Chapter 5.4, Problem 5E

(a)

To determine

To calculate: The $g^{\prime }\left(x\right)$, using part 1 of fundamental theorem of calculus.

Answer to Problem 5E

The $g^{\prime }\left(x\right)$ value is $\underset{\_}{x^2}$ by using part 1 of fundamental theorem of calculus method.

Explanation of Solution

Given information:

$g\left(x\right)={\displaystyle \underset{1}{\overset{x}{\int }}{t}^{2}dt}$

Calculation:

Show the integral function as shown below:

$g\left(x\right)={\displaystyle \underset{1}{\overset{x}{\int }}{t}^{2}dt}$ (1)

Here, $g\left(x\right)$ is function of x andt is derivative function.

Draw the area represented by $g\left(x\right)$ as shown below;

Show the part 1 of fundamental theorem of calculus as below:

$g\left(x\right)={\displaystyle \underset{a}{\overset{x}{\int }}f\left(t\right)dt}$ (2)

$g^{\prime }\left(x\right)=\frac{d}{dx}{\displaystyle \underset{a}{\overset{x}{\int }}f\left(t\right)dt}$ (3)

Comparing Equation (1), (2) and (3).

$\begin{array}{c}g\left(x\right)={\displaystyle \underset{1}{\overset{x}{\int }}{t}^{2}dt}\\ g^{\prime }\left(x\right)=f\left(x\right)\\ =x^2.\end{array}$

Therefore, we get the value of $g^{\prime }\left(x\right)$ as $\underset{\_}{x^2}$ by using the part 1 of fundamental theorem of calculus.

(b)

To determine

The value for $g^{\prime }\left(x\right)$ by using the integral evaluation method.

Answer to Problem 5E

The $g^{\prime }\left(x\right)$ value is $\underset{\_}{x^2}$ by using the integral evaluation method.

Explanation of Solution

Given information:

$g\left(x\right)={\displaystyle \underset{1}{\overset{x}{\int }}{t}^{2}dt}$

Calculation:

$\begin{array}{c}g\left(x\right)={\displaystyle \underset{1}{\overset{x}{\int }}{t}^{2}dt}\\ ={\left[\frac{t^3}{3}\right]}_1^x\\ =\frac{x^3}{3}-\frac{1}{3}\end{array}$

Differentiate with respect to x.

$\begin{array}{c}g\left(x\right)=\frac{x^3}{3}-\frac{1}{3}\\ g^{\prime }\left(x\right)=\frac{1}{3}\left(3x^2\right)\\ =x^2\end{array}$

Therefore, get the value of $g^{\prime }\left(x\right)$ as $\underset{\_}{x^2}$ by using the integral evaluation method.