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Average Value of a Function Worksheet

Average Value of a Function Worksheet
  • Page 1
 1.  
The average value of f(x) = x4 on [1, 5] is:
a.
781
b.
624.8
c.
124
d.
156.2


Solution:

y = f(x) = x4

The average value of y on [1, 5] = y = 1 / (5-1) 15 x4 dx
[Definition.]

= 1 / 4 (x55) 1 5

= 1 / 20 (55 - 15)

= 3124 / 20 = 156.2


Correct answer : (4)
 2.  
The average value of f(x)= 7x6 - 6x5 + 4x on [0, 5] is:
a.
62502
b.
62550
c.
90645
d.
12510


Solution:

y = f(x) = 7x6 - 6x5 + 4x

The average value of y on [0, 5]

= y = 1 / (5-0)05(7x6 - 6x5 + 4x) dx
[Definition.]

= 1 / 5 [x7 - x6 + 2x2 ] 0 5

= 1 / 5 [(78125 - 15625 + 50) - (0 - 0 + 0)] = 12510
[Simplify.]


Correct answer : (4)
 3.  
What is the average value of f(x) = x5 on [- 32, - 1]?
a.
105 62
b.
- 63 31
c.
- 105 62
d.
325 186


Solution:

y = f(x) = x5= x15

The average value of y on [- 32, - 1]

= y = 1 / ((-1)-(-32))- 32 - 1 x15 dx
[Definition.]

= 1 / 31({5} / 6x65 ) - 32 - 1

= 5 / 186[(-1)65 - (-32)65]

= 5 / 186(1 - 64)

= - 105 / 62.


Correct answer : (3)
 4.  
Find the average value of g(x) = /ex on [0, 7].
a.
(1 7) e7
b.
(1 7) (e7 + 1)
c.
1 56(e8-8)
d.
(1 7) (/e7 - 1)


Solution:

y = g(x) = /ex

The average value of /ex on [0, 7]

= y = (1 / 7-0) 07 ex dx
[Definition.]

= (1 / 7) (ex) 0 7

= (1 / 7) (e7 - e0)

= (1 / 7) (e7 - 1)


Correct answer : (4)
 5.  
What is the average value of h(x) = 3x on [0, 3]?
a.
26 3(1 ln3)
b.
26 3(1 log3)
c.
26 3ln3
d.
- 26 3


Solution:

y = h(x) = 3x

The average value of y on [0, 3]

= y = 1 / (3-0) 033x dx
[Definition.]

= 1 / 3 × (3xln 3) 0 3

= 1 / 3ln3 (33 - 30)

= 26 / 3(1 / ln3)
[Simplify.]


Correct answer : (1)
 6.  
The average value of f(x)= x + 3 on [6, 13] is:
a.
74 21
b.
2 21
c.
37
d.
1 21


Solution:

y = f(x)= x + 3 = (x + 3)12

The average value of y on [6, 13]

= y = 1 / (13-6) 613 (x+3)12 dx
[Definition.]

= 1 / 7 [2 / 3(x + 3)32 ] 6 13

= (1 / 7) (2 / 3) [(13 + 3)32 - (6 + 3)32 ]

= (2 / 21) (64 - 27)

= 74 / 21


Correct answer : (1)
 7.  
The average value of g(x) = sin 11x on [0, π2] is
a.
(11π2)
b.
- 2π
c.
(- 211π)
d.
(211π)


Solution:

y = g(x) = sin 11x

The average value of y on [0, π2]

y = 1(π2-0) 0 π/2 sin 11x dx
[Definition.]

= (2π) [- cos 11x11 ] 0 π/2

= (- 211π) (0 - 1)

= (211π)


Correct answer : (4)
 8.  
The average value of h(x) = cos 9x on [0, π] is
a.
19π
b.
1π
c.
1


Solution:

y = h(x) = cos 9x

The average value of y on [0, π]

= y = 1(π- 0) 0 π cos 9x dx
[Definition.]

= 1π [sin 9x9] 0 π

= 19π (0 - 0)

= 0


Correct answer : (4)
 9.  
What is the average value of the function f(x)= 5sin 22πx on [- 1 2, 1 2]?
a.
- 122π
b.
122π
c.
111π


Solution:

y = f(x)= 5sin 22πx

The average value of f(x) on [ - 1 / 2, 1 / 2]

= y = 112-(-12) 5- ½ ½ sin 22πx dx
[Definition.]

= 5[- cos 22πx22π] - 1/2 1/2

= (- 522π) [cos 11π - cos (- 11π)]

= 5 / 22(- 1π)(- 1 + 1)

= 0


Correct answer : (3)
 10.  
The average value of g(x) = 21sec2 9x on [- π36, π36]:
a.
1 84(1π)
b.
84 (1π)
c.
1 42(1π)
d.
42 (1π)


Solution:

y = g(x) = 21sec2 9x

The average value of y on [- π36, π36]

= y = 1π36-(-π36) 21-π/36 π/36sec2 9x dx
[Definition.]

= 378 (1π) [tan 9x9]- π/36 π/36

= 42 (1π) [tan (9π36) - tan (- 9π36)]

= 42 (1π) [1 - (- 1)]

= 42 (1π) (2)

= 84 (1π)


Correct answer : (2)

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