Question

Use the definition of the derivative to find the equations of the lines described in Exercises 59-64.

The line that passes through the point $(3,2)$and is parallel to the tangent line to $f\left(x\right)=\frac{1}{x}$ at $x=-1$.

Short answer

The equation is$y=-x+5$

Step-by-step solution

Step 1.Given information

Given the function$f\left(x\right)=\frac{1}{x}$and the point$x=-1$

Calculate the tangent line

Calculating, we get

$$\begin{gathered} y=-1+\left(\frac{-1}{x^2}\right)(x+1) \\ y=-1-x-\frac{1}{x^2} \end{gathered}$$

Calculate the line

Calculating, we get

$$\begin{gathered} y-2=-1(x-3) \\ y-2=-x+3 \\ y=-x+5 \end{gathered}$$