Find the average value of the function f(x) = 8 x - 4 x^2 on the interval [0,5]

Question

Find the average value of the function f(x) = 8 x - 4 x^2 on the interval [0,5] and determine a number c in this interval for which f(x) is equal to the average value. a) Average value = -200/3, c = 1 + root 39/3 b) Average value = 0, c = 0 c) Average value= -40/3, c = 1 - root 39/3 c) Average value= -40/3, c = 1 + root 39/3 e) Average value= -200/3, c = 0 Question 4 Find the avenge value of the function f(x) = sin(x) on the interval [pi/2, 3pi/2] and determine a number c in this interval for which f(X) is equal to the average value. a) Average value = 1, c = pi/2 b) Average value= 1/2, c = 0 c) Average value = 0, c = pi d) Average value= 1/2, c = pi e) Average value= 0, c = 3pi/

Explanation / Answer

3)

1/(5-0) * (int from 0 to 5) 8x - 4x^2 * dx

(1/5)*(int from 0 to 5) 4x^2 - 4x^3/3

(1/5)*(4*5^2 - 4*5^3/3)

20 - 100/3

-40/3

Average = -40/3

8x - 4x^2 = -40/3
8c - 4c^2 = -40/3

c = 1 + sqrt(39) / 3

So, option D

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1/(3pi/2 - pi/2) * (int from pi/2 to 3pi/2) sinxdx

1/pi * (int from pi/2 to 3pi/2) * -cosx

0

Average = 0

sinx = 0
x = sininmverse(0)
x = pi

Option C

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