Chapter 3.5, Problem 40E

To determine

To find:Find the equation for the line.

Answer to Problem 40E

Equation for this line is $y=x$ ..

Explanation of Solution

Given information: The graph of function $y=\sin x$ close to origin.

Calculation:

  $\begin{array}{l}y=\sin x\\ \frac{dy}{dx}=\cos x\end{array}$

Put $x=0\mathrm{in}\frac{dy}{dx}$ we get:

  $\begin{array}{l}\frac{dy}{dx}=\cos x\\ \frac{dy}{dx}=\cos 0\\ \frac{dy}{dx}=1\end{array}$

Equation of line is $y=mx+b$

Put $x=0\mathrm{and}y=0$ we get:

  $\begin{array}{l}y=mx+b\\ 0=m(0)+b\\ 0=0+b\\ 0=b\end{array}$

Put $m=1\mathrm{and}b=0$ in equation $y=mx+b$ we get:

  $\begin{array}{l}y=mx+b\\ y=(1)x+0\\ y=x\end{array}$

Therefore equation for this line is $y=x$ .