1
Suppose \(f(x)\) is a function given by
\begin{equation*}
f(x)= \frac{g(x)}{x^2-9}
\end{equation*}
where \(g(x)\) is also a function. True or false: \(f(x)\) has a vertical asymptote at \(x=-3\text{.}\)
Subsection 3.6.7 Exercises
Subsubsection Exercises for § 3.6.1
Exercises — Stage 1
Exercises — Stage 2
2
Match the functions \(f(x)\text{,}\) \(g(x)\text{,}\) \(h(x)\text{,}\) and \(k(x)\) to the curves \(y=A(x)\) through \(y=D(x)\text{.}\)
\begin{align*}
f(x) & =\sqrt{x^2+1} &
g(x) &=\sqrt{x^2-1} \\
h(x) &=\sqrt{x^2+4} &
k(x) &=\sqrt{x^2-4}
\end{align*}
3
Below is the graph of
\begin{equation*}
y=f(x)=\sqrt{\log^2(x+p)}
\end{equation*}
- What is \(p\text{?}\)
- What is \(b\) (marked on the graph)?
- What is the \(x\)-intercept of \(f(x)\text{?}\)
Remember \(\log(x+p)\) is the natural logarithm of \(x+p\text{,}\) \(\log_e(x+p)\text{.}\)
4
Find all asymptotes of \(f(x)=\dfrac{x(2x+1)(x-7)}{3x^3-81}\text{.}\)
5
Find all asymptotes of \(f(x)=10^{3x-7}\text{.}\)
Subsubsection Exercises for § 3.6.2
Exercises — Stage 1
1
Match each function graphed below to its derivative from the list. (For example, which function on the list corresponds to \(A'(x)\text{?}\))
The \(y\)-axes have been scaled to make the curve's behaviour clear, so the vertical scales differ from graph to graph.
\(l(x)=(x-2)^4\)
\(m(x)=(x-2)^4(x+2)\)
\(n(x)=(x-2)^2(x+2)^2\)
\(o(x)=(x-2)(x+2)^3\)
\(p(x)=(x+2)^4\)
Exercises — Stage 2
2 (✳)
Find the largest open interval on which \(f(x)=\dfrac{e^x}{x+3}\) is increasing.
3 (✳)
Find the largest open interval on which \(f(x)=\dfrac{\sqrt{x-1}}{2x+4}\) is increasing.
4 (✳)
Find the largest open interval on which \(f(x)=2\arctan (x) - \log(1+x^2)\) is increasing.
Subsubsection Exercises for § 3.6.3
Exercises — Stage 1
1
On the graph below, mark the intervals where \(f''(x) \gt 0\) (i.e. \(f(x)\) is concave up) and where \(f''(x) \lt 0\) (i.e. \(f(x)\) is concave down).
2
Sketch a curve that is:
- concave up when \(|x| \gt 5\text{,}\)
- concave down when \(|x| \lt 5\text{,}\)
- increasing when \(x \lt 0\text{,}\) and
- decreasing when \(x \gt 0\text{.}\)
3
Suppose \(f(x)\) is a function whose second derivative exists and is continuous for all real numbers.
True or false: if \(f''(3)=0\text{,}\) then \(x=3\) is an inflection point of \(f(x)\text{.}\)
Remark: compare to Question 3.6.7.7
Exercises — Stage 2
4 (✳)
Find all inflection points for the graph of \(f(x)=3x^5-5x^4+13x\text{.}\)
Exercises — Stage 3
Questions 3.6.7.5 through 3.6.7.7 ask you to show that certain things are true. Give a clear explanation using concepts and theorems from this textbook.
5 (✳)
Let
\begin{equation*}
f(x)=\frac{x^5}{20}+\frac{5x^3}{6}-10x^2+500x+1000
\end{equation*}
Show that \(f(x)\) has exactly one inflection point.
6 (✳)
Let \(f(x)\) be a function whose first two derivatives exist everywhere, and \(f''(x) \gt 0\) for all \(x\text{.}\)
- Show that \(f(x)\) has at most one critical point and that any critical point is an absolute minimum for \(f(x)\text{.}\)
- Show that the maximum value of \(f(x)\) on any finite interval occurs at one of the endpoints of the interval.
7
Suppose \(f(x)\) is a function whose second derivative exists and is continuous for all real numbers, and \(x=3\) is an inflection point of \(f(x)\text{.}\) Use the Intermediate Value Theorem to show that \(f''(3)=0\text{.}\)
Remark: compare to Question 3.6.7.3.
Subsubsection Exercises for § 3.6.4
Exercises — Stage 1
1
What symmetries (even, odd, periodic) does the function graphed below have?
2
What symmetries (even, odd, periodic) does the function graphed below have?
