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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) = x - 3 with respect to x over [39, 52]?
a.
- 13
b.
- 1 13
c.
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


Correct answer : (3)
 2.  
Find the average rate of change of y = 13x + 4 with respect to x as x changes from a = 4 to b = 12.
a.
3 640
b.
- 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


Correct answer : (2)
 3.  
What is the average rate of change of f(x) = 5x + 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


Correct answer : (4)
 4.  
What is the average rate of change of f(x) = 3x2 + 4x + 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


Correct answer : (3)
 5.  
Find the average rate of change of f(x) = 2x3 - 3x2 + 4x + 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


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


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.]


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


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.]


Correct answer : (4)
 8.  
f(x) changes from α to 4 as x changes from 8 to α . If the average rate of change of f(x) with respect to x as x changes from 8 to α is 4, then find the value of α.
a.
- 36 5
b.
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.]


Correct answer : (2)
 9.  
What is the average rate of change of f(x) = sin x with respect to x over [0, 2]?
a.
π
b.
2
c.
1 π
d.
2


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 /


Correct answer : (4)
 10.  
What is the average rate of change of cos x with respect to x as x changes from 9π to 10π?
a.
3 π
b.
4 π
c.
1 π
d.
2 π


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


Correct answer : (4)

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