Question 1

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Accepted Answer

The function $g(t) = \frac{t^2}{t^2 + 2}$ has a derivative $g'(t) = \frac{2t(t^2 + 2) - 2t(t^2)}{(t^2 + 2)^2} = \frac{4t}{(t^2 + 2)^2}$, which is zero at $t = 0$. This is the only critical point in the interval $[-3,3]$. At $t = 0$, $g(0) = 0$, which is the minimum value of the function on the interval. The function increases as $|t|$ increases, reaching a maximum at the endpoints $t = \pm 3$, where $g(3) = g(-3) = \frac{9}{11}$.

Question 2

Locate the absolute extrema of the function

Accepted Answer

D

Question 3

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Accepted Answer

To find the absolute extrema of $f(x) = x^3 - 12x$ on the interval $[0,4]$, we first find the critical points by taking the derivative of $f(x)$ and setting it equal to zero. The derivative is $f'(x) = 3x^2 - 12$. Setting $f'(x) = 0$ gives $3x^2 - 12 = 0$, or $x^2 = 4$, so $x = \pm 2$. However, only $x = 2$ is within the interval $[0,4]$. Evaluating $f(x)$ at the critical point and the endpoints of the interval:- $f(0) = 0^3 - 12(0) = 0$- $f(2) = 2^3 - 12(2) = 8 - 24 = -16$- $f(4) = 4^3 - 12(4) = 64 - 48 = 16$Thus, the absolute minimum is at $(2, -16)$ and the absolute maximum is at $(4, 16)$.

Question 4

extremum.

Accepted Answer

The answer of   extremum.    ...

Question 5

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Accepted Answer

The function $f(x) = \cos(\pi x)$ achieves its maximum value at $x = 0$, where $f(0) = \cos(0) = 1$, and its minimum value at $x = \frac{1}{2}$, where $f\left(\frac{1}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0$.

Question 6

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Accepted Answer

First, we find any critical points in the interval $[-60,60]$ by finding where the derivative of $f(x)$ is equal to $0$ or undefined:

\begin{align*}
    f(x) &= \frac{60x}{x^2+36} \
    f'(x) &= \frac{60(x^2+36) - 60x(2x)}{(x^2+36)^2} \
    &= \frac{-120x^2+2160}{(x^2+36)^2} \
    &= \frac{-120(x^2-18)}{(x^2+36)^2}
\end{align*}

$f'(x)$ is undefined when $x = \pm 6\sqrt{5}$. Notice that these values are outside of the interval $[-60,60]$.

$f'(x) = 0$ when $x = 0$.

Next, we check the endpoints of the interval:
\begin{align*}
    f(-60) &= \frac{-3600}{-3594} \approx -1.0022 \
    f(60) &= \frac{3600}{3594} \approx 1.0022 \
    f(0) &= 0
\end{align*}

Therefore, $f$ attains an absolute maximum at $x = 6$, and $f$ attains an absolute minimum at $x = -6$ (since $f$ is an even function). The respective values of these extrema are $f(6) = \frac{180}{17}$ and $f(-6) = -\frac{180}{17}$.

Thus, the answer is $\boxed{	extbf{(C)}}$.

Question 7

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Accepted Answer

First, we need to find the critical points of the function in the given closed interval. These critical points are the endpoints, which are $x=0$ and $x=\frac{1}{3}$. Next, we evaluate the function at these points and find which one gives the highest and which one gives the lowest value. 
$f(0) = \sin(0) = 0$
$f(\frac{1}{3}) = \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$
Therefore, the absolute minimum is 0 and it occurs at the left endpoint $x=0$. The absolute maximum is $\frac{\sqrt{3}}{2}$ and it occurs at the right endpoint $x=\frac{1}{3}$. Choice A is the only one that matches this result.

Question 8

Determine whether Rolle's Theorem can be applied to the function

Accepted Answer

For Rolle's Theorem to apply, the function must be continuous on the closed interval $( - 1,3 )$ and differentiable on the open interval $ ( - 1,3 )$. 
$f(x) = x^2 - 2x - 3$ is a polynomial function and is therefore continuous and differentiable for all real numbers.
Furthermore, $f(-1) = 6$ and $f(3)=-6$ so the function takes on different values at the endpoints of the interval. By Rolle's Theorem, there exists at least one point $c$ in the open interval $( - 1,3 )$ such that $f'(c) = 0$.
Now, 
$$f(x) = x^2 - 2x - 3$$
$$\Rightarrow f'(x) = 2x -2$$
Setting $f'(x) = 0$ and solving for $x$ gives us 
$$2x-2 = 0 \Rightarrow x = 1$$
Since $x = 1$ lies in the interval $( - 1,3 )$, we can conclude that Rolle's Theorem applies and $c = 1$ is the only point in the open interval $( - 1,3 )$ such that $f'(c) = 0$. Thus, the answer is A.

Question 9

any critical numbers of the function

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The answer of any critical numbers of the function ...

Question 10

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Accepted Answer

To find the absolute extrema of the function $y = x^2 - 2\cos(x)$ on the interval $[-3,5]$, we first evaluate the function at the endpoints and then find its critical points by taking the derivative and setting it equal to zero. The derivative is $y' = 2x + 2\sin(x)$. Setting $y' = 0$ gives critical points, one of which is at $x = 0$, leading to $y(0) = -2$, which is the absolute minimum within the interval. Evaluating the function at the endpoints, $y(-3) = 9 - 2\cos(-3) \approx 10.9800$ and $y(5) = 25 - 2\cos(5) \approx 24.4327$, confirms the absolute maximum at the right endpoint.

Question 11

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Accepted Answer

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Question 12

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Accepted Answer

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Question 13

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Accepted Answer

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Question 14

extremum point .