3
Suppose \(f(x)\) is an even function defined for all real numbers. Below is the curve \(y=f(x)\) when \(x \gt 0\text{.}\) Complete the sketch of the curve.
4
Suppose \(f(x)\) is an odd function defined for all real numbers. Below is the curve \(y=f(x)\) when \(x \gt 0\text{.}\) Complete the sketch of the curve.
Exercises — Stage 2
5
\begin{equation*}
f(x)=\frac{x^4-x^6}{e^{x^2}}
\end{equation*}
Show that \(f(x)\) is even.
6
\begin{equation*}
f(x)=\sin(x)+\cos\left(\frac{x}{2}\right)
\end{equation*}
Show that \(f(x)\) is periodic.
7
\begin{equation*}
f(x)=x^4+5x^2+\cos\left(x^3\right)
\end{equation*}
What symmetries (even, odd, periodic) does \(f(x)\) have?
8
\begin{equation*}
f(x)=x^5+5x^4
\end{equation*}
What symmetries (even, odd, periodic) does \(f(x)\) have?
9
\begin{equation*}
f(x)=\tan\left(\pi x\right)
\end{equation*}
What is the period of \(f(x)\text{?}\)
Exercises — Stage 3
10
\begin{equation*}
f(x)=\tan\left(3 x\right)+\sin\left(4 x\right)
\end{equation*}
What is the period of \(f(x)\text{?}\)
Subsubsection Exercises for § 3.6.6
Exercises — Stage 1
In Questions 3.6.7.6 and 3.6.7.7, you will sketch the graphs of functions with an exponential component. In the next section, you will learn how to find their horizontal asymptotes, but for now these are given to you.
In Questions 3.6.7.8 and 3.6.7.9, you will sketch the graphs of functions that have a trigonometric component.
1 (✳)
Let \(f(x) = x\sqrt{3 - x}\text{.}\)
- Find the domain of \(f(x)\text{.}\)
- Determine the \(x\)-coordinates of the local maxima and minima (if any) and intervals where \(f(x)\) is increasing or decreasing.
- Determine intervals where \(f(x)\) is concave upwards or downwards, and the \(x\) coordinates of inflection points (if any). You may use, without verifying it, the formula \(f''(x) = (3x -12)(3 - x)^{-3/2}/4\text{.}\)
- There is a point at which the tangent line to the curve \(y = f(x)\) is vertical. Find this point.
- Sketch the graph \(y = f(x)\text{,}\) showing the features given in items (a) to (d) above and giving the \((x, y)\) coordinates for all points occurring above.
2 (✳)
Sketch the graph of
\begin{equation*}
f(x)= \dfrac{x^3-2}{x^4}.
\end{equation*}
Indicate the critical points, local and absolute maxima and minima, vertical and horizontal asymptotes, inflection points and regions where the curve is concave upward or downward.
3 (✳)
The first and second derivatives of the function \(f(x)=\dfrac{x^4}{1+x^3}\) are:
\begin{equation*}
f'(x)=\frac{4x^3+x^6}{(1+x^3)^2}\qquad\hbox{and}\qquad f''(x)=\frac{12x^2-6x^5}{(1+x^3)^3}
\end{equation*}
Graph \(f(x)\text{.}\) Include local and absolute maxima and minima, regions where \(f(x)\) is increasing or decreasing, regions where the curve is concave upward or downward, and any asymptotes.
4 (✳)
The first and second derivatives of the function \(f(x)=\dfrac{x^3}{1-x^2}\) are:
\begin{equation*}
f'(x)=\frac{3x^2-x^4}{(1-x^2)^2}\qquad\hbox{and}\qquad f''(x)=\frac{6x+2x^3}{(1-x^2)^3}
\end{equation*}
Graph \(f(x)\text{.}\) Include local and absolute maxima and minima, regions where the curve is concave upward or downward, and any asymptotes.
5 (✳)
The function \(f(x)\) is defined by
\begin{equation*}
f(x) =
\left\{\begin{array}{lc}
e^x &x \lt 0\\
\frac{x^2+3}{3(x+1)} & x \ge 0
\end{array}\right.
\end{equation*}
- Explain why \(f(x)\) is continuous everywhere.
-
Determine all of the following if they are present:
- \(x\)--coordinates of local maxima and minima, intervals where \(f(x)\) is increasing or decreasing;
- intervals where \(f(x)\) is concave upwards or downwards;
- equations of any horizontal or vertical asymptotes.
- Sketch the graph of \(y = f(x)\text{,}\) giving the \((x, y)\) coordinates for all points of interest above.
6 (✳)
The function \(f(x)\) and its derivative are given below:
\begin{equation*}
f(x)=(1+2x)e^{-x^2}\qquad\hbox{and}\qquad f'(x)=2(1-x-2x^2)e^{-x^2}
\end{equation*}
Sketch the graph of \(f(x)\text{.}\) Indicate the critical points, local and/or absolute maxima and minima, and asymptotes. Without actually calculating the inflection points, indicate on the graph their approximate location.
