Chapter 12.3, Problem 46E

a.

To determine

Find the slope of the graph.

Answer to Problem 46E

  $\boxed{10}$

Explanation of Solution

Given information:

Find the slope of the graph of $f$ at a given point.

  $f(x)=x^3-x$ , $(2,6)$

Calculation:

Consider the function, $f(x)=x^3-x$

The slope of the graph $f(x)$ of at a point $(x,y)$

  $f\text{'}(x)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f(x)}{h}$

So, the slope of the graph $f(x)=x^3-x$ is,

  $f\text{'}(x)=\underset{h\to 0}{\lim }\frac{{\left(x+h\right)}^3-\left(x+h\right)-x^3+x}{h}$

Use mathematical identity ${\left(a+b\right)}^3=a^3+3a^{2}b+3a{b}^2+b^3$

  $\begin{array}{l}f\text{'}(x)=\underset{h\to 0}{\lim }\frac{x^3-h^3-3x{h}^2-3x{h}^2-x-h+x^3+x}{h}\\ =\underset{h\to 0}{\lim }(h^2-3x^2+3xh-1)\\ =3x^2-1\end{array}$

At point $(2,6)$ , the slope will be ,

  $\begin{array}{l}f\text{'}\left(2\right)=12-2\\ =10\end{array}$

Hence, the slope of the graph $f(x)=x^3-x$ at $(2,6)$ is $\boxed{10}$ .

b.

To determine

Find an equaion of the tangent line to the graph at the point.

Answer to Problem 46E

  $\boxed{y=10x-14}$

Explanation of Solution

Given information:

Find an equaion of the tangent line to the graph at the point.

  $f(x)=x^3-x$ , $(2,6)$

Calculation:

Since the equation of a standard line is given by,

  $y=mx+c$

Where $m$ is the slope of the line and $c$ is the $y$ intercept made by the line.

Slope of $f(x)=x^3-x$ is $10$ .

So,

Since the equation of the line is to be found on point $(2,6)$ ,so it should satisfy the line $y=10x+c$

Now, put the values, $x=2$ and $y=6$ in the equation $y=10x+c$

  $6=10\times 2+c$

  $c=-14$

Hence, the equation of the tangent line at a point $(2,6)$ of the graph $f(x)=x^3-x$ is

  $\boxed{y=10x-14}$ .

c.

To determine

Graph the function and the tangent line.

Answer to Problem 46E

  

Explanation of Solution

Given information:

Graph the function and the tangent line.

  $f(x)=x^3-x$ , $(2,6)$

Calculation:

Use maple to draw the graph of the function $f(x)=x^3-x$ and the tangent line $y=10x-14$ can be drawn as below,

  

Hence, the result shown in the graph.