Chapter 13, Problem 30RE

To determine

To determine an equation of the tangent shown

Answer to Problem 30RE

The equation of the tangent shown is $x-2y=-1$

Explanation of Solution

Given information:

Consider the given information,

  $y=\sqrt{x}\text{ at }\left(1,1\right)$

CALCULATION:

To find the equation is,

  $\begin{array}{l}y=\sqrt{x}\\ f\left(x\right)=y=\sqrt{x}\\ f\text{'}\left(x\right)=\frac{1}{2\sqrt{x}}\end{array}$

  $\begin{array}{l}\text{At}\left(1,1\right)f\text{'}\left(1\right)=\frac{1}{2\sqrt{1}}=\frac{1}{2}\\ \therefore m=\frac{1}{2}\end{array}$

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

Therefore, the equation of the tangent is $x-2y=-1$