Q. 1 - Calculus

Question

Consider the differential equation $f\text{'}\left(x\right)=g\left(x\right)$. What has to be true about $g\left(x\right)$ for the Second Fundamental Theorem of Calculus to guarantee that a function $f$ exists that satisfies $f\text{'}\left(x\right)=g\left(x\right)$?

Step-by-Step Solution

Answer

For, $f\text{'}\left(x\right)=g\left(x\right)$ to be true two conditions need to be satisfied:

$F$is continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$.

$F$is an antiderivative of $f$, that is, $F\text{'}\left(x\right)=f\left(x\right)$.

Step 1. Given Information.

The objective to be true is to explain the second fundamental theorem of calculus.

Step 2. The conditions.

For, $f\text{'}\left(x\right)=g\left(x\right)$ to be true two conditions need to be satisfied:

$F$is continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$.

$F$is an antiderivative of $f$ that is, $F\text{'}\left(x\right)=f\left(x\right)$.