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Let
f(x)=x^(3)-6x^(2)+12 x and let
c be the number that satisfies the Mean Value Theorem for
f on the interval
[0,3].
What is
c ?
Choose 1 answer:
(A) 0
(B) 1
(C) 2
(D) 3

Let $f(x)=x^{3}-6 x^{2}+12 x$ and let $c$ be the number that satisfies the Mean Value Theorem for $f$ on the interval $[0,3]$ . $\newline$ What is $c$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $0$ $\newline$ (B) $1$ $\newline$ (C) $2$ $\newline$ (D) $3$

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Full solution

Q. Let

f(x)=x^{3}-6 x^{2}+12 x

and let

c

be the number that satisfies the Mean Value Theorem for

f

on the interval

[0,3]

.

\newline

What is

c

?

\newline

Choose

1

answer:

\newline

(A)

0

\newline

(B)

1

\newline

(C)

2

\newline

(D)

3

State MVT: State the Mean Value Theorem (MVT). The Mean Value Theorem states that if a function $f$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ , then there exists at least one number $c$ in the interval $(a, b)$ such that $f'(c)$ is equal to the average rate of change of the function over $[a, b]$ . Mathematically, this is expressed as: $f'(c) = \frac{f(b) - f(a)}{b - a}$
Calculate $f(a)$ and $f(b)$ : Calculate $f(a)$ and $f(b)$ where $a=0$ and $b=3$ .
$f(x) = x^3 - 6x^2 + 12x$
$f(0) = 0^3 - 6\cdot0^2 + 12\cdot0 = 0$
$f(3) = 3^3 - 6\cdot3^2 + 12\cdot3 = 27 - 54 + 36 = 9$
Calculate average rate of change: Calculate the average rate of change of $f(x)$ over the interval $[0, 3]$ . Using the values from Step $2$ : $(f(3) - f(0)) / (3 - 0) = (9 - 0) / 3 = 9 / 3 = 3$
Find $f'(x)$ : Find $f'(x)$ , the derivative of $f(x)$ .
$f(x) = x^3 - 6x^2 + 12x$
$f'(x) = 3x^2 - 12x + 12$
Set $f'(c)$ equal to rate of change: Set $f'(c)$ equal to the average rate of change and solve for $c$ . $\newline$ $f'(c) = 3c^2 - 12c + 12$ $\newline$ Set $f'(c)$ equal to $3$ (from Step $3$ ): $\newline$ $3c^2 - 12c + 12 = 3$
Solve equation for $c$ : Solve the equation $3c^2 - 12c + 12 = 3$ for $c$ . $\newline$ Subtract $3$ from both sides: $\newline$ $3c^2 - 12c + 9 = 0$ $\newline$ Divide by $3$ : $\newline$ $c^2 - 4c + 3 = 0$ $\newline$ Factor the quadratic: $\newline$ $(c - 3)(c - 1) = 0$
Find values of c: Find the values of c that satisfy the equation. $\newline$ $c - 3 = 0$ or $c - 1 = 0$ $\newline$ $c = 3$ or $c = 1$
Determine $c$ in interval: Determine which value of $c$ is in the open interval $(0, 3)$ . $\newline$ Since $c$ must be in the interval $(0, 3)$ , we exclude $c = 3$ and choose $c = 1$ .

More problems from Euler's method

Question

g(x)=x^{3}+12 x^{2}+36 x

c

be the number that satisfies the Mean Value Theorem for

g

on the interval

-8 \leq x \leq-2

\newline

c

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1

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Question

h(x)=x^{3}-6 x^{2}-10 x

c

be the number that satisfies the Mean Value Theorem for

h

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\newline

c

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Question

g(x)=2 x^{3}-21 x^{2}+60 x

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What is the absolute maximum value of

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1

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25

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42

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36

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f(x)=2 x^{3}+21 x^{2}+36 x

\newline

What is the absolute maximum value of

f

over the closed interval

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1

\newline

32

\newline

\mathbf{1 8 0}

\newline

\mathbf{1 0 8}

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Question

h(x)=x^{3}+6 x^{2}+2

\newline

What is the absolute minimum value of

h

over the closed interval

-6 \leq x \leq 2

\newline

1

\newline

34

\newline

-34

\newline

2

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Question

1,000=20z^{2}

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How many distinct real solutions does the given equation have?

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1

\newline

0

\newline

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2

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Question

Dante commutes to work

4

mornings a week. For his commute each morning, he walks for

10

minutes, waits and rides the bus for

x

minutes, and waits and rides the train for

y

minutes. If Dante spends at least

3.5

hours on his morning commute each week, which of the following inequalities best describes Dante's weekly morning commute?

\newline

1

\newline

x+y+10 \geq 3.5(60)

\newline

x+y+10 \geq 3.5(60)(4)

\newline

4(x+y)+10 \geq 3.5(60)

\newline

4(x+y+10) \geq 3.5(60)

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Question

\frac{1}{3}x+3y=4

\newline

2x-4=2y

\newline

Consider the given system of equations. If

(x,y)

is the solution to the system, then what is the value of

x-y

\newline

\square

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Question

Which of the following is the derivative

\frac{d y}{d x}

for the plane curve defined by the equations

x(t)=\frac{1}{2} \sin 2 t

y(t)=2 \cos 2 t

0 \leq t \leq 2 \pi

\newline

Select the correct answer below:

\newline

4 \cot 2 t

\newline

\tan 4 t

\newline

-\tan 4 t

\newline

-4 \tan 2 t

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Question

You pick a card at random, put it back, and then pick another card at random.

\newline

4

\newline

5

\newline

6

\newline

7

\newline

What is the probability of picking a number greater than

5

and then picking a

4

\newline

Write your answer as a percentage.

Posted 1 month ago

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