Question

4.4 the mean value theorem: problem 3
(6 points)
let ( f(x)=4 x^{2}+2 x - 8 ). answer the following questions.

1. find the average slope of the function ( f ) on the interval ( -1,1 ).

average slope: ( overline{m}= )

1. verify the mean value theorem by finding a number ( c ) in ( (-1,1) ) such that ( f^{prime}(c)=overline{m} ).

answer: ( c= )
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Explanation:

Step1: Calculate the average slope

The formula for the average slope (average rate of change) of a function \(y = f(x)\) on the interval \([a,b]\) is \(\overline{m}=\frac{f(b)-f(a)}{b - a}\).
Here, \(a=-1\), \(b = 1\), and \(f(x)=4x^{2}+2x - 8\).
First, find \(f(-1)\):
\(f(-1)=4\times(-1)^{2}+2\times(-1)-8=4 - 2-8=-6\).
Then, find \(f(1)\):
\(f(1)=4\times(1)^{2}+2\times(1)-8=4 + 2-8=-2\).
Now, calculate \(\overline{m}\):
\(\overline{m}=\frac{f(1)-f(-1)}{1-(-1)}=\frac{-2-(-6)}{2}=\frac{-2 + 6}{2}=\frac{4}{2}=2\).

Step2: Find the derivative of \(f(x)\)

Differentiate \(f(x)=4x^{2}+2x - 8\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).
\(f^\prime(x)=\frac{d}{dx}(4x^{2}+2x - 8)=4\times2x+2=8x + 2\).

Step3: Solve \(f^\prime(c)=\overline{m}\)

Set \(f^\prime(c)=8c + 2\) equal to \(\overline{m} = 2\).
\(8c+2 = 2\).
Subtract \(2\) from both sides: \(8c=2 - 2=0\).
Divide both sides by \(8\): \(c = 0\).

Answer:

1. Average Slope: \(\overline{m}=2\)
2. \(c = 0\)