Question

Ap calc multiple choice question; attached below
please explain how, please!

Answer 1

C.
`(f(b)-f(1))/(b-1)=20`

General Formulas and Concepts:

Calculus

• Mean Value Theorem (MVT) - If f is continuous on interval [a, b], then there is a c∈[a, b] such that
  `f'(c)=(f(b)-f(a))/(b-a)`

• MVT is also Average Value

Explanation:

Step 1: Define

`f(x)=e^(2x)`

f'(c) = 20

Interval [1, b]

Step 2: Check/Identify

Function [1, b] is continuous.

Derivative [1, b] is continuous.

∴ There exists a c∈[1, b] such that
`f'(c)=(f(b)-f(a))/(b-a)`

Step 3: Mean Value Theorem

1. Substitute:
   `20=( f(b)-f(1))/(b-1)`
2. Rewrite:
   `( f(b)-f(1))/(b-1)=20`

And we have our final answer!