The Derivative as a Rate of Change Worksheet

**Page 1**

1.

What is the average rate of change of $f$($x$) = $\sqrt{x-3}$ with respect to $x$ over [39, 52]?

a. | - 13 | ||

b. | - $\frac{1}{13}$ | ||

c. | $\frac{1}{13}$ | ||

d. | 13 |

The average rate of change of

[Definition.]

=

=

[Simplify.]

=

Correct answer : (3)

2.

Find the average rate of change of $y$ = $\frac{1}{3x+4}$ with respect to $x$ as $x$ changes from $a$ = 4 to $b$ = 12.

a. | $\frac{3}{640}$ | ||

b. | - $\frac{3}{640}$ | ||

c. | 3 |

The average rate of change of

[Definition.]

=

=

[Simplify.]

= ( -

= -

Correct answer : (2)

3.

What is the average rate of change of $\mathrm{f(x}$) = 5^{$x$} + 4 over [5, 6] ?

a. | - 12500 | ||

b. | 3129 | ||

c. | 15629 | ||

d. | 12500 |

The average rate of change of

[Definition.]

=

=

[Simplify.]

= 12500

Correct answer : (4)

4.

What is the average rate of change of $f$($x$) = 3$x$^{2} + 4$x$ + 4 with respect to $x$ as $x$ changes from 0 to 3?

a. | - 13 | ||

b. | 39 | ||

c. | 13 | ||

d. | 12 |

The average rate of change of

[Definition.]

=

=

[Simplify.]

=

Correct answer : (3)

5.

Find the average rate of change of $f$($x$) = 2$x$^{3} - 3$x$^{2} + 4$x$ + 2 between $x$ = - 1 and $x$ = 1.

a. | - 12 | ||

b. | - 6 | ||

c. | 6 | ||

d. | 12 |

The average rate of change of

[Definition.]

=

=

[Simplify.]

= 6

Correct answer : (3)

6.

$f$($x$) changes from - 10 to $l$ as $x$ changes from - 2 to 2. If the average rate of change of $f$($x$) with respect to $x$ over [ - 2, 2 ] is 5, then what is the value of $l$?

a. | 10 | ||

b. | - 10 | ||

c. | - 20 | ||

d. | 20 |

The average rate of change of

[Definition.]

[Substitute

[Solve for

Correct answer : (1)

7.

$f$($x$) changes from $k$ to 65 as $x$ changes from $k$ to $k$ + 65. If the average rate of change of $f$($x$) with respect to $x$ over [$k$, $k$ + 65] is 8, then find the value of $k$.

a. | 65 | ||

b. | 520 | ||

c. | 8 | ||

d. | -455 |

The average rate of change of

[Definition.]

[Substitute

[Simplify.]

[Solve for

Correct answer : (4)

8.

$f$($x$) changes from $\alpha $ to 4 as $x$ changes from 8 to $\alpha $ . If the average rate of change of $f$($x$) with respect to $x$ as $x$ changes from 8 to $\alpha $ is 4, then find the value of $\alpha $.

a. | - $\frac{36}{5}$ | ||

b. | $\frac{36}{5}$ | ||

c. | 36 | ||

d. | 5 |

The average rate of change of

[Definition.]

[Substitute

4 -

[Simplify.]

[Solve for

Correct answer : (2)

9.

What is the average rate of change of $f$($x$) = sin $x$ with respect to $x$ over [0, $\frac{\mathrm{9\pi}}{2}$]?

a. | π | ||

b. | $\frac{\mathrm{9\pi}}{2}$ | ||

c. | $\frac{1}{\mathrm{\pi}}$ | ||

d. | $\frac{2}{\mathrm{9\pi}}$ |

The average rate of change of

[Definition.]

=

=

[Simplify.]

=

Correct answer : (4)

10.

What is the average rate of change of cos $x$ with respect to $x$ as $x$ changes from 9$\pi $ to 10$\pi $?

a. | $\frac{3}{\mathrm{\pi}}$ | ||

b. | $\frac{4}{\mathrm{\pi}}$ | ||

c. | $\frac{1}{\mathrm{\pi}}$ | ||

d. | $\frac{2}{\mathrm{\pi}}$ |

The average rate of change of

[Definition.]

=

=

[Simplify.]

=

Correct answer : (4)