Chapter 12, Problem 57RE

To determine

To sketch: a graph of the given function, and to approximate the slope of the tangent line.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\frac{6}{x-4}$

The tangent line at the point $\left(2,f\left(2\right)\right)$ .

Calculation:

To draw graph the function $f$ and the tangent line on it at the point $\left(2,f\left(2\right)\right)=\left(2,-3\right)$ using Maple. The following Maple commands will produce the graph of the function $f$ and the tangent at $\left(2,-3\right)$ :

  

Now, visually approximate the slope of the graph of the function $f$ at the point $\left(2,-3\right)$ .

Because a tangent line approximates the slope of the graph at a point, the slope of the graph of $f$ at $\left(2,-3\right)$ is same to the slope of the tangent at that point.

From the graph, the tangent line at $\left(2,-3\right)$ falls approximately three units for each two unit change in $x$ . So, estimate the slope of the tangent line at $\left(2,-3\right)$ to be

  $\begin{array}{c}\text{Slope}=\frac{\text{Change}\text{in}\text{y}}{\text{Change}\text{in}\text{x}}\\ =\frac{-3}{2}\\ =-\frac{3}{2}\end{array}$

Because the tangent line at the point $\left(2,-3\right)$ has a slope of about $\left(-\frac{3}{2}\right)$ , this concludes that the graph of $f$ has a slope of about $\left(-\frac{3}{2}\right)$ at the point $\left(2,-3\right)$ .