Question

Consider the function `f(x) = 1/x` on the interval `[4, 11]` and answer the following?

Consider the function `f(x) = 1/x` on the interval `[4, 11]`.

(A) Find the average or mean slope of the function on this interval.

(B) By the Mean Value Theorem, we know there exists a `c` in the open interval `(4, 11)` such that `f'(c)` is equal to this mean slope. Find all values of `c` that work and list them (separated by commas) in a box.

Answer 1

See below.

Explanation:

A)

The Average Rate of change over an interval #[a,b] is given by:

`(f(b)-f(a))/(b-a)`

From example:

`(f(b)-f(a))/(b-a)=((1/11)-(1/4))/(11-4)=(-7/44)/7=color(blue)(-1/44)`

B)

The mean value theorem:

`f^'(c)=(f(b)-f(a))/(b-a)`

We already have the average rate of change `color(blue)(-1/44)`

So we need to differentiate `f(x)=1/x`, and then solve:

`f^'(x)=-1/44`

`dy/dx(1/x)=-x^-2=-1/x^2`

`:.`

`-1/x^2=-1/44`

`1/x^2=1/44`

`x^2=44=>x=+-sqrt(44)=+-2sqrt(11)~~+-6.6332`

For interval `(4,11)`

`color(blue)(c=2sqrt(11))`