﻿ Average Value of a Function Worksheet | Problems & Solutions

Average Value of a Function Worksheet

Average Value of a Function Worksheet
• Page 1
1.
The average value of $f$($x$) = $x$4 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

2.
The average value of $f$($x$)= 7$x$6 - 6$x$5 + 4$x$ 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.]

3.
What is the average value of $f$($x$) = $\sqrt[5]{x}$ on [- 32, - 1]?
 a. $\frac{105}{62}$ b. - $\frac{63}{31}$ c. - $\frac{105}{62}$ d. $\frac{325}{186}$

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.

4.
Find the average value of $g$($x$) = $/{e}^{x}$ on [0, 7].
 a. ($\frac{1}{7}$) ${e}^{7}$ b. ($\frac{1}{7}$) (${e}^{7}$ + 1) c. $\frac{1}{56}$(${e}^{8}-8$) d. ($\frac{1}{7}$) ($/{e}^{7}$ - 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)

5.
What is the average value of $h$($x$) = ${3}^{x}$ on [0, 3]?
 a. $\frac{26}{3}$($\frac{1}{ln3}$) b. $\frac{26}{3}$($\frac{1}{log3}$) c. $\frac{26}{3}$ln3 d. - $\frac{26}{3}$

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.]

6.
The average value of $f$($x$)= on [6, 13] is:
 a. $\frac{74}{21}$ b. $\frac{2}{21}$ c. 37 d. $\frac{1}{21}$

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

7.
The average value of $g$($x$) = sin 11$x$ on [0, $\frac{\pi }{2}$] is
 a. ($\frac{11\pi }{2}$) b. - $\frac{2}{\pi }$ c. (- $\frac{2}{11\pi }$) d. ($\frac{2}{11\pi }$)

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)

8.
The average value of $h$($x$) = cos 9$x$ on [0, $\pi$] is
 a. $\frac{1}{9\pi }$ b. $\frac{1}{\pi }$ 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

9.
What is the average value of the function $f$($x$)= 5sin 22$\pi$$x$ on [- $\frac{1}{2}$, $\frac{1}{2}$]?
 a. - $\frac{1}{22\pi }$ b. $\frac{1}{22\pi }$ c. $\frac{1}{11\pi }$

Solution:

y = f(x)= 5sin 22πx

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

= y = 112-(-12) 5- Ã‚Â½ Ã‚Â½ sin 22πx dx
[Definition.]

= 5[- cos 22πx22π] - 1/2 1/2

= (- 522π) [cos 11π - cos (- 11π)]

= 5 / 22(- 1π)(- 1 + 1)

= 0

10.
The average value of $g$($x$) = 21 on [- $\frac{\pi }{36}$, $\frac{\pi }{36}$]:
 a. $\frac{1}{84}$($\frac{1}{\pi }$) b. 84 ($\frac{1}{\pi }$) c. $\frac{1}{42}$($\frac{1}{\pi }$) d. 42 ($\frac{1}{\pi }$)

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)