Chapter 3.4, Problem 47E

a.

To determine

To calculate:Thederivatives of three functions.

Answer to Problem 47E

Derivatives of three given function is same that is $g^{\prime }\left(x\right)=h^{\prime }\left(x\right)=t^{\prime }\left(x\right)=3x^2$ .

Explanation of Solution

Given information:

Three functions for which derivative needs to be determine are as follows

1. $g\left(x\right)=x^3$ .
2. $h\left(x\right)=x^3-2$
3. $t\left(x\right)=x^3+3$

Formula used:

1. $\frac{d}{dx}c=0$ where c is constant function.
2. $\frac{d}{dx}{x}^n=n{x}^{n-1}$

Calculation:

Derivative of $g\left(x\right)=x^3$

  $g^{\prime }\left(x\right)=\frac{d}{dx}{x}^3$

  $=3x^2$

Derivative of $h\left(x\right)=x^3-2$

  $h^{\prime }\left(x\right)=\frac{d}{dx}\left(x^3-2\right)$

  $=\frac{d}{dx}{x}^3-\frac{d}{dx}2$

  $=3x^2-0$

  $=3x^2$

Derivative of $t\left(x\right)=x^3+3$

  $t^{\prime }\left(x\right)=\frac{d}{dx}\left(x^3+3\right)$

  $=\frac{d}{dx}{x}^3+\frac{d}{dx}3$

  $=3x^2+0$

  $=3x^2$

Hence derivative of three given function is same that is $g^{\prime }\left(x\right)=h^{\prime }\left(x\right)=t^{\prime }\left(x\right)=\underset{\_}{3x^2}$ .

b.

To determine

To graph:The derivative of three given function.

Explanation of Solution

Given information:

1. Three functions for which derivative needs to be determine are as follows
1. $g\left(x\right)=x^3$ .
2. $h\left(x\right)=x^3-2$
3. $t\left(x\right)=x^3+3$
2. Derivative of three functions are same that is $g^{\prime }\left(x\right)=h^{\prime }\left(x\right)=t^{\prime }\left(x\right)=3x^2$

Graph:

The graph of $f\left(x\right)=3x^2$ which is derivative of $g\left(x\right)$ , $h\left(x\right)$ and $t\left(x\right)$ is as follows

  

c.

To determine

To find:The family of the function when derivative of the function is provided.

Answer to Problem 47E

Required family function $f\left(x\right)=x^3+c$ .

Explanation of Solution

Given information:

1. Derivative of the function $f\left(x\right)$ that is $f^{\prime }\left(x\right)=3x^2$ .
2. Need to determine family of $f\left(x\right)$ .
3. $g^{\prime }\left(x\right)=h^{\prime }\left(x\right)=t^{\prime }\left(x\right)=3x^2$

Calculation:

Observe the three given function in $g\left(x\right)=x^3,h\left(x\right)=x^3-2\text{and}t\left(x\right)=x^3+3$ .

As $g^{\prime }\left(x\right)=h^{\prime }\left(x\right)=t^{\prime }\left(x\right)=3x^2$ so need to identify family function $f\left(x\right)$ of $g\left(x\right),h\left(x\right)\text{and}t\left(x\right)$ .

Family function $f\left(x\right)$ is as follows

  $f\left(x\right)=x^3+c$ where $c$ is any constant .

  $f^{\prime }\left(x\right)=\frac{d}{dx}{x}^3+\frac{d}{dx}c$

  $=3x^2+0$

  $=3x^2$

Hence it can be concluded that required family function $\underset{\_}{f\left(x\right)=x^3+c}$ .

d.

To determine

To find:The function of given derivative and condition.

Answer to Problem 47E

There exist a function whose derivative is $3x^2$ and $f\left(0\right)=0$ and the function is $\underset{\_}{f\left(x\right)=x^3}$ .

Explanation of Solution

Given information:

1. Derivative of the function $f\left(x\right)$ that is $f^{\prime }\left(x\right)=3x^2$ .
2. $f\left(0\right)=0$ .
3. Evaluated that family function when $f^{\prime }\left(x\right)=3x^2$ is $f\left(x\right)=x^3+c$

Calculation:

Consider the family function $f\left(x\right)=x^3+c$ having $f^{\prime }\left(x\right)=3x^2$

  $f\left(x\right)=x^3+c$

  $f\left(0\right)=0^3+c$

  $f\left(0\right)=c$

Required function must have $f\left(0\right)=0$

That is $c=0$

Substitute $c=0$ in family function.

  $f\left(x\right)=x^3+0$

  $=x^3$

Hence it can be concluded that there exist a function whose derivative is $3x^2$ and $f\left(0\right)=0$ and the function is $\underset{\_}{f\left(x\right)=x^3}$ .

e.

To determine

To find:The function of given derivative and condition.

Answer to Problem 47E

There exist a function whose derivative is $3x^2$ and $f\left(0\right)=3$ and the function is $\underset{\_}{f\left(x\right)=x^3+3}$ .

Explanation of Solution

Given information:

1. Derivative of the function $f\left(x\right)$ that is $f^{\prime }\left(x\right)=3x^2$ .
2. $f\left(0\right)=3$ .
3. Evaluated that family function when $f^{\prime }\left(x\right)=3x^2$ is $f\left(x\right)=x^3+c$

Calculation:

Consider the family function $f\left(x\right)=x^3+c$ having $f^{\prime }\left(x\right)=3x^2$

  $f\left(x\right)=x^3+c$

  $f\left(0\right)=0^3+c$

  $f\left(0\right)=c$

Required function must have $f\left(0\right)=3$

That is $c=3$

Substitute $c=3$ in family function.

  $f\left(x\right)=x^3+3$

Hence it can be concluded that there exist a function whose derivative is $3x^2$ and $f\left(0\right)=3$ and the function is $\underset{\_}{f\left(x\right)=x^3+3}$ .