Chapter 11.3, Problem 46E

(a)

To determine

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

Answer to Problem 46E

The slope of the graph is $-4$

Explanation of Solution

Given:

  $f\left(x\right)=x^2+2x+1,\left(-3,4\right)$

Calculation:

To find the slop of the graph at $\left(-3,4\right)$ , find the derivative at $-3$ .

  $\begin{array}{c}{f}^{\prime }\left(-3\right)=\underset{\Delta x\to 0}{\lim }\frac{f\left(-3+\Delta x\right)-f\left(-3\right)}{\Delta x}\\ =2\left(-3\right)+2\\ =2-6\\ m=-4\end{array}$

Conclusion:

The slope of the graph is $-4$

(b)

To determine

To find: an equation of the tangent line

Answer to Problem 46E

An equation of the tangent line is $y=-4x-8$

Explanation of Solution

Calculation:

Use the point-slope form of the equation for a line to find the tangent line.

  $\begin{array}{c}y-y_1=m\left(x-x_1\right)\\ y-\left(4\right)=-4\left[x-\left(-3\right)\right]\\ y-4=-4x-12\\ y=-4x-8\end{array}$

Conclusion:

an equation of the tangent line is $y=-4x-8$

(c)

To determine

To graph: the function and the tangent line.

Answer to Problem 46E

  

Explanation of Solution

Calculation:

To graph the equation and the tangent line:

1. Press " $y=$ " on the calculator

2. Input the equation given for f and the equation for the tangent line in $Y_1$ and $Y_2$ respectively

3. Press "window" and adjust Xmin to $-5$ , Xmax to 1, Ymin to $-1$ , and Yman to 10

  

Conclusion:

Thus, the required graph is drawn.