Chapter 4, Problem 54RE

a.

To determine

For what values of the constant m is f(x) continuous at x =0? Explain.

Answer to Problem 54RE

All values of m .

Explanation of Solution

Given information: Let $\begin{array}{l}f(x)=\sin 2x,\text{if}x\le 0\text{and}\\ f(x)=mx,\text{if}x>0.\end{array}$

Calculation:

fis continuous for all values of m because no matter what m is, f(x) will approach 0 from the right. This allows the graph to be continuous because it approaches 0 from the left as well and f(0) = 0.

b.

To determine

For what values of the constant m is f(x) differentiable at x =0? Explain.

Answer to Problem 54RE

  $m=2.$

Explanation of Solution

Given information: Let $\begin{array}{l}f(x)=\sin 2x,\text{if}x\le 0\text{and}\\ f(x)=mx,\text{if}x>0.\end{array}$

Calculation:

  $\begin{array}{l}f(x)=\sin 2x,\text{if}x\le 0\text{and}\\ f(x)=mx,\text{if}x>0.\end{array}$

For f(x) to be differentiable at x =0 ,it must be continuous at x =0 so substitute b =3 that we found from part (a ).

  $\begin{array}{l}f(x)=\sin 2x,\text{if}x\le 0\text{and}\\ f(x)=mx,\text{if}x>0.\\ \text{Take}\text{the}\text{derivative}\text{of}\text{each}\text{piece}\text{of}\text{the}\text{function}\\ f\text{'}(x)=2\cos 2x,ifx\le 0.\\ f\text{'}(x)=m,ifx>0\\ \text{If}f\text{contiunous}\text{at}x=0\text{for}\text{any}\text{value}\text{of}m.t\text{The}\text{function}\text{will}\text{then}\text{be}\text{differentiable}\text{at}x=0\text{if}\text{the}\text{limit}\text{of}\text{the}\text{derivatives}\text{on}\text{left}\text{and}\text{right}\text{hand}\text{side}\text{of}x=0\text{must}\text{be}\text{equal}\text{.Set}\text{the}\text{limits}\text{equal}\text{to}\text{each}\text{other}\text{.}\\ \underset{x\to 0^{-}}{\lim }f\text{'}(x)=\underset{x\to 0^{+}}{\lim }f\text{'}(x).\\ \underset{x\to 0^{-}}{\lim }(2\cos 2x)=\underset{x\to 0^{+}}{\lim }m.\\ 2\cos 0=m.\\ 2(1)=m.\\ m=2.\end{array}$