negative.

d)

When a graph is increasing, so is its derivative.

13.

What do we know about f'(x) when the graph of f(x) is increasing?

a)

f'(x) < 0 (negative)

b)

f'(x) > 0 (positive)

c)

f'(x) is increasing

d)

Can't be determined

14.

f' is given, which could be f?

a)

A

b)

B

c)

C

15.

Find the second derivative of
f(x) = x²+ e^x - cosx

a)

f"(x) = 2 + e^x + cosx

b)

f"(x) = 2x + e^x + cosx

c)

f"(x) = 2x + xe^x - cosx

d)

f"(x) = 2x + e^x + sinx

16.

Find the derivative
f(t) = (t² + 2t)⁵

a)

f'(t) = 5(2t+2)⁴

b)

f'(t) = 5(t² + 2t)⁴

c)

f'(t) = 5(t² + 2t)⁴(2t)

d)

f'(t) = 5(t² + 2t)⁴(2t + 2)

17.

If the position of a particle is represented by x(t) = -t² + 1, what is its instantaneous velocity at t = 1?

a)

v = 0

b)

v = 1

c)

v = -1

d)

v = -2

18.

Set up the derivative of y=(3x⁴- 7)*(5x²+ 1)

a)

(12x³)(10x)

b)

(3x⁴ - 7)(10x) + (12x³)(5x² + 1)

c)

(3x⁴ - 7)(10x) - (12x³)(5x² + 1)

d)

$\frac{3x^4-7}{10x}+\frac{12x^3}{5x^2+1}$

19.

The position of an object is given as a function of time by x = 3t² + 5t³- 2t
What is the acceleration of the object at time t = 2 s?

a)

64 m/s/s

b)

60 m/s/s

c)

66 m/s/s

d)

70 m/s/s

20.

a)

A

b)

B

c)

C

d)

D

21.

At which x-value is f continuous but not differentiable?

a)

a

b)

b

c)

c

d)

d

22.

What is f'(x) if f(x) = cos(5x⁴)?

a)

f'(x) = sin(20x³)

b)

f'(x) = 20x³ sin(5x⁴)

c)

f'(x) = -sin(20x³)

d)

f'(x) = -20x³ sin(5x⁴)

23.

Find h'(3)

a)

-2

b)

0

c)

1

d)

3

24.

a)

1/2

b)

-1/2

c)

0

d)

3/2

25.

Find the derivative.

a)

x⁴cosx - 4x³sinx

b)

x⁴cosx + 4x³sinx

c)

4x³cosx

d)

-4x³cosx

26.

Write the equation of the normal line of: $f\left(x\right)=2x^2-\frac{1}{x}$ at x = 1

a)

$y=-\frac{1}{5}x+\frac{4}{5}$

b)

$y=5x-6$

c)

$y-1=5\left(x-1\right)$

d)

$y-1=-\frac{1}{5}\left(x-1\right)$

27.

Find the equation of the tangent line to the curve

$y=\left(5x-2\right)^2\$ at $x=\frac{1}{2}$ .

28.

Label each of the graph as f(x), f'(x) and f''(x).

a)

f(x)

b)

f'(x)

c)

f''(x)

1

2

3

1)

2)

3)

29.

A population of 500 bacteria is introduced into a culture and grows in number according to the equation $P\left(t\right)=500\left(1+\frac{4t}{50+t^2}\right)$ where t is measured in hours. Find the rate at which the population is growing when t = 2. (round to three decimal places)

(a)

30.

Find the equation of the tangent line to the graph of $f\left(x\right)=\left(9-x^2\right)^{\frac{2}{3}}$ at x = 1. Write your equation in the form $y=m(x-x_{1\ })+y_1$ .

31.

A particle moves along the x-axis so that at time its position is given by $x(t)=2t^3-21t^2+72t-53$ . At what time is the particle at rest?

(hint: a particle is at rest when the velocity is 0).

a)

t = 1 only

b)

t = 3 only

c)

t = 7/2 only

d)

t = 3 and t = 7/2

e)

t = 3 and t = 4

32.

a)

-7

b)

0

c)

1

d)

DNE

33.

Find y' if y = tan(3x²+2).

a)

sec²(3x²+2)

b)

6xsec²(3x²+2)

c)

6x(sec²x)(3x²+2)

d)

sec²(6x)

34.

Find the derivative:
y=5x²e^3x

a)

y'=10xe^3x(2x+3)

b)

y'=5xe^3x(3x+2)

c)

y'=10ex^3x(3x+2)

d)

y'=5xe^3x(2x+3)

35.

A 5 ft ladder is leaning against a wall and sliding towards the floor. The top of the ladder is sliding down the wall at a rate of 2 ft/sec. How fast is the base of the ladder sliding away from the wall when the base of the ladder is 3 ft from the wall?

a)

4/3 ft/sec

b)

8/7 ft/sec

c)

1 ft/sec

d)

8/3 ft/sec

36.

A conical paper cup is 10 cm tall with a radius of 30 cm. The bottom of the cup is punctured so that the water level goes down at a rate of 4 cm/sec. At what rate is the volume of water in the cup changing when the

water level is 2 cm?

a)

-144π cm³/sec

b)

-72π cm³/sec

c)

-288π cm³/sec

d)

-154π cm³/sec

37.

A certain medical procedure requires that a balloon be inserted into the stomach and then inflated. Model the shape of the balloon by a sphere of radius r. If r is increasing at the rate of 0.3 cm/min, how fast is the volume changing when the radius is 4 cm?

a)

15.08 cm³/min

b)

268.08 cm³/min

c)

60.32 cm³/min

d)

6.03 cm³/min

38.

Suppose f(x) = x³ – x.

Use a linear approximation at x = 2 to estimate f(2.5).

a)

10.5

b)

11

c)

11.5

d)

12

39.

Let f be a differentiable function such that f(3)=2 and f'(3)=5. If the tangent line at x = 3 is used to find an approximation to a zero of f, that approximation is

a)

0.4

b)

0.5

c)

2.6

d)

3.4

e)

5.5

40.

The approximate value of $y=\sqrt{\ 4+\sin x}$ at x = 0.12, obtained from the tangent ot the graph at x = 0 is

a)

2.00

b)

2.03

c)

2.06

d)

2.12

e)

2.24

41.

An objects distance from its starting point at time t is given by the equation
s(t) = t³ - 6t² - 4.
What is the speed of the object when its acceleration is 0?

a)

2

b)

-24

c)

12

d)

44

42.

The position of an object is given as a function of time by x(t) = 3t² + 5t³- 2t
What is the acceleration of the object at time t = 2 s?

a)

64 m/s/s

b)

60 m/s/s

c)

66 m/s/s

d)

70 m/s/s

43.

A particle moves along the x-axis so that at time $t\ge0$ its position is given by $x\left(t\right)=-t^3+8t^2-16t$ . Determine if the particle is moving to the right or to the left at t = 5.

a)

left

b)

right

c)

cannot be determined

44.

a)

-1

b)

1

c)

0

d)

12

45.

a)

0

b)

5/3

c)

e^5x

d)

DNE

Calculus AB: Units 1 - 4 Review

Total questions: 45

Worksheet time: 23mins

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