Chapter 11.3, Problem 47E

(a)

To determine

To find: the slope of the graph at the given point

Answer to Problem 47E

The slope of the graph is $\frac{1}{4}$

Explanation of Solution

Given:

  $f\left(x\right)=\sqrt{x+3},\left(6,3\right)$

Calculation:

In order to find the slope of the graph at the point $\left(c,f\left(c\right)\right)$ , use the formula

  $m=\underset{h\to 0}{\lim }\frac{f\left(c+h\right)f\left(c\right)}{h}$

Therefore, using the given function $f\left(x\right)$ and plugging in $x=3$ , expand the terms in the difference quotient, and then simplify, divide out, and evaluate the limit by direct substitution to obtain the slope:

  $\begin{array}{c}m=\underset{h\to 0}{\lim }\frac{\sqrt{3+h+1}-\sqrt{3+1}}{h}\\ =\underset{h\to 0}{\lim }\frac{\sqrt{4+h}-2}{h}*\frac{\sqrt{4+h}+2}{\sqrt{4+h}+2}\\ =\underset{h\to 0}{\lim }\frac{\left(4+h\right)-\left(4\right)}{h\left(\sqrt{4+h}+2\right)}\\ =\underset{h\to 0}{\lim }\frac{h}{h\left(\sqrt{4+h}+2\right)}\\ =\underset{h\to 0}{\lim }\frac{1}{\sqrt{4+h}+2}\\ =\frac{1}{4}\left(\text{By}\text{direct}\text{substitution}\text{.}\right)\end{array}$

Conclusion:

(b)

To determine

To find: an equation of the tangent line

Answer to Problem 47E

An equation of the tangent line is $y=\frac{1}{4}x+\frac{5}{4}$

Explanation of Solution

Calculation:

The slope m of the graph at this point that obtained in part (a) is also the slope of the tangent line to the graph at this point. Therefore, simply plug this slope, and the given point into the equation $y-y_1=m\left(x-x_1\right)$ in order to find an equation for this line:

  $\begin{array}{c}y-y_1=m\left(x-x_1\right)\\ y-2=\frac{1}{4}\left(x-3\right)\\ y-2=\frac{1}{4}x-\frac{3}{4}\\ y=\frac{1}{4}x+\frac{5}{4}\end{array}$

Conclusion:

An equation of the tangent line is $y=\frac{1}{4}x+\frac{5}{4}$

(c)

To determine

To graph: the function and the tangent line.

Answer to Problem 47E

  

Explanation of Solution

Calculation:

Now, graph the given function and the equation of the tangent line just found in the same viewing window:

  

Conclusion:

Thus, the required graph is drawn.