# If f(4) = 12, f ' is continuous, and 5 f '(x) dx 4 = 18, what is the value of f(5)?

## Question:

If f(4) = 12, f ' is continuous, and 5 f '(x) dx 4 = 18, what is the value of f(5)?

## Fundamental Theorem of Calculus:

The fundamental theorem of calculus condenses the concepts of finding the derivative of a function and integrating that function. Mathematically, it implies the existence of the anti-derivatives for the continuous function. From the fundamental theorem of calculus, we have:

{eq}\Rightarrow \displaystyle\int_{a}^{b}f'(x)=f(b)-f(a) {/eq}

As per the given piece of information, we have:

{eq}f(4)=12 {/eq}, {eq}f'(x) {/eq} is continuous and {eq}\displaystyle\int_{4}^{5}f'(x)=18 {/eq}

To find: {eq}f(5) {/eq}.

Now, from the fundamental theorem of calculus, we have:

{eq}\Rightarrow \displaystyle\int_{a}^{b}f'(x)=f(b)-f(a) {/eq}

{eq}\Rightarrow f(5)-f(4)=18\\\Rightarrow f(5)=18+f(4)=18+12=30\\\Rightarrow f(5)=30. {/eq}