Question

Consider the function f(x) = 6 - 7x ^ 2 on the interval [- 6, 7] Find the average or mean slope of the function on this interval , (7)-f(-6) 7-(-6) = boxed |

Answer 1

• Mean Slope = -7

,

• c=0.5

Explanation:

Given the function:

`f\mleft(x\mright)=6-7x^2`

Part A

We want to find the mean slope on the interval [-6, 7].

First, evaluate f(7) and f(-6):

`\begin{gathered} f(7)=6-7(7^2)=6-7(49)=6-343=-337 \\ f(-6)=6-7(-6)^2=6-7(36)=6-252=-246 \end{gathered}`

Next, substitute these values into the formula for the mean slope.

`\begin{gathered} \text{ Mean Slope}=(f(7)-f(-6))/(7-(-6))=(-337-(-246))/(7+6)=(-337+246)/(13) \\ =-(91)/(13) \\ =-7 \end{gathered}`

The mean slope of the function over the interval [-6,7] is -7.

Part B

Given the function, f(x):

`f\mleft(x\mright)=6-7x^2`

Its derivative, f'(x) will be:

`f^(\prime)(x)=-14x`

Replace c for x:

`f^(\prime)(c)=-14c`

Equate f'(c) to the mean slope obtained in part a.

`-14c=-7`

Solve for c:

`\begin{gathered} (-14c)/(-14)=(-7)/(-14) \\ c=0.5 \end{gathered}`

The value of c that satisfies the mean value theorem is 0.5.