Chapter 11, Problem 51RE

To determine

To calculate:

The slop of the graph at the point $\left(x,y\right)$ using given figure.

Answer to Problem 51RE

The slop of the graph at the point $\left(x,y\right)$ is $2$ .

Explanation of Solution

Given information:

  

Calculation:

Consider the function $f$ , $f$ has graph and the tangent on it at the point $\left(x,y\right)=\left(2,-\frac{3}{2}\right)$ is represented by the graph below,

  

Visually approximate the slope of the graph of the function $f$ at the point $\left(2,-\frac{3}{2}\right)$

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

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

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

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