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

Question

Consider the function f(x)=2√(x)+1 on the interval [2,7]. Find the average or mean slope of the function on this interval.
By the mean value theorem, we know there exists a c in the open interval (2,7) such that f^'(c) is equal to this mean slope. There is only one c that works for this problem, what is it?

Answer 1

Find f'(x) first  as 2 times one half divided by square root of x or one divided by square root of x. Then f'(c) according to mean value theorem is f(7)-f(2) all divided by 7-2. In this case: 2 times square root of seven minus 2 times square root of two , all divided by five. Complete the conjugate, multiplying top and bottom by square root of smini plus square root of two (with a constant of two) and you get two times five divided by five times (square root of seven plus square root of two). The fives cancel out. Punch the rest in a calculator. We will call that m. 

 

On the other side plug in c for f'(x) or 1/square root of c. 

 

Bring c to the other side. Now m^2*c=1. Solve for c.