Chapter 4.1, Problem 76E

To determine

To calculate: To graph the function $y=2\cos 2x$ for $-2\le x\le 3.5$ and graph $y=\frac{\sin 2\left(x+h\right)-\sin 2x}{h}$ for $h=1,0.5,0.2$ . What happens as $h\to 0$

Answer to Problem 76E

As $h\to 0$, the function approaches to $2\cos 2x$.

Explanation of Solution

Given Information: The functions are $y=2\cos 2x$ and $y=\frac{\sin 2\left(x+h\right)-\sin 2x}{h}$

Calculation:

The graph of the above equations is,

  

The black curve represents the graph of the curve $y=2\cos 2x$ .

Now, if the negative values are used for h, the values will approach the function from right- hand side instead from the left-hand side.

Using the definition of the derivative of the function,

  $\begin{array}{c}\frac{d}{dx}f\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ \frac{d}{dx}\sin 2x=\underset{h\to 0}{\lim }\frac{\sin 2\left(x+h\right)-\sin 2x}{h}\\ 2\cos 2x=\underset{h\to 0}{\lim }\frac{\sin 2\left(x+h\right)-\sin 2x}{h}\end{array}$

Thus, as $h\to 0$ , the function approaches to $2\cos 2x$ .