Thomas Calculus

Answer the following questions about the functions whose derivatives are given:
\begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation}
\begin{equation}f^{\prime}(x)=x(x-1)\end{equation}

Answer the following questions about the functions whose derivatives are given:
\begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation}
\begin{equation}f^{\prime}(x)=(x-1)(x+2)\end{equation}

Answer the following questions about the functions whose derivatives are given:
\begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation}
\begin{equation}f^{\prime}(x)=(x-1)^{2}(x+2)\end{equation}

Answer the following questions about the functions whose derivatives are given:
\begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation}
\begin{equation}f^{\prime}(x)=(x-1)^{2}(x+2)^{2}\end{equation}

Answer the following questions about the functions whose derivatives are given:
\begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation}
\begin{equation}f^{\prime}(x)=(x-1)(x+2)(x-3)\end{equation}

Answer the following questions about the functions whose derivatives are given:
\begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation}
\begin{equation}f^{\prime}(x)=(x-7)(x+1)(x+5)\end{equation}

Answer the following questions about the functions whose derivatives are given:
\begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation}
\begin{equation}f^{\prime}(x)=\frac{x^{2}(x-1)}{x+2}, \quad x \neq-2\end{equation}

Answer the following questions about the functions whose derivatives are given:
\begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation}
\begin{equation}f^{\prime}(x)=\frac{(x-2)(x+4)}{(x+1)(x-3)}, \quad x \neq-1,3\end{equation}

Answer the following questions about the functions whose derivatives are given:
\begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation}
\begin{equation}f^{\prime}(x)=1-\frac{4}{x^{2}}, \quad x \neq 0\end{equation}

Answer the following questions about the functions whose derivatives
are given.
a. What are the critical points of $f ?$
b. On what open intervals is $f$ increasing or decreasing?
c. At what points, if any, does $f$ assume local maximum and minimum values?
$$
f^{\prime}(x)=3-\frac{6}{\sqrt{x}}, \quad x \neq 0
$$

Answer the following questions about the functions whose derivatives
are given.
a. What are the critical points of $f ?$
b. On what open intervals is $f$ increasing or decreasing?
c. At what points, if any, does $f$ assume local maximum and minimum values?
$$
f^{\prime}(x)=x^{-1 / 3}(x+2)
$$

Answer the following questions about the functions whose derivatives
are given.
a. What are the critical points of $f ?$
b. On what open intervals is $f$ increasing or decreasing?
c. At what points, if any, does $f$ assume local maximum and minimum values?
$$
f^{\prime}(x)=x^{-1 / 2}(x-3)
$$

Answer the following questions about the functions whose derivatives
are given.
a. What are the critical points of $f ?$
b. On what open intervals is $f$ increasing or decreasing?
c. At what points, if any, does $f$ assume local maximum and minimum values?
$$
f^{\prime}(x)=(\sin x-1)(2 \cos x+1), 0 \leq x \leq 2 \pi
$$

Answer the following questions about the functions whose derivatives
are given.
a. What are the critical points of $f ?$
b. On what open intervals is $f$ increasing or decreasing?
c. At what points, if any, does $f$ assume local maximum and minimum values?
$$
f^{\prime}(x)=(\sin x+\cos x)(\sin x-\cos x), 0 \leq x \leq 2 \pi
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
g(t)=-t^{2}-3 t+3
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
g(t)=-3 t^{2}+9 t+5
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
h(x)=-x^{3}+2 x^{2}
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
h(x)=2 x^{3}-18 x
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
f(\theta)=3 \theta^{2}-4 \theta^{3}
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
f(\theta)=6 \theta-\theta^{3}
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
f(r)=3 r^{3}+16 r
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
h(r)=(r+7)^{3}
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
f(x)=x^{4}-8 x^{2}+16
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
g(x)=x^{4}-4 x^{3}+4 x^{2}
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
H(t)=\frac{3}{2} t^{4}-t^{6}
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
K(t)=15 t^{3}-t^{5}
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
f(x)=x-6 \sqrt{x-1}
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
g(x)=4 \sqrt{x}-x^{2}+3
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
g(x)=x \sqrt{8-x^{2}}
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
g(x)=x^{2} \sqrt{5-x}
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
f(x)=\frac{x^{2}-3}{x-2}, \quad x \neq 2
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
f(x)=\frac{x^{3}}{3 x^{2}+1}
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
f(x)=x^{1 / 3}(x+8)
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
g(x)=x^{2 / 3}(x+5)
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$
h(x)=x^{1 / 3}\left(x^{2}-4\right)
$$

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.$$
k(x)=x^{2 / 3}\left(x^{2}-4\right)
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
f(x)=2 x-x^{2}, \quad-\infty<x \leq 2
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
f(x)=(x+1)^{2}, \quad-\infty<x \leq 0
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
g(x)=x^{2}-4 x+4, \quad 1 \leq x<\infty
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
g(x)=-x^{2}-6 x-9, \quad-4 \leq x<\infty
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
f(t)=12 t-t^{3}, \quad-3 \leq t<\infty
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
f(t)=t^{3}-3 t^{2}, \quad-\infty< t \leq 3
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
h(x)=\frac{x^{3}}{3}-2 x^{2}+4 x, \quad 0 \leq x<\infty
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
k(x)=x^{3}+3 x^{2}+3 x+1, \quad-\infty<x \leq 0
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
f(x)=\sqrt{25-x^{2}}, \quad-5 \leq x \leq 5
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
f(x)=\sqrt{x^{2}-2 x-3}, \quad 3 \leq x<\infty
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
g(x)=\frac{x-2}{x^{2}-1}, \quad 0 \leq x<1
$$

