Chapter 12, Problem 58RE

To determine

Find the slope the tangent of line and graph at given point.

Answer to Problem 58RE

The tangent line at the point (2,1) has a slope of about $1$

Explanation of Solution

Given:

  $f(x)=\frac{1}{3-x}$

Consider the following function

  $f(x)=\frac{1}{3-x}$

To draw graph the function $f$ and the tangent line on it at the point $(2,f(2))=(2,1)$ using Maple.

The following Maple commands will produce the graph of the function $f$ and the tangent at (2,0)

  

Now, visually approximate the slope of the graph of the function $f$ at the point (2,1)

Because a tangent line approximates the slope of the graph at a point, the slope of the graph of $f$ at (2,1) is same to the slope of the tangent at that point.

From the graph, the tangent line at (2,1) rises approximately two units for each eight-unit

change in $x.$ So, estimate the slope of the tangent line at (2,1) to be

  $\begin{array}{cc}\text{Slope}& =\frac{\text{Change in}y}{\text{Change in}x}\\ & =\frac{2}{2}\\ & =1\end{array}$

Because the tangent line at the point (2,1) has a slope of about $1$ this concludes that the graph of $f$ has a slope of about $1$ at the point (2,1)