Q. 53 - Calculus

Question

Find a function f that has the given derivative f  and value f(c). Find an antiderivative of f  by hand, if

possible; if it is not possible to anti differentiate by hand, use the Second Fundamental Theorem of Calculus to write down an antiderivative.

$f^{\text{'}}\left(x\right)=e^{-x^2}, f\left(1\right)=0 .$

Step-by-Step Solution

Answer

The function $f\left(x\right)={\int }_1^x{e}^{-t^2}dt$ .

Step 1.Given Information

Given function $f^{\text{'}}\left(x\right)=e^{-x^2}, f\left(1\right)=0 .$

Step 2.The formula

The antiderivative of f is $$\begin{gathered} F\left(x\right)={\int }_a^{x}f\left(t\right)dt \\ ={\int }_a^x{e}^{-t^2}dt ;x\in [a,b] \end{gathered}$$

Using the fact $f\left(1\right)=0$ the antiderivative is,

$f\left(x\right)={\int }_1^x{e}^{-t^2}dt$ .

Step 3.The solution

The function $f\left(x\right)={\int }_1^x{e}^{-t^2}dt$ .