Chapter 15.2, Problem 3CFU

To determine

To find: The difference between given functions.

Answer to Problem 3CFU

The functions $f\left(x+h\right)$ and $f\left(x\right)+h$ are equal for $f\left(x\right)=x$ .

Explanation of Solution

Given information:

The given functions are $f\left(x+h\right)$ and $f\left(x\right)+h$ .

Calculation:

  $f\left(x+h\right)$ is a function of $x+h$ , whereas $f\left(x\right)+h$ is a function of $x$ to which $h$ is been added.

They are totally two different functions.

The derivative of the function is given as.

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

Replace $f\left(x+h\right)$ by $f\left(x\right)+h$ .

  $\begin{array}{c}f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x\right)+h-f\left(x\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{h}{h}\\ =1\end{array}$

From above observation it is clear that $f\left(x+h\right)$ and $f\left(x\right)+h$ are equal if $f\left(x\right)=x$ .

Therefore, the functions $f\left(x+h\right)$ and $f\left(x\right)+h$ are equal for $f\left(x\right)=x$ .