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$
g(x)=\frac{x^{2}}{4-x^{2}}, \quad-2< x \leq 1
$$

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}. $
$$
f(x)=\sin 2 x, \quad 0 \leq x \leq \pi
$$

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}. $
$$
f(x)=\sin x-\cos x, \quad 0 \leq x \leq 2 \pi
$$

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}. $
$$
f(x)=\sqrt{3} \cos x+\sin x, \quad 0 \leq x \leq 2 \pi
$$

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}. $
$$
f(x)=-2 x+\tan x, \quad \frac{-\pi}{2}< x <\frac{\pi}{2}
$$

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}. $
$$
f(x)=\frac{x}{2}-2 \sin \frac{x}{2}, \quad 0 \leq x \leq 2 \pi
$$

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}. $
$$
f(x)=-2 \cos x-\cos ^{2} x, \quad-\pi \leq x \leq \pi
$$

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}. $
$$
f(x)=\csc ^{2} x-2 \cot x, \quad 0<x<\pi
$$

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}. $
$$
f(x)=\sec ^{2} x-2 \tan x, \quad \frac{-\pi}{2}< x <\frac{\pi}{2}
$$

The graph of $f^{\prime}$ is given. Assume that $f$ is continuous and determine the $x$ -values corresponding to local minima and local maxima.

The graph of $f^{\prime}$ is given. Assume that $f$ is continuous and determine the $x$ -values corresponding to local minima and local maxima.

Show that the functions have local extreme values at the given values of $\theta,$ and say which kind of local extreme the function has.
$$
h(\theta)=3 \cos \frac{\theta}{2}, \quad 0 \leq \theta \leq 2 \pi, \quad \text { at } \theta=0 \text { and } \theta=2 \pi
$$

Show that the functions have local extreme values at the given values of $\theta,$ and say which kind of local extreme the function has.
$$
h(\theta)=5 \sin \frac{\theta}{2}, \quad 0 \leq \theta \leq \pi, \quad \text { at } \theta=0 \text { and } \theta=\pi
$$

Sketch the graph of a differentiable function $y=f(x)$ through the
point $(1,1)$ if $f^{\prime}(1)=0$ and
$$
\begin{array}{l}{\text { a. } f^{\prime}(x)>0 \text { for } x<1 \text { and } f^{\prime}(x)<0 \text { for } x>1} \\ {\text { b. } f^{\prime}(x)<0 \text { for } x<1 \text { and } f^{\prime}(x)>0 \text { for } x>1} \\ {\text { c. } f^{\prime}(x)>0 \text { for } x \neq 1} \\ {\text { d. } f^{\prime}(x)<0 \text { for } x \neq 1}\end{array}
$$

Sketch the graph of a differentiable function $y=f(x)$ that has
a. a local minimum at $(1,1)$ and a local maximum at $(3,3)$
b. a local maximum at $(1,1)$ and a local minimum at $(3,3)$
c. 1 ocal maxima at $(1,1)$ and $(3,3)$
d. local minima at $(1,1)$ and $(3,3)$ .

Sketch the graph of a continuous function $y=g(x)$ such that
$$\begin{array}{c}{\text { a. } g(2)=2,0 < g^{\prime}<1 \text { for } x < 2, g^{\prime}(x) \rightarrow 1^{-} \text { as } x \rightarrow 2^{-}} \\ {-1< g^{\prime} < 0 \text { for } x > 2, \text { and } g^{\prime}(x) \rightarrow-1^{+} \text { as } x \rightarrow 2^{+}}\end{array}$$
$$\begin{array}{l}{\text { b. } g(2)=2, g^{\prime}<0 \text { for } x<2, g^{\prime}(x) \rightarrow-\infty \text { as } x \rightarrow 2^{-}} \\ {g^{\prime}>0 \text { for } x>2, \text { and } g^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 2^{+} \text { . }}\end{array}$$

Sketch the graph of a continuous function $y=h(x)$ such that
$$
\begin{array}{l}{\text { a. } h(0)=0,-2 \leq h(x) \leq 2 \text { for all } x, h^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 0^{-}} \\ {\text { and } h^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 0^{+}} \\ {\text { b. } h(0)=0,-2 \leq h(x) \leq 0 \text { for all } x, h^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 0^{-}} \\ {\quad \text { and } h^{\prime}(x) \rightarrow-\infty \text { as } x \rightarrow 0^{+} \text { . }}\end{array}
$$

Discuss the extreme-value behavior of the function $f(x)=$ $x \sin (1 / x), x \neq 0 .$ How many critical points does this function have? Where are they located on the $x$ -axis? Does $f$ have an absolute minimum? An absolute maximum? (See Exercise 49 in Section $2.3 .$ .

Find the open intervals on which the function $f(x)=a x^{2}+$ $b x+c, a \neq 0,$ is increasing and decreasing. Describe the reasoning behind your answer.

Determine the values of constants $a$ and $b$ so that $f(x)=a x^{2}+b x$
has an absolute maximum at the point $(1,2) .$

Determine the values of constants $a, b, c,$ and $d$ so that
$f(x)=a x^{3}+b x^{2}+c x+d$ has a local maximum at the point
$(0,0)$ and a local minimum at the point $(1,-1)$ .