Free Printable Chapter 12: Limits and Derivatives

1.

Use a graph to find $\lim_{x\rightarrow0}\ \frac{1-\cos x}{x}$

a)

1

b)

-1/3

c)

0

d)

-1

2.

Use a graph to estimate $\lim_{x\rightarrow-2}\left(4x^2+3x-1\right)$

a)

-23

b)

-15

c)

21

d)

9

3.

Use a graph to estimate the above limit.

a)

-2

b)

6

c)

4

d)

does not exist

4.

Find $\lim_{x\rightarrow\infty}\frac{2+5n^2}{n^2}$

a)

0

b)

7

c)

5

d)

does not exist

5.

Evaluate $\lim_{x\rightarrow2}\ \left(x^3-x^2+2x+1\right)$

a)

7

b)

9

c)

12

d)

11

6.

Evaluate $\lim_{x\rightarrow1}\frac{x^2-1}{x-1}$

a)

-2

b)

0

c)

does not exist

d)

2

7.

$Evaluate\ \lim_{h\rightarrow0}\ \frac{h^3+2h^2-4h}{h}$

a)

0

b)

-4

c)

5

d)

does not exist

8.

Find the slope of the line tangent to the graph of y = x³ - 3 at (-2, -11).

a)

6

b)

12

c)

8

d)

9

9.

Find an equation for the slope of the graph of y = 10x − 2x².

a)

m = 10 - 4x

b)

m = -4x

c)

m = 10

d)

m = 6x

10.

Find an equation for the slope of the graph of y = x³ - 7

a)

m = 7

b)

m = 3x

c)

m = 3x²

d)

m = 3x - 7

11.

The position of an object in feet after t seconds is given by $s\left(t\right)=16t^2-\frac{1}{2}t^3$ . Find the average velocity of the object in feet per second for 2 ≤ t ≤ 6.

a)

468 ft/s

b)

60 ft/s

c)

408 ft/s

d)

98 ft/s

12.

Use the derivative rules to find the derivative of the function f(x) = (3x - 4)²

a)

f '(x) = 12x - 16

b)

f '(x) = 9

c)

f '(x) = 18x - 24

d)

f '(x) = 6x - 8

13.

Acceleration is the rate at which the velocity of a moving object changes. That is, acceleration is the derivative of velocity. If time is measured in seconds and velocity in feet per second, then acceleration is measured in feet per second squared. If a car's velocity is described by the function $v\left(t\right)=16+3t+\frac{1}{4}t^2$ , what is the car's acceleration at t = 4?

a)

32 feet per second squared

b)

18 feet per second squared

c)

37 feet per second squared

d)

5 feet per second squared

14.

Find the derivative of the function f (x) = 2x⁴ - 3x³ - 7x² + 3x - 7.

a)

f ' (x) = 2x³ - 3x² - 7x +3

b)

f ' (x) = 4x³ - 3x² - 2x +1

c)

f ' (x) = 8x³ - 9x² - 14x +3

d)

f ' (x) = 8x⁴ - 9x³ - 14x² +3x

15.

Andrés throws a ball straight upwards from a height of 1 meter with a velocity of 20 meters per second. Since the acceleration due to gravity on Earth is 9.8 meters per second squared, the function for the height of the ball at time t is $f\left(t\right)=-4.9t^2+20t+1$ . Find the height of the ball at its highest point, when its velocity is zero.

a)

21.41 meters

b)

20.41 meters

c)

2.04 meters

d)

4.13 meters

16.

Use limits to find the area of the shaded region in the graph.

a)

27 units²

b)

26 units²

c)

20 units²

d)

81/4 units²

17.

Evauate $\int_{-2}^1\ x^2dx$

a)

$\frac{x^3}{3}$

b)

2x

c)

2

d)

3

18.

Use limits to find the area between the graph of y = 1/2x³ and the x-axis from x = 0 to x = 4.

a)

32

b)

64

c)

128

d)

32/3

19.

Approximate the area of the shaded region for f(x) = −x² + 8x using 6 rectangles and right endpoints to determine the heights of the rectangles.

a)

$\approx$ 84 $units^2$

b)

$\approx$ 70 $units^2$

c)

$\approx$ 80 $units^2$

d)

$\approx$ 77 $units^2$

20.

Find the antiderivative of the function f(x) = 2x⁵.

a)

f(x) = x⁶

b)

f(x) = 1/3x⁶

c)

f(x) = 1/3x⁶ + C

d)

f(x) = 1/3x3 + C

21.

What is the antiderivative of the function $f\left(x\right)=x^2\left(x^2-1\right)$ ?

a)

$f\left(x\right)\ =\ \frac{1}{5}x^5+\frac{1}{3}x^3+C$

b)

$f\left(x\right)\ =\ \frac{1}{5}x^5-\frac{1}{3}x^3+C$

c)

$f\left(x\right)\ =\ \frac{1}{5}x^3-\frac{1}{3}x^5+C$

d)

$f\left(x\right)\ =\ \frac{1}{5}x^3+\frac{1}{3}x^5+C$

22.

Find the antiderivative of f(x) = x³ - 2x² + 4x - 5.

a)

$f\left(x\right)=3x^2-4x+4$

b)

$f\left(x\right)=\frac{1}{4}x^4-\frac{2}{3}x^3+2x^2+4x$

c)

$f\left(x\right)=\frac{1}{4}x^4-\frac{2}{3}x^3+2x^2-5x+C$

d)

$f\left(x\right)=x^4+\frac{1}{3}x^3+2x^2-\frac{5}{2}x+C$

23.

$Evaluate\ \int_1^5x^3dx$

a)

156

b)

41

c)

156.25

d)

41.33

Chapter 12: Limits and Derivatives

Total questions: 23

Worksheet time: 12mins

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