Note: \(\ds\lim_{x \to \pm\infty}f(x)=0\text{.}\)
7 (✳)
Consider the function \(f(x) = xe^{-x^2/2}\text{.}\)
Note: \(\ds\lim_{x \to \pm\infty}f(x)=0\text{.}\)
- Find all inflection points and intervals of increase, decrease, convexity up, and convexity down. You may use without proof the formula \(f''(x) = (x^3-3x)e^{-x^2/2}\text{.}\)
- Find local and global minima and maxima.
- Use all the above to draw a graph for \(f\text{.}\) Indicate all special points on the graph.
8
Use the techniques from this section to sketch the graph of \(f(x)=x+2\sin x\text{.}\)
9 (✳)
\begin{equation*}
f(x) = 4\sin x - 2\cos 2x
\end{equation*}
Graph the equation \(y = f(x)\text{,}\) including all important features. (In particular, find all local maxima and minima and all inflection points.) Additionally, find the maximum and minimum values of \(f(x)\) on the interval \([0,\pi]\text{.}\)
10
Sketch the curve \(y=\sqrt[3]{\dfrac{x+1}{x^2}}\text{.}\)
You may use the facts \(y'(x)=\dfrac{-(x+2)}{3x^{5/3}(x+1)^{2/3}}\) and \(y''(x)=\dfrac{4x^2+16x+10}{9x^{8/3}(x+1)^{5/3}}\text{.}\)
Exercises — Stage 3
11 (✳)
A function \(f(x)\) defined on the whole real number line satisfies the following conditions
\begin{equation*}
f(0)=0\qquad
f(2)=2\qquad
\lim_{x\rightarrow+\infty}f(x)=0\qquad
f'(x)=K(2x-x^2)e^{-x}
\end{equation*}
for some positive constant \(K\text{.}\) (Read carefully: you are given the derivative of \(f(x)\text{,}\) not \(f(x)\) itself.)
- Determine the intervals on which \(f\) is increasing and decreasing and the location of any local maximum and minimum values of \(f\text{.}\)
- Determine the intervals on which \(f\) is concave up or down and the \(x\)--coordinates of any inflection points of \(f\text{.}\)
- Determine \(\lim\limits_{x\rightarrow-\infty}f(x)\text{.}\)
- Sketch the graph of \(y=f(x)\text{,}\) showing any asymptotes and the information determined in parts 3.6.7.11.a and 3.6.7.11.b.
12 (✳)
Let \(f(x) = e^{-x}\) , \(x \ge 0\text{.}\)
- Sketch the graph of the equation \(y = f(x)\text{.}\) Indicate any local extrema and inflection points.
- Sketch the graph of the inverse function \(y = g (x)=f^{-1}(x)\text{.}\)
- Find the domain and range of the inverse function \(g(x)= f^{-1}(x)\text{.}\)
- Evaluate \(g'(\half)\text{.}\)
13 (✳)
- Sketch the graph of \(y=f(x)=x^5-x\text{,}\) indicating asymptotes, local maxima and minima, inflection points, and where the graph is concave up/concave down.
- Consider the function \(f(x)=x^5-x+k\text{,}\) where \(k\) is a constant, \(-\infty \lt k \lt \infty\text{.}\) How many roots does the function have? (Your answer might depend on the value of \(k\text{.}\))
14 (✳)
The hyperbolic trigonometric functions \(\sinh(x)\) and \(\cosh(x)\) are defined by
\begin{equation*}
\sinh(x)=\dfrac{e^x-e^{-x}}{2}\qquad
\cosh(x)=\dfrac{e^x+e^{-x}}{2}
\end{equation*}
They have many properties that are similar to corresponding properties of \(\sin(x)\) and \(\cos(x)\text{.}\) In particular, it is easy to see that
\begin{equation*}
\diff{}{x} \sinh(x)=\cosh(x)\qquad
\diff{}{x} \cosh(x)=\sinh(x)\qquad
\cosh^2(x)-\sinh^2(x)=1
\end{equation*}
You may use these properties in your solution to this question.
- Sketch the graphs of \(\sinh(x)\) and \(\cosh(x)\text{.}\)
- Define inverse hyperbolic trigonometric functions \(\sinh^{-1}(x)\) and \(\cosh^{-1}(x)\text{,}\) carefully specifing their domains of definition. Sketch the graphs of \(\sinh^{-1}(x)\) and \(\cosh^{-1}(x)\text{.}\)
- Find \(\diff{}{x}\left\{ \cosh^{-1}(x)\right\}\text{.}\)