Chapter 4, Problem 53RE

a.

To determine

If f is continuous at x =0,find the value of b. Justify your answer.

Answer to Problem 53RE

  $b=3.$

Explanation of Solution

Given information: Let $\begin{array}{l}f(x)=\sin ax+b\cos x,\text{if}x<0\text{and}\\ f(x)=5x+3,\text{if}x\ge 0.\end{array}$

Calculation:

Set the two equations equal to each other since being continuous means their limits are equal.

  $\begin{array}{l}\sin ax+b\cos x=5x+3.\\ Givenatx=0.\\ \sin a(0)+b\cos (0)=5(0)+3.\\ 0+b(1)=0+3.\\ b=3.\end{array}$

If the right side equation equals 3 then, in order to be continuous, the left side equation must equal 3 as well.

  $b=3.$

b.

To determine

If f is differentiable atx =0,find the value of a. Justify your answer.

Answer to Problem 53RE

  $a=5.$

Explanation of Solution

Given information: Let $\begin{array}{l}f(x)=\sin ax+b\cos x,\text{if}x<0\text{and}\\ f(x)=5x+3,\text{if}x\ge 0.\end{array}$

Calculation:

  $\begin{array}{l}f(x)=\sin ax+b\cos x,\text{if}x<0\text{and}\\ f(x)=5x+3,\text{if}x\ge 0.\end{array}$

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 ).

  $\begin{array}{l}f(x)=\sin ax+3\cos x,\text{if}x<0\text{and}\\ f(x)=5x+3,\text{if}x\ge 0.\\ \text{Take}\text{the}\text{derivative}\text{of}\text{each}\text{piece}\text{of}\text{the}\text{function}\\ f\text{'}(x)=a\cos ax-3\sin x,ifx<0.\\ f\text{'}(x)=5,ifx>0\\ \text{If}f\text{differentiable}\text{at}x=0\text{then}\text{limit}\text{of}\text{the}\text{derivatives}\text{on}\text{left}\text{and}\text{right}\text{hand}\text{side}\text{of}x=0\text{must}\text{be}\text{equal}\text{.Set}\text{the}\text{limits}\text{equal}\text{to}\text{each}\text{other}\text{and}\text{then}\text{solve}\text{for}a.\\ \underset{x\to 0^{-}}{\lim }f\text{'}(x)=\underset{x\to 0^{+}}{\lim }f\text{'}(x).\\ \underset{x\to 0^{-}}{\lim }(a\cos ax-3\sin x)=\underset{x\to 0^{+}}{\lim }5.\\ a\cos (0)-3\sin (0)=5.\\ a(1)-3(0)=5\\ a-0=5.\\ a=5.\end{array}$

c.

To determine

Is f is differentiable at x =0,if a =5 and b =4. Justify your answer.

Answer to Problem 53RE

No,

Explanation of Solution

Given information: Let $\begin{array}{l}f(x)=\sin ax+b\cos x,\text{if}x<0\text{and}\\ f(x)=5x+3,\text{if}x\ge 0.\end{array}$

Calculation:

No, From part (a), f(x) is only continuous at x =0 if b =3 .For a function to be differentiable ,it must be continuous so if b =4 , f(x) is discontinuous at x =0 and therefore not differentiable at x =0.