Q. 1 - Calculus

Question

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

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

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

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

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

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

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

(g) True or False: If f is not continuous at x = c, then f is not 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

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

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

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

d. The $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3\text{ then }\mathrm{f}(\mathrm{x}+\mathrm{h})=(\mathrm{x}+\mathrm{h}{)}^3$ given statement is incorrect.

e. The  $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3\text{ then }{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=\underset{\mathrm{h}\to 0}{lim} \frac{\mathrm{f}\left({\mathrm{x}}^3+\mathrm{h}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}$statement is incorrect.

f. A function f is differentiable at x = c if and only if  ${{\mathrm{f}}^{\text{'}}}_{+}\left(\mathrm{c}\right) \text{and }{{\mathrm{f}}^{\text{'}}}_{-}\left(\mathrm{c}\right)$statement is incorrect.

g. If f is not continuous at x = c, then f is not differentiable at x = c  statement is incorrect. 

h. If f is not continuous at x = c, then f is not differentiable at x = c statement is correct.

Part (a) Step 1: Given Information

The given function is $\mathrm{f}\text{'}\left(\mathrm{x}\right)=\frac{\mathrm{f}\left(\mathrm{x}+\mathrm{h}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}$

Part (a) Step 2: Calculations

The term "derivative" is defined as "a thing that is derived from something else."

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

Hence,

${\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=\left(\frac{\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}\right)$is a false assertion

Part (b) Step 1: Given Information

The given function is ${\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=\underset{\mathrm{x}\to 0}{lim} \left(\frac{\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}\right)$

Part (b) Step 2: Calculations

The term derivative is defined as a thing that is derived from something else.

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

Hence,

${\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=\underset{\mathrm{x}\to 0}{lim} \left(\frac{\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}\right)$is a false assertion

Part (c) Step 1: Given Information

The given function is $\mathrm{f}\text{'}\left(\mathrm{x}\right)=\underset{\mathrm{z}\to 0}{lim} \left(\frac{\mathrm{f}\left(\mathrm{z}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{z}-\mathrm{x}}\right)$

Part (c) Step 2: Calculations

The term derivative is defined as a thing that is derived from something else.

${\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=\underset{\mathrm{z}\to \mathrm{x}}{lim} \left(\frac{\mathrm{f}\left(\mathrm{z}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{z}-\mathrm{x}}\right)$

Hence,

${\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=\underset{\mathrm{z}\to 0}{lim} \left(\frac{\mathrm{f}\left(\mathrm{z}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{z}-\mathrm{x}}\right)$is a false assertion 

Part (d) Step 1: Given Information

The given function is $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3\text{ then }\mathrm{f}(\mathrm{x}+\mathrm{h})={\mathrm{x}}^3+{\mathrm{h}}^{\mathrm{\prime }}$

Part (d) Step 2: Calculations

As a result of the fundamental attribute of function, $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3\text{ then }\mathrm{f}(\mathrm{x}+\mathrm{h})=(\mathrm{x}+\mathrm{h}{)}^3$ is a false assertion 

Part (e) Step 1: Given Information

The given function is $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3\text{ then }{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=\underset{\mathrm{h}\to 0}{lim} \left(\frac{\mathrm{f}\left({\mathrm{x}}^3+\mathrm{h}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}\right)$

Part (e) Step 2: Calculations

The term derivative is defined as a thing that is derived from something else.

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

Hence,

${\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=\underset{\mathrm{h}\to 0}{lim} \left(\frac{\mathrm{f}\left({\mathrm{x}}^3+\mathrm{h}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}\right)$is a false assertion 

Part (f) Step 1: Given Information

The given function is $\mathrm{f}\left(\mathrm{x}\right)$at x=c is differentiable if and only if $\mathrm{f}{\text{'}}_{+}\left(\mathrm{c}\right) \mathrm{and} \mathrm{f}{\text{'}}_{-}\left(\mathrm{c}\right)$both are real

Part (f) Step 2: Calculations

At x=c, a function f(x) is differentiable if ${\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{c}\right)=\underset{\mathrm{h}\to 0}{lim} \left(\frac{\mathrm{f}(\mathrm{c}+\mathrm{h})-\mathrm{f}\left(\mathrm{c}\right)}{\mathrm{h}}\right)$existed.

Part (g) Step 1: Given Information

At x=c, if f(x) is continuous, then f(x) is differentiable.

Part (g) Step 2: Calculations

At x=c, if f(x) is continuous, then f(x) is differentiable and It is not necessary for the converse to be true. 

The statement is a false assertion 

Part (h) Step 1: Given Information

If f(x) is not continuous at x=c, f(x) is not differentiable at x=c as well.

Part (h) Step 2: Calculations

If f(x) is not continuous at x=c, f(x) is not differentiable at x=c as well.

This means that if f(x) is not continuous at some point, If x=c, then f(x) will not be differentiable.