Answer:
C. [tex]\frac{f(b)-f(1)}{b-1}=20[/tex]
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 [tex]f'(c)=\frac{f(b)-f(a)}{b-a}[/tex]
- MVT is also Average Value
Step-by-step explanation:
Step 1: Define
[tex]f(x)=e^{2x}[/tex]
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 [tex]f'(c)=\frac{f(b)-f(a)}{b-a}[/tex]
Step 3: Mean Value Theorem
- Substitute: [tex]20=\frac{ f(b)-f(1)}{b-1}[/tex]
- Rewrite: [tex]\frac{ f(b)-f(1)}{b-1}=20[/tex]
And we have our final answer!
