Chapter 7.3, Problem 46E

a.

To determine

To write:and expand the expression for the difference quotient.

Answer to Problem 46E

  $\frac{\sin \left(x\right)\cos \left(h\right)+\cos \left(x\right)\sin \left(h\right)-\sin \left(x\right)}{h}$

Explanation of Solution

Given information:

Difference quotient $=\frac{f\left(x+h\right)-f\left(x\right)}{h}$

  $f\left(x\right)=\sin x$

Calculation:

Replace $f$ by $\sin$ and put it in the difference quotient as shown below.

  $\begin{array}{c}\frac{f\left(x+h\right)-f\left(x\right)}{h}=\frac{\sin \left(x+h\right)-\sin \left(x\right)}{h}\\ =\frac{\sin \left(x\right)\cos \left(h\right)+\cos \left(x\right)\sin \left(h\right)-\sin \left(x\right)}{h}\left[\text{Sum identity of sine}\right]\end{array}$

Thus, the difference quotient is $\frac{\sin \left(x\right)\cos \left(h\right)+\cos \left(x\right)\sin \left(h\right)-\sin \left(x\right)}{h}$ .

b.

To determine

To graph: the given function.

Explanation of Solution

Given information:

Difference quotient $=\frac{\sin \left(x\right)\cos \left(h\right)+\cos \left(x\right)\sin \left(h\right)-\sin \left(x\right)}{h}$

Difference quotient $=y$

  $h=0.1$

Calculation:

The difference quotient will be as follows:

  $\begin{array}{c}y=\frac{\sin \left(x\right)\cos \left(h\right)+\cos \left(x\right)\sin \left(h\right)-\sin \left(x\right)}{h}\\ y=\frac{\sin \left(x\right)\cos \left(0.1\right)+\cos \left(x\right)\sin \left(0.1\right)-\sin \left(x\right)}{0.1}\left[\text{Put }h=0.1\right]\\ y=\frac{\sin \left(x\right)\left(1\right)+\cos \left(x\right)\left(0.1\right)-\sin \left(x\right)}{0.1}\\ y=10\sin x+\cos x-\sin x\end{array}$

Now, use a graphing calculator to plot the function $y$ as shown below.

c.

To determine

To find: the function for the given situation.

Answer to Problem 46E

  $y=-10\sin \left(x+3.1\right)-\cos \left(x+3.1\right)+\sin \left(x+3.1\right)$

Explanation of Solution

Given information:

Graph of function should be similar to graph of function in part b.

  $y=10\sin x+\cos x-\sin x$

Calculation:

Use the graphing calculator to invert the graph first for function $y=-10\sin x-\cos x+\sin x$ . Now the obtained will be reciprocal of that obtained in part b and is shifted to 3.1 units left. So, add 3.1 units to the trigonometric function to get the graph same as that of part b. Thus, the required function is $y=-10\sin \left(x+3.1\right)-\cos \left(x+3.1\right)+\sin \left(x+3.1\right)$ .

The graph for the function is shown below.