Chapter 11, Problem 60CR

To determine

To evaluate: the formula for the slope of the graph of f at the point ( x , f ( x )) and then use it to find the slopes at the two specified points.

Answer to Problem 60CR

The slope at the (−2, 4) is −8 and the slope at the (1, 1/4) is 1.

Explanation of Solution

Given:

  $f\left(x\right)=\frac{1}{4}{x}^4$

To find the slope of the graph, using limit definition of the derivative,

  $f\text{'}\left(x\right)=\frac{d}{dx}\left(f\left(x\right)\right)$

So,

  $\begin{array}{c}f\text{'}\left(x\right)=\frac{d}{dy}\left(\frac{1}{4}{x}^4\right)\\ =x^3\end{array}$

At (−2, 4), the slope,

  $\begin{array}{l}f\text{'}\left(x\right)=x^3\\ f\text{'}\left(-2\right)={\left(-2\right)}^3\\ f\text{'}\left(-2\right)=-8\end{array}$

At (1, 1/4 ), the slope,

  $\begin{array}{l}f\text{'}\left(x\right)=x^3\\ f\text{'}\left(1\right)={\left(1\right)}^3\\ f\text{'}\left(1\right)=1\end{array}$

The slope at the (−2, 4) is −8 and the slope at the (1, 1/4) is 1.