Chapter 11, Problem 56RE

To determine

To calculate:

The slop of the tangent line using graph.

Answer to Problem 56RE

The slop of the tangent line is $\frac{2}{3}$ .

Explanation of Solution

Given information:

Point of tangent line: $\left(2,f\left(2\right)\right)$

Function: $f\left(x\right)=\sqrt{x^2+5}$

Calculation:

Consider the function $f\left(x\right)=\sqrt{x^2+5}$

To sketch the 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 above graph shows the function $f$ and tangent at $\left(2,3\right)$ .

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

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

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