Accepted Answer

The answer of   extremum point .  ...

Question 15

a graphing utility to graph the function   absolute extrema of the function on the interval

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The answer of a graphing utility to graph the function...

Question 16

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Accepted Answer

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Question 17

all critical numbers of of the function

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The answer of all critical numbers of of the function...

Question 18

a computer algebra system to graph the function   absolute extrema on the interval

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The answer of a computer algebra system to graph the...

Question 19

Locate the absolute extrema of the function

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The answer of Locate the absolute extrema of the function...

Question 20

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Accepted Answer

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Question 21

height of an object t seconds after it is dropped from a height of 550 meters is

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The answer of height of an object t seconds after...

Question 22

Identify the open intervals where the function   decreasing.

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The answer of Identify the open intervals where the function...

Question 23

company introduces a new product for which the number of units sold S is

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The answer of company introduces a new product for which...

Question 24

Identify the open intervals on which the function   increasing or decreasing.

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The answer of Identify the open intervals on which the...

Question 25

company introduces a new product for which the number of units sold S is

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The answer of company introduces a new product for which...

Question 26

Determine whether the Mean Value Theorem can be applied to the function

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The answer of Determine whether the Mean Value Theorem can...

Question 27

plane begins its takeoff at 2:00 P.M. on a 2200-mile flight. After 12.5 hours, the plane arrives at its destination. Explain why there are at least two times during the flight when the
Speed of the plane is 100 miles per hour.

A)By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 303 mi/hr.The speed was 100 mi/hr when the plane was accelerating to
303 mi/hr and decelerating from 303 mi/hr.
B)By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 152 mi/hr.The speed was 100 mi/hr when the plane was accelerating to
152 mi/hr and decelerating from 152 mi/hr.
C)By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 88 mi/hr.The speed was 100 mi/hr when the plane was accelerating to 88
Mi/hr and decelerating from 88 mi/hr.
D)By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 117 mi/hr.The speed was 100 mi/hr when the plane was accelerating to
117 mi/hr and decelerating from 117 mi/hr.
E)By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 176 mi/hr.The speed was 100 mi/hr when the plane was accelerating to
176 mi/hr and decelerating from 176 mi/hr.

Accepted Answer

The answer of plane begins its takeoff at 2:00 P.M....

Question 28

height of an object t seconds after it is dropped from a height of 250 meters is

Accepted Answer

The answer of height of an object t seconds after...

Question 29

Determine whether Rolle's Theorem can be applied to the function

Accepted Answer

The answer of Determine whether Rolle's Theorem can be applied...

Question 30

Determine whether the Mean Value Theorem can be applied to the function

Accepted Answer

The answer of Determine whether the Mean Value Theorem can...

Question 31

Determine whether the Mean Value Theorem can be applied to the function

Accepted Answer

The answer of Determine whether the Mean Value Theorem can...

Question 32

Determine whether Rolle's Theorem can be applied to the function

Accepted Answer

The answer of Determine whether Rolle's Theorem can be applied...

Question 33

decreasing.

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The answer of   decreasing.  ...

Question 34

Which of the following functions passes through the point

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The answer of Which of the following functions passes through...

Question 35

Determine whether Rolle's Theorem can be applied to the function

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The answer of Determine whether Rolle's Theorem can be applied...

Question 36

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Accepted Answer

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Question 37

Determine whether Rolle's Theorem can be applied to

Accepted Answer

The answer of Determine whether Rolle's Theorem can be applied...

Question 38

a function f that has derivative   the point (5,6).

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The answer of a function f that has derivative ...

Question 39

ordering and transportation cost C for components used in a manufacturing

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The answer of ordering and transportation cost C for components...

Question 40

intervals on which the function is increasing or decreasing.

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The answer of   intervals on which the function...

Question 41

interval Round numerical values in your answer to three decimal places.

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The answer of   interval Round numerical values in...

Question 42

the points of inflection and discuss the concavity of the function.

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The answer of the points of inflection and discuss the...

Question 43

Derivative Test.

Accepted Answer

The answer of   Derivative Test.  ...

Question 44

Determine the open intervals on which the graph of   is concave downward or concave upward.

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The answer of Determine the open intervals on which the...

Question 45

the points of inflection and discuss the concavity of the function

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The answer of the points of inflection and discuss the...

Question 46

resistance R of a certain type of resistor is   where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra&#10;System to find

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The answer of resistance R of a certain type of...

Question 47

all points of inflection on the graph of the function

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The answer of all points of inflection on the graph...

Question 48

Determine the open intervals on which the graph of the function   concave upward or concave downward.

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The answer of Determine the open intervals on which the...

Question 49

the First Derivative Test. Round numerical values in your answer to three decimal places.

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The answer of   the First Derivative Test. Round...

Question 50

Then use a graphing utility to confirm your results.

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The answer of     Then use a...

Question 51

ball bearing is placed on an inclined plane and begins to roll. The angle of

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The answer of ball bearing is placed on an inclined...

Question 52

(a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and&#10;(c) Apply the First Derivative Test to identify all relative extrema.

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The answer of   (a) Find the critical numbers...

Question 53

Determine the open intervals on which the graph of the function

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The answer of Determine the open intervals on which the...

Question 54

downward or concave upward.

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The answer of   downward or concave upward. ...

Question 55

the open interval(s) on which the function   is increasing in the interval Round numerical values in your answer to three decimal places.

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The answer of the open interval(s) on which the function...

Question 56

concave downward or concave upward.

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The answer of   concave downward or concave upward....

Question 57

the relative minima of   the First Derivative Test. Round numerical values in your answer to three decimal places.

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The answer of the relative minima of   the...

Question 58

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Accepted Answer

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Question 59

measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to find the critical number of the function. Round numerical values in your answer to the
Nearest whole number.