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: \(f^{\prime}(x)=\frac{f(x+h)-f(x)}{h}\).

(b) True or False: \(f^{\prime}(x)=\lim _{x \rightarrow 0} \frac{f(x+h)-f(x)}{h}\).

(c) True or False: \(f^{\prime}(x)=\lim _{z \rightarrow 0} \frac{f(z)-f(x)}{z-x}\).

(d) True or False: If \(f(x)=x^{3}\), then \(f(x+h)=x^{3}+h\).

(e) True or False: If \(f(x)=x^{3}\), then \(f^{\prime}(x)=\) \(\lim _{h \rightarrow 0} \frac{f\left(x^{3}+h\right)-f(x)}{h}\)

(f) True or False: A function \(f\) is differentiable at \(x=c\) if and only if \(f_{-}^{\prime}(c)\) and \(f_{+}^{\prime}(c)\) both exist.

(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 \(x=c\)

Short answer

(a). False

(b). True

(c). False

(d). False

(e). False

(f). False

(g). False

(h). True

Step-by-step solution

Part (a) Step 1: Given information

Here the objective is to state whether the statement is true or false.

Part (a) Step 2: Simplification

The given statement is false as the there exist two types of definition of derivative using limit. The derivative of a function is given by using $h \rightarrow 0$ is:

$$

f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}

$$

Part (b) Step 1: Given information

Here the objective is to state whether the statement is true or false.

Part (b) Step 2: Simplification

The given statement is true as the there exist two types of definition of derivative using limit. The derivative of a function is given by using $h \rightarrow 0$ is:

$$

f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}

$$

Part (c) Step 1: Given information

Here the objective is to state whether the statement is true or false.

Part (c) Step 2: Simplification

The given statement is false as the there exist two types of definition of derivative using limit. The derivative of a function is given by using $z \rightarrow x$ is:

$$

f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}

$$

Part (d) Step 1: Given information

Here the objective is to state whether the statement is true or false.

Part (d) Step 2: Simplification

In $f(x)=x^{3}$ if $x$ is replaced with $x+h$, then the function becomes $f(x+h)=(x+h)^{3}$.

Since $(x+h)^{3} \neq x^{3}+h$, therefore it is a false statement.

Part (e) Step 1: Given information

Here the objective is to state whether the statement is true or false.

Part (e) Step 2: Simplification

Since the given function is $f(x)=x^{3}$, therefore the derivative of the function according to definition is:

$$

f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}

$$

Part (f) Step 1: Given information

Here the objective is to state whether the statement is true or false.

Part (f) Step 2: Simplification

The given statement is false as the function will said to be differentiable at $x=c$ only when left derivative at $x=c$ that mean $f_{-}^{\prime}(c)$ and right derivative at $x=c$ that mean $f_{+}^{\prime}(c)$ exist and $f_{+}^{\prime}(c)=f_{-}^{\prime}(c)$.

Part (g) Step 1: Given information

Here the objective is to state whether the statement is true or false.

Part (g) Step 2: Simplification

The given statement is false as all continuous function at any given point may not differentiable at the same point.

For example the function

$$

f(x)= \begin{cases}2 x+1, & \text { if } x>1 \\ x^{3}+2 & \text { if } x \leq 1\end{cases}

$$

Is continuous at $x=1$ but it is not differentiable at $x=1$.

Part (h) Step 1: Given information

Here the objective is to state whether the statement is true or false.

Part (h) Step 2: Simplification

The given statement is true as the function is differentiable at $x=c$, therefore the left and right derivative of the function must be equal for that particular point which indicates that the functions left and right limit will exist and hence it is continuous.