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Thomas Calculus

George B. Thomas Jr.

Chapter 4

Applications of Derivatives

Educators


Problem 1

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}

Runpeng L.
Numerade Educator

Problem 2

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}

Matt J.
Numerade Educator

Problem 3

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}

Runpeng L.
Numerade Educator

Problem 4

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}

Matt J.
Numerade Educator

Problem 5

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}

Runpeng L.
Numerade Educator

Problem 6

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}

Matt J.
Numerade Educator

Problem 7

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}

Runpeng L.
Numerade Educator

Problem 8

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}

Matt J.
Numerade Educator

Problem 9

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}

Runpeng L.
Numerade Educator

Problem 10

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?
$$
f^{\prime}(x)=3-\frac{6}{\sqrt{x}}, \quad x \neq 0
$$

Matt J.
Numerade Educator

Problem 11

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?
$$
f^{\prime}(x)=x^{-1 / 3}(x+2)
$$

Runpeng L.
Numerade Educator

Problem 12

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?
$$
f^{\prime}(x)=x^{-1 / 2}(x-3)
$$

Matt J.
Numerade Educator

Problem 13

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?
$$
f^{\prime}(x)=(\sin x-1)(2 \cos x+1), 0 \leq x \leq 2 \pi
$$

Check back soon!

Problem 14

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?
$$
f^{\prime}(x)=(\sin x+\cos x)(\sin x-\cos x), 0 \leq x \leq 2 \pi
$$

Matt J.
Numerade Educator

Problem 15

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.

Runpeng L.
Numerade Educator

Problem 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.

Matt J.
Numerade Educator

Problem 17

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.

Runpeng L.
Numerade Educator

Problem 18

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.

Matt J.
Numerade Educator

Problem 19

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.
$$
g(t)=-t^{2}-3 t+3
$$

Runpeng L.
Numerade Educator

Problem 20

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.
$$
g(t)=-3 t^{2}+9 t+5
$$

Matt J.
Numerade Educator

Problem 21

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.
$$
h(x)=-x^{3}+2 x^{2}
$$

Runpeng L.
Numerade Educator

Problem 22

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.
$$
h(x)=2 x^{3}-18 x
$$

Matt J.
Numerade Educator

Problem 23

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.
$$
f(\theta)=3 \theta^{2}-4 \theta^{3}
$$

Runpeng L.
Numerade Educator

Problem 24

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.
$$
f(\theta)=6 \theta-\theta^{3}
$$

Matt J.
Numerade Educator

Problem 25

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.
$$
f(r)=3 r^{3}+16 r
$$

Runpeng L.
Numerade Educator

Problem 26

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.
$$
h(r)=(r+7)^{3}
$$

Matt J.
Numerade Educator

Problem 27

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.
$$
f(x)=x^{4}-8 x^{2}+16
$$

Runpeng L.
Numerade Educator

Problem 28

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.
$$
g(x)=x^{4}-4 x^{3}+4 x^{2}
$$

Matt J.
Numerade Educator

Problem 29

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.
$$
H(t)=\frac{3}{2} t^{4}-t^{6}
$$

Runpeng L.
Numerade Educator

Problem 30

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.
$$
K(t)=15 t^{3}-t^{5}
$$

Matt J.
Numerade Educator

Problem 31

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.
$$
f(x)=x-6 \sqrt{x-1}
$$

Runpeng L.
Numerade Educator

Problem 32

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.
$$
g(x)=4 \sqrt{x}-x^{2}+3
$$

Matt J.
Numerade Educator

Problem 33

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.
$$
g(x)=x \sqrt{8-x^{2}}
$$

Runpeng L.
Numerade Educator

Problem 34

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.
$$
g(x)=x^{2} \sqrt{5-x}
$$

Matt J.
Numerade Educator

Problem 35

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.
$$
f(x)=\frac{x^{2}-3}{x-2}, \quad x \neq 2
$$

Runpeng L.
Numerade Educator

Problem 36

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.
$$
f(x)=\frac{x^{3}}{3 x^{2}+1}
$$

Matt J.
Numerade Educator

Problem 37

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.
$$
f(x)=x^{1 / 3}(x+8)
$$

Check back soon!

