Q. 67 - Calculus

Question

Prove Theorem 4.34: If $f$ is continuous on $\left[a,b\right]$, then for all $x\in [a,b]$, $\frac{d}{dx}{\int }_a^{x}f\left(t\right)dt=f\left(x\right)$. The proof follows directly from the Second Fundamental Theorem of Calculus.

Step-by-Step Solution

Answer

If $f$ is continuous on $\left[a,b\right]$, then for all $x\in \left[a,b\right]$, $\frac{d}{dx}{\int }_a^{x}f\left(t\right)dt=f\left(x\right)$.

Step 1. Given information

We have to prove that if $f$ is continuous on $\left[a,b\right]$, $\frac{d}{dx}{\int }_a^{x}f\left(t\right)dt=f\left(x\right)$.

It is supposed that $f\text{'}=F$.

Step 2. Proof of the given question.

$$\begin{gathered} \frac{d}{dx}{\int }_a^{x}f\left(t\right)dt \\ =\frac{d}{dx}{\left[F\left(t\right)\right]}_a^x \\ =\frac{d}{dx}\left[F\left(x\right)-F\left(a\right)\right] \\ =F\text{'}\left(x\right)-0 \\ =f\left(x\right) \end{gathered}$$

Therefore, $\frac{d}{dx}{\int }_a^{x}f\left(t\right)dt=f\left(x\right)$ is proved.