Chapter 12.3, Problem 26E

To determine

Sketch a graph of the given and the tangent line at point $(1,f(1))$ , find the approximate slope of the tangent line.

Answer to Problem 26E

The approximate slope of the curve at point $(1,2)$ is $\text{Slope}=-\frac{1}{4}$

Explanation of Solution

Given:

The function given is:

  $f(x)=\sqrt{x}+3$

Consider the function:

  $f(x)=\sqrt{x}+3$

Since, the tangent is drawn to be on the point $(1,f(1))$ .

  $\begin{array}{cc}f(1)& =\sqrt{1+3}\\ & =\sqrt{4}\\ & =2\end{array}$

The point (1, 2) as marked in the graph below. Now, draw the tangent at (1, 2).

  

By the above graph the slope at point $(1,2)$ will be as,

  $\text{Slope}=\frac{\text{change in y}}{\text{change in x}}$

Now, from the above graph, the tangent crosses the $x$ axis at $3$ and $y$ at $3$ .

So, the change in y is equal to final position- initial position

  $\Delta y=2-0=2$

Similarly the change in x

  $\Delta x=1-(-7)=8$

Hence, by the above formula of slope of the tangent

  $\begin{array}{l}\text{Slope}=\frac{2}{8}\\ =\frac{1}{4}\end{array}$

Since, the graph is concave down therefore the slope will be negative.

Therefore, the approximate slope of the curve at point $(1,2)$ is $\text{Slope}=-\frac{1}{4}$