1
The function \(f(x)\) is shown. Select all options below that describe its derivative, \(\ds\diff{f}{x}\text{:}\)
- (a) constant
- (b) increasing
- (c) decreasing
- (d) always positive
- (e) always negative
Subsection 2.2.4 Exercises
Exercises — Stage 1
2
The function \(f(x)\) is shown. Select all options below that describe its derivative, \(\ds\diff{f}{x}\text{:}\)
- (a) constant
- (b) increasing
- (c) decreasing
- (d) always positive
- (e) always negative
3
The function \(f(x)\) is shown. Select all options below that describe its derivative, \(\ds\diff{f}{x}\text{:}\)
- (a) constant
- (b) increasing
- (c) decreasing
- (d) always positive
- (e) always negative
4 (✳)
State, in terms of a limit, what it means for \(f(x) = x^3\) to be differentiable at \(x = 0\text{.}\)
5
For which values of \(x\) does \(f'(x)\) not exist?
6
Suppose \(f(x)\) is a function defined at \(x=a\) with
\begin{equation*}
\lim_{h \to 0^-}\frac{f(a+h)-f(a)}{h}=\lim_{h \to 0^+}\frac{f(a+h)-f(a)}{h}=1.
\end{equation*}
True or false: \(f'(a)=1\text{.}\)
7
Suppose \(f(x)\) is a function defined at \(x=a\) with
\begin{equation*}
\lim_{x \to a^-}f'(x)=\lim_{x \to a^+}f'(x)=1.
\end{equation*}
True or false: \(f'(a)=1\text{.}\)
8
Suppose \(s(t)\) is a function, with \(t\) measured in seconds, and \(s\) measured in metres. What are the units of \(s'(t)\text{?}\)
Exercises — Stage 2
9
Use the definition of the derivative to find the equation of the tangent line to the curve \(y(x)=x^3+5\) at the point \((1,6)\text{.}\)
10
Use the definition of the derivative to find the derivative of \(f(x)=\frac{1}{x}\text{.}\)
11 (✳)
Let \(f(x) = x|x|\text{.}\) Using the definition of the derivative, show that \(f(x)\) is differentiable at \(x = 0\text{.}\)
12 (✳)
Use the definition of the derivative to compute the derivative of the function \(f(x)=\frac{2}{x+1}\text{.}\)
13 (✳)
Use the definition of the derivative to compute the derivative of the function \(f(x)=\frac{1}{x^2+3}\text{.}\)
14
Use the definition of the derivative to find the slope of the tangent line to the curve \(f(x)=x\log_{10}(2x+10)\) at the point \(x=0\text{.}\)
15 (✳)
Compute the derivative of \(f(x)=\frac{1}{x^2}\) directly from the definition.
16 (✳)
Find the values of the constants \(a\) and \(b\) for which
\begin{align*}
f(x) = \left\{
\begin{array}{lc}
x^2 & x\le 2\\
ax + b & x \gt 2
\end{array}\right.
\end{align*}
is differentiable everywhere.
Remark: In the text, you have already learned the derivatives of \(x^2\) and \(ax+b\text{.}\) In this question, you are only asked to find the values of \(a\) and \(b\)—not to justify how you got them—so you don't have to use the definition of the derivative. However, on an exam, you might be asked to justify your answer, in which case you would show how to differentiate the two branches of \(f(x)\) using the definition of a derivative.
17 (✳)
Use the definition of the derivative to compute \(f'(x)\) if \(f(x) = \sqrt{1 + x}\text{.}\) Where does \(f'(x)\) exist?
Exercises — Stage 3
18
Use the definition of the derivative to find the velocity of an object whose position is given by the function \(s(t)=t^4-t^2\text{.}\)
19 (✳)
Determine whether the derivative of following function exists at \(x=0\text{.}\)
\begin{align*}
f(x) &=\begin{cases}
x \cos x & \text{ if } x\ge 0\\
\sqrt{x^2+x^4} & \text{ if } x \lt 0
\end{cases}
\end{align*}
You must justify your answer using the definition of a derivative.
20 (✳)
Determine whether the derivative of the following function exists at \(x=0\)
\begin{align*}
f(x) &=\begin{cases}
x \cos x & \text{ if } x\le 0\\
\sqrt{1+x}-1 & \text{ if } x \gt 0
\end{cases}
\end{align*}
You must justify your answer using the definition of a derivative.
21 (✳)
Determine whether the derivative of the following function exists at \(x=0\)
\begin{align*}
f(x) &=\begin{cases}
x^3-7x^2 & \text{ if } x\le 0\\
x^3 \cos\left(\frac{1}{x}\right) & \text{ if } x \gt 0
\end{cases}
\end{align*}
You must justify your answer using the definition of a derivative.
22 (✳)
Determine whether the derivative of the following function exists at \(x=1\)
\begin{align*}
f(x) &=\begin{cases}
4x^2-8x+4 & \text{ if } x\le 1\\
(x-1)^2\sin\left(\dfrac{1}{x-1}\right) & \text{ if } x \gt 1
\end{cases}
\end{align*}
You must justify your answer using the definition of a derivative.
23
Sketch a function \(f(x)\) with \(f'(0)=-1\) that takes the following values:
| \(\mathbf{x}\) | \(-1\) | \(-\frac{1^{ }}{2_{ }}\) | \(-\frac{1}{4}\) | \(-\frac{1}{8}\) | \(0\) | \(\frac{1}{8}\) | \(\frac{1}{4}\) | \(\frac{1}{2}\) | \(1\) |
| \(\mathbf{f(x)}\) | \(-1\) | \(-\frac{1^{ }}{2_{ }}\) | \(-\frac{1}{4}\) | \(-\frac{1}{8}\) | \(0\) | \(\frac{1}{8}\) | \(\frac{1}{4}\) | \(\frac{1}{2}\) | \(1\) |
Remark: you can't always guess the behaviour of a function from its points, even if the points seem to be making a clear pattern.
24
Let \(p(x)=f(x)+g(x)\text{,}\) for some functions \(f\) and \(g\) whose derivatives exist. Use limit laws and the definition of a derivative to show that \(p'(x)=f'(x)+g'(x)\text{.}\)
Remark: this is called the sum rule, and we'll learn more about it in Lemma 2.4.1.
25
Let \(f(x)=2x\text{,}\) \(g(x)=x\text{,}\) and \(p(x)=f(x) \cdot g(x)\text{.}\)
- Find \(f'(x)\) and \(g'(x)\text{.}\)
- Find \(p'(x)\text{.}\)
- Is \(p'(x)=f'(x) \cdot g'(x)\text{?}\)
In Theorem 2.4.3, you'll learn a rule for calculating the derivative of a product of two functions.
26 (✳)
There are two distinct straight lines that pass through the point \((1,-3)\) and are tangent to the curve \(y = x^2\text{.}\) Find equations for these two lines.
Remark: the point \((1,-3)\) does not lie on the curve \(y=x^2\text{.}\)
27 (✳)
For which values of \(a\) is the function
\begin{equation*}
f(x) =\left\{\begin{array}{ll}
0 & x\le 0\\
x^a \sin\frac{1}{x} & x \gt 0\end{array}\right.
\end{equation*}
differentiable at 0?