Chapter 11, Problem 77CR

a)

To determine

To evaluate: the slope of the graph of f at the given point.

Answer to Problem 77CR

Slope at (2, 14) is 4.

Explanation of Solution

Given:

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

Calculation: -

It is given that,

  $\begin{array}{l}f\left(x\right)=x^2+1\\ f\text{'}\left(x\right)=2x\\ f\text{'}\left(1\right)=2\end{array}$

Hence, Slope at (1, 2) is 2.

b)

To determine

To evaluate: the equation of the tangent line to the graph at the point.

Answer to Problem 77CR

The equation of tangent is $4x-y+6=0$ .

Explanation of Solution

Given:

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

Calculation: -

It is given that,

  $\begin{array}{l}f\left(x\right)=x^2+1\\ f\text{'}\left(x\right)=2x\\ f\text{'}\left(1\right)=2\end{array}$

Hence, Slope at (1, 2) is 2.

Using one-point formula to find the equation of line,

  $\begin{array}{c}y-y_1=m\left(x-x_1\right)\\ y-2=2\left(x-1\right)\\ y-2=2x-2\\ 2x-y=0\end{array}$

Hence, the equation of tangent is $2x-y=0$ .

c)

To determine

To evaluate: the graph of the function and the tangent line on it.

Answer to Problem 77CR

  

Explanation of Solution

Given:

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

Calculation: -It is given that,

  $\begin{array}{l}f\left(x\right)=x^2+1\\ f\text{'}\left(x\right)=2x\\ f\text{'}\left(1\right)=2\end{array}$

Hence, Slope at (1, 2) is 2.

Using one-point formula to find the equation of line,

  $\begin{array}{c}y-y_1=m\left(x-x_1\right)\\ y-2=2\left(x-1\right)\\ y-2=2x-2\\ 2x-y=0\end{array}$

Hence, the equation of tangent is $2x-y=0$ .

Using graphing calculator to plot the graph,