Chapter 3, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

1. a. The function f(x) = |2x + 1| is continuous for all x; therefore, it is differentiable for all x.
2. b. If $\frac{d}{dx}(f(x))=\frac{d}{dx}(g(x))$ then f = g.
3. c. For any function f, $\frac{d}{dx}|f(x)|=|f^{\prime }(x)|$.
4. d. The value of f′(a) fails to exist only if the curve y = f(x) has a vertical tangent line at x = a.
5. e. An object can have negative acceleration and increasing speed.

To determine

Whether the statement “The function $f\left(x\right)=\left|2x+1\right|$ is continuous for all x; therefore, it is differentiable for all x” is true or not.

Answer to Problem 1RE

The given statement is false.

Explanation of Solution

It is known that modulus function is continuous everywhere.

Therefore, the function $f\left(x\right)=\left|2x+1\right|$ is continuous at all points.

Note that modulus function $f\left(x\right)=\left|2x+1\right|$ has a corner point at $x=\frac{-1}{2}$.

It is known that at corner points, the function is not differentiable.

Therefore, the function is continuous at all points and it is not differentiable at $x=\frac{-1}{2}$.

Thus, the given statement is false.

To determine

Whether the statement “If $\frac{d}{dx}\left(f\left(x\right)\right)=\frac{d}{dx}\left(g\left(x\right)\right)$, then $f=g$” is true or not.

Answer to Problem 1RE

The given statement is false.

Explanation of Solution

Suppose the function $f\left(x\right)=x^2+9$ and $g\left(x\right)=x^2+123$.

Note that the derivative of the function $f\left(x\right)=x^2+9$ is $2x$ and $g\left(x\right)=x^2+123$ is $2x$.

Also, note that the functions $f\left(x\right)\text{ and }g\left(x\right)$ are different.

Thus, the given statement is false.

To determine

Whether the statement “For any function f, $\frac{d}{dx}\left|f\left(x\right)\right|=\left|f^{\prime }\left(x\right)\right|$” is true or false.

Answer to Problem 1RE

The given statement is false.

Explanation of Solution

The example given below shows that the statement is false.

Consider the given function $f\left(x\right)=e^{-x}$.

The derivative of the function $\left|e^{-x}\right|$ is computed as follows,

$\begin{array}{c}\frac{d}{dx}\left(\left|e^{-x}\right|\right)=\frac{d}{dx}\left(e^{-x}\right)\\ =-e^{-x}\\ \ne \left|-e^{-x}\right|\end{array}$

Therefore, the given statement is false.

To determine

Whether the statement “The value of $f^{\prime }\left(a\right)$ fails to exist only if the curve $y=f\left(x\right)$ has a vertical tangent line at $x=a$” is true or not.

Answer to Problem 1RE

The given statement is false.

Explanation of Solution

The example given below shows that the statement is false.

The function $f\left(x\right)=\left|x\right|$ has no derivative at $x=0$.

Also, note that there is no vertical tangent there.

Thus, the given statement is false.

To determine

Whether the statement “An object can have negative acceleration and increasing speed” is true or not.

Answer to Problem 1RE

The given statement is true.

Explanation of Solution

The example given below shows that the statement is true.

A ball dropping from a high tower has acceleration due to gravity which is negative but it is speeding up as it falls because the velocity is in the same direction as the acceleration.

Thus, the given statement is true.