Q. 1C - Calculus

Question

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: $f\text{'}\left(x\right)=\frac{f(x+h)-f\left(x\right)}{h}$

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

(c) True or False: $f\text{'}\left(x\right)=\underset{z\to 0}{\lim }\frac{f\left(z\right)-f\left(x\right)}{z-x}$

(d) True or False: If $f\left(x\right)=x^3, then f(x+h)=x^3+h$

(e) True or False: If $f\left(x\right)=x^3, then f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f(x^3+h)-f\left(x\right)}{h}$

(f) True or False: A function $f$ is differentiable at $x=c$ if and only if $f\text{'}\_\left(c\right) and f{\text{'}}_{+}\left(c\right)$ both exists.

(g) True or False: If $f$ is continuous at $x=c$, then $f$is differentiable at $x=c$.

(h) True or False: If $f$ is not continuous at $x=c, then f$ is not differentiable at $x=c$.

Step-by-Step Solution

Answer

Part (a). Given statement is false.

Part (b). Given statement is false.

Part (c). Given statement is false.

Part (d). Given statement is false.

Part (e). Given statement is false.

Part (f). Given statement is false.

Part (g). Given statement is false.

Part (h). Given statement is true.

Part (a) Step 1. Explanation

According to the definition of derivative,

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

Therefore, 

$f\text{'}\left(x\right)=\left(\frac{f(x+h)-f\left(x\right)}{h}\right)$ is an incorrect statement.

Part (b) Step 1. Explanation

According to the definition of derivative,

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

Therefore,

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

Part (c) Step 1. Explanation

According to the definition of derivative,

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

Therefore,

$f\text{'}\left(x\right)=\underset{z\to 0}{\lim }\left(\frac{f\left(z\right)-f\left(x\right)}{z-x}\right)$ is an incorrect statement.

Part (d) Step 1. Explanation

From the basic property of function, if $f\left(x\right)=x^3 then f(x+h)={(x+h)}^3$

Hence, given statement is incorrect.

Part (e) Step 1. Explanation

According to the definition of derivative,

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

Therefore,

$f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\left(\frac{f(x^3+h)-f\left(x\right)}{h}\right)$ is an incorrect statement.

Part (f) Step 1. Explanation

A function $f\left(x\right)$ is differentiable at $x=c$ if and only if $f{\text{'}}_{+}\left(c\right) and f\text{'}\_\left(c\right)$ both exist and both should be equal. In other words, a function $f\left(x\right)$ is differentiable at $x=c$ if

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

Hence, given statement is false.

Part (g) Step 1. Explanation

It is known that,

If $f\left(x\right)$ is differentiable at $x=c$, then $f\left(x\right)$ is continuous at $x=c$ 

and converse need not to be true. Hence, given statement is false.

Part (h) Step 1. Explanation

It is known that,

If $f\left(x\right)$ is differentiable at $x=c$, then $f\left(x\right)$ is continuous at $x=c$. This implies that if $f\left(x\right)$is not continuous at $x=c$, then $f\left(x\right)$ is not differentiable at $x=c.$

Hence, given statement is true.