Problem 38

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.
$$
g(x)=x^{2 / 3}(x+5)
$$

Matt J.
Numerade Educator

Problem 39

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.
$$
h(x)=x^{1 / 3}\left(x^{2}-4\right)
$$

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Problem 40

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.$$
k(x)=x^{2 / 3}\left(x^{2}-4\right)
$$

Matt J.
Numerade Educator

Problem 41

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.
$$
f(x)=2 x-x^{2}, \quad-\infty<x \leq 2
$$

Runpeng L.
Numerade Educator

Problem 42

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.
$$
f(x)=(x+1)^{2}, \quad-\infty<x \leq 0
$$

Matt J.
Numerade Educator

Problem 43

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.
$$
g(x)=x^{2}-4 x+4, \quad 1 \leq x<\infty
$$

Runpeng L.
Numerade Educator

Problem 44

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.
$$
g(x)=-x^{2}-6 x-9, \quad-4 \leq x<\infty
$$

Matt J.
Numerade Educator

Problem 45

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.
$$
f(t)=12 t-t^{3}, \quad-3 \leq t<\infty
$$

Runpeng L.
Numerade Educator

Problem 46

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.
$$
f(t)=t^{3}-3 t^{2}, \quad-\infty< t \leq 3
$$

Matt J.
Numerade Educator

Problem 47

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.
$$
h(x)=\frac{x^{3}}{3}-2 x^{2}+4 x, \quad 0 \leq x<\infty
$$

Runpeng L.
Numerade Educator

Problem 48

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.
$$
k(x)=x^{3}+3 x^{2}+3 x+1, \quad-\infty<x \leq 0
$$

Matt J.
Numerade Educator

Problem 49

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.
$$
f(x)=\sqrt{25-x^{2}}, \quad-5 \leq x \leq 5
$$

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Problem 50

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.
$$
f(x)=\sqrt{x^{2}-2 x-3}, \quad 3 \leq x<\infty
$$

Matt J.
Numerade Educator

Problem 51

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.
$$
g(x)=\frac{x-2}{x^{2}-1}, \quad 0 \leq x<1
$$

Check back soon!

Problem 52

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.
$$
g(x)=\frac{x^{2}}{4-x^{2}}, \quad-2< x \leq 1
$$

Matt J.
Numerade Educator

Problem 53

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}. $
$$
f(x)=\sin 2 x, \quad 0 \leq x \leq \pi
$$

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Problem 54

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}. $
$$
f(x)=\sin x-\cos x, \quad 0 \leq x \leq 2 \pi
$$

Matt J.
Numerade Educator

Problem 55

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}. $
$$
f(x)=\sqrt{3} \cos x+\sin x, \quad 0 \leq x \leq 2 \pi
$$

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Problem 56

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}. $
$$
f(x)=-2 x+\tan x, \quad \frac{-\pi}{2}< x <\frac{\pi}{2}
$$

Matt J.
Numerade Educator

Problem 57

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}. $
$$
f(x)=\frac{x}{2}-2 \sin \frac{x}{2}, \quad 0 \leq x \leq 2 \pi
$$

Check back soon!

Problem 58

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}. $
$$
f(x)=-2 \cos x-\cos ^{2} x, \quad-\pi \leq x \leq \pi
$$

Matt J.
Numerade Educator

Problem 59

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}. $
$$
f(x)=\csc ^{2} x-2 \cot x, \quad 0<x<\pi
$$

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Problem 60

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}. $
$$
f(x)=\sec ^{2} x-2 \tan x, \quad \frac{-\pi}{2}< x <\frac{\pi}{2}
$$

Matt J.
Numerade Educator

Problem 61

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

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Problem 62

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

Matt J.
Numerade Educator

Problem 63

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

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Problem 64

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

Matt J.
Numerade Educator

Problem 65

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

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Problem 66

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 $(3,3)$ .

Matt J.
Numerade Educator

Problem 67

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

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Problem 68

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

Matt J.
Numerade Educator

Problem 69

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 $2.3 .$ .

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Problem 70

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.

Matt J.
Numerade Educator

Problem 71

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

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Problem 72

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 $(1,-1)$ .

Matt J.
Numerade Educator