Question

If\(f'\)is continuous on \((1,3)\), then \(\int\limits_1^3 {f'(v)dv = f(3) - f(1)} \).

Short answer

The answer is TRUE

Step-by-step solution

Step 1: The Definitions of The Evaluation Theorem And The Net Change Theorem. 

Evaluation theorem states that:\(\int\limits_a^b {f(x)dx} = F(b) - F(a)\)

Net Change Theorem states that:\(\int\limits_a^b {F'(x)dx} = F(b) - F(a)\)

\(F(b) - F(a)\) Is The Net Change Of Function y In The Interval (a, b)

 Step 2: From Above Definitions the Final Answer

From the Evaluation Theorem and the Net Change Theorem, we know that the integral of the rate of change is simply the net change.

In this case:

Is simply a use case of the Theorem and since the rate of change function is continuous on and is the net change, then the theorems hold and the given statement is TRUE.