# The Derivative as a Rate of Change Worksheet

The Derivative as a Rate of Change Worksheet
• Page 1
1.
What is the average rate of change of $f$($x$) = with respect to $x$ over [39, 52]?
 a. - 13 b. - $\frac{1}{13}$ c. $\frac{1}{13}$ d. 13

#### Solution:

Here f(x) = x - 3, x changes from a = 39 to b = 52

The average rate of change of f(x) = f(b) - f(a)b - a
[Definition.]

= f(52) - f(39)49 - 36

= 7 - 613
[Simplify.]

= 1 / 13

2.
Find the average rate of change of $y$ = with respect to $x$ as $x$ changes from $a$ = 4 to $b$ = 12.
 a. $\frac{3}{640}$ b. - $\frac{3}{640}$ c. 3

#### Solution:

Here y = f(x) = 13x + 4, x changes from a = 4 to b = 12

The average rate of change of y = y(b) - y(a)b - a
[Definition.]

= y(12) - y(4)12 - 4

= 140 -1168
[Simplify.]

= ( - 3 / 640 )

= - 3 / 640

3.
What is the average rate of change of $\mathrm{f\left(x}$) = 5$x$ + 4 over [5, 6] ?
 a. - 12500 b. 3129 c. 15629 d. 12500

#### Solution:

Here f(x) = 5x + 4, x changes from a = 5 to b = 6.

The average rate of change of f(x) = f(b) - f(a)b - a
[Definition.]

= f(6) - f(5)6 - 5

= 15629  - 31291
[Simplify.]

= 12500

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

#### Solution:

Here f(x) = 3x2 + 4x + 4 , x changes from a = 0 to b = 3

The average rate of change of f(x) = f(b) - f(a)b - a
[Definition.]

= f(3) - f(0)3 - 0

= 43 - 43
[Simplify.]

= 39 / 3= 13

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

#### Solution:

Here f(x) = 2x3 - 3x2 + 4x + 2 , x changes from a = -1 to b = 1

The average rate of change of f(x) = f(b) - f(a)b - a
[Definition.]

= f(1) - f(-1)1 - (-1)

= (5) - (-7)2
[Simplify.]

= 6

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

#### Solution:

Here f(x) changes from - 10 to l as x changes from a = - 2 to b = 2 and the average rate of change of f(x) over [ - 2, 2 ] = 5

The average rate of change of f(x) = f(b) - f(a)b - a = 5
[Definition.]

l - (- 10)2 - (- 2) = 5
[Substitute f (b) = l, f (a) = - 10, a = - 2, b = 2.]

l + 10 = 20

l = 20 - 10 = 10
[Solve for l.]

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

#### Solution:

f(x) changes from k to 65 as x changes from a = k to b = k + 65 and the average rate of change of f(x) over [k, k + 65] = 8

The average rate of change of f(x) = f(b) - f(a)b - a = 8
[Definition.]

65 -  kk + 65 - k = 8
[Substitute f (b) = 65, f(a) = k, a = k, b = k + 65.]

65 -  k65 = 8

65 - k = 520
[Simplify.]

k = -455
[Solve for k.]

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

#### Solution:

Here f(x) changes from α to 4 as x changes from 8 to α and the average rate of change of f(x) with respect to x from 8 to α = 4

The average rate of change of f(x) = f(b) - f(a)b - a = 4
[Definition.]

4 - αα - 8 = 4
[Substitute f(b) = 4, f (a) = α, a = 8, b = α.]

4 - α = 4 α - 32
[Simplify.]

α = 36 / 5
[Solve for α.]

9.
What is the average rate of change of $f$($x$) = sin $x$ with respect to $x$ over [0, $\frac{9\pi }{2}$]?
 a. π b. $\frac{9\pi }{2}$ c. $\frac{1}{\pi }$ d. $\frac{2}{9\pi }$

#### Solution:

Here f(x) = sin x, x changes from a = 0 to b = / 2

The average rate of change of f(x) = f(b) - f(a)b - a
[Definition.]

= f(9π2) - f(0)9π2 - 0

= 1 - 09π2
[Simplify.]

= 2 /

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. $\frac{3}{\pi }$ b. $\frac{4}{\pi }$ c. $\frac{1}{\pi }$ d. $\frac{2}{\pi }$

#### Solution:

Here f(x) = cos x, x changes from a = 9π to b = 10π

The average rate of change of f(x) = f(b) - f(a)b - a
[Definition.]

= f(10π) - f(9π)10π - 9π

= 1 - (- 1)π
[Simplify.]

= 2 / π