Chapter 12, Problem 54RE

To determine

To approximate: the slope of the graph at the point $\left(x,y\right)$ .

Answer to Problem 54RE

The graph of $f$ has slope 0 at the point $\left(2,4\right)$ .

Explanation of Solution

Given information:

Given graph is

Calculation:

Consider the function $f$ whose graph and the tangent on it at the point $\left(x,y\right)=\left(2,4\right)$ is represented by the figure below.

  

Visually approximate the slope of the graph of the function $f$ at the point $\left(2,4\right)$ .

Because a tangent line approximates the slope of the graph at a point, the slope of the graph of $f$ at $\left(2,4\right)$ is same to the slope of the tangent at that point.

From the graph, the tangent line at $\left(2,4\right)$ remains constant at $y=4$ for each unit change in $x$ .

So, estimate the slope of the tangent line at $\left(2,4\right)$ to be

  $\begin{array}{c}\text{Slope=}\frac{\text{Change}\text{in}\text{y}}{\text{Change}\text{in}\text{x}}\\ =\frac{0}{1}\\ =0\end{array}$

Because the tangent line at the point $\left(2,4\right)$ has slope 0, this conclude that the graph of $f$ has slope 0 at the point $\left(2,4\right)$ .