Chapter 3, Problem 56RE

(a).

To determine

To find: Where the function is differentiable.

Answer to Problem 56RE

The function is differentiable everywhere.

Explanation of Solution

Given:

The given function is:

  $g\left(x\right)=\sin \left(x^2+1\right)$

Concept Used:

A function is said to be continuous at $a$ if limit of that function exist at $a$:

This implies:

  $\underset{x\to a^{-}}{\lim }f\left(a\right)=\underset{x\to a^{+}}{\lim }f\left(a\right)$

Calculation:

  

From the graph, the function is differentiable at $\left(-\infty ,\infty \right)$

Conclusion:

The function is differentiable at $\left(-\infty ,\infty \right)$ .

(b).

To determine

Where the function is continuous but not differentiable

Answer to Problem 56RE

The function is continuous but not differentiable at nowhere.

Explanation of Solution

Given:

The function is:

  $g\left(x\right)=\sin \left(x^2+1\right)$

Concept Used:

  $f\left(x\right)$ is differentiable if $f\text{'}\left(0\right)$ exist.

This implies:

  $\underset{x\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=\underset{x\to 0}{\lim }\frac{f\left(x-h\right)-f\left(x\right)}{-h}$

Calculation:

  

From the graph, the function is continuous everywhere but not differentiable at nowhere.

Conclusion:

The function is continuous but not differentiable at nowhere.

(c).

To determine

Where the function is neither continuous nor differentiable

Answer to Problem 56RE

The function is neither continuous nor differentiable at nowhere.

Explanation of Solution

Given:

The function is:

  $g\left(x\right)=\sin \left(x^2+1\right)$

Concept Used:

• A function is said to be continuous at $a$ if limit of that function exist at $a$:

This implies:

$\underset{x\to a^{-}}{\lim }f\left(a\right)=\underset{x\to a^{+}}{\lim }f\left(a\right)$

• $f\left(x\right)$ is differentiable if $f\text{'}\left(0\right)$ exist.

This implies:

$\underset{x\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=\underset{x\to 0}{\lim }\frac{f\left(x-h\right)-f\left(x\right)}{-h}$

Calculation:

  

From the graph, there is no point in the graph where the function is neither continuous nor differentiable.

Conclusion:

The function is neither continuous nor differentiable at nowhere.