What value of x satisfies the conclusion of the mean value theorem for f(x) = ln(x3) over the interval [1, e2]?

Answer:

[tex]x \approx 3.195[/tex] satisfies the conclusion of the Mean Value Theorem for [tex]f(x) = \ln x^{3}[/tex] over the interval [tex][1,e^{2}][/tex].

Step-by-step explanation:

According to the Mean Value Theorem, for all function that is differentiable over the interval [tex][a, b][/tex], there is at a value [tex]c[/tex] within the interval such that:

[tex]f'(c) = \frac{f(b)-f(a)}{b-a}[/tex] (1)

Where:

[tex]a[/tex], [tex]b[/tex] - Lower and upper bounds.

[tex]f(a)[/tex], [tex]f(b)[/tex] - Function evaluated at lower and upper bounds.

[tex]f'(c)[/tex] - First derivative of the function evaluated at [tex]c[/tex].

If we know that [tex]f(x) = \ln x^{3} = 3\cdot \ln x[/tex], [tex]f'(x) = \frac{3}{x}[/tex], [tex]a = 1[/tex] and [tex]b = e^{2}[/tex], then we find that:

[tex]\frac{3}{c} = \frac{3\cdot \ln e^{2}-3\cdot \ln 1}{e^{2}-1}[/tex]

[tex]\frac{3}{c} = \frac{6\cdot \ln e-3\cdot \ln 1 }{e^{2}-1 }[/tex]

[tex]\frac{3}{c} = \frac{6}{e^{2}-1}[/tex]

[tex]c = \frac{1}{2}\cdot (e^{2}-1)[/tex]

[tex]c \approx 3.195[/tex]

[tex]x \approx 3.195[/tex] satisfies the conclusion of the Mean Value Theorem for [tex]f(x) = \ln x^{3}[/tex] over the interval [tex][1,e^{2}][/tex].

Answer:

C. 1/2(e^2-1)

Step-by-step explanation:

Edge AP Cal 2022

Let N* be the total number of ranchers in Uruguay, and N(t) be the number of ranchers who have adopted an improved pasture technology there. Assume that the rate of adoption, , is proportional to both the number who have adopted the technology and the fraction of the ranchers who have not (and so are susceptible to conversion). Let a be the proportionality constant.

a. Write down the differential equation modeling N(t).
b. According to Banks (1993), N* = 17000, N(0) = 170, a = 0.5 per year.

Determine how long it takes for the improved pasture technology to spread to 80% of the population of the ranchers.

Convert 7% as a decimal

Solve the problem using long division

3 to 4 is equivalent to the ratio of x to 36. What will x be?

(28 points) In the drawing, a regular polygon is partially covered by a rectangle. what is the number of sides of this polygon?

A: 6
B: 7
C: 8
D :60


​

What’s the answer pls

A 2-pound package of hamburger costs $7.00, and you pay $10.50 for a 3-pound package of hamburger. Use a double number line to complete the table.

Weight (lb)
2
3
4
5
Cost ($)
7.00
10.50
14.00
17.50

What is the unit rate in this relationship?

calculate the pay for the following day of a weekly time card given a wage of $16/hr

Convert 6 months to​ years

Snowflake the snowman wants to buy new scarves if each scarf cost $4.27 how much did snowflake spend if he buys 5

28800000, 1440000, 72000, 3600, 180, ...

What is the solution to (4/2)4 + (1 + 3)?

Find the product of (1)(3)(x).
Your
answer

Iman’s favorite colors are \blue{\text{blue}}bluestart color #6495ed, start text, b, l, u, e, end text, end color #6495ed and \pink{\text{pink}}pinkstart color #ff00af, start text, p, i, n, k, end text, end color #ff00af.
She has \blue{\text{1 blue shirt}}1 blue shirtstart color #6495ed, start text, 1, space, b, l, u, e, space, s, h, i, r, t, end text, end color #6495ed, \pink{\text{1 pink shirt}}1 pink shirtstart color #ff00af, start text, 1, space, p, i, n, k, space, s, h, i, r, t, end text, end color #ff00af, \blue{\text{1 blue hat}}1 blue hatstart color #6495ed, start text, 1, space, b, l, u, e, space, h, a, t, end text, end color #6495ed, \blue{\text{1 blue belt}}1 blue beltstart color #6495ed, start text, 1, space, b, l, u, e, space, b, e, l, t, end text, end color #6495ed, \blue{\text{1 blue pair of pants}}1 blue pair of pantsstart color #6495ed, start text, 1, space, b, l, u, e, space, p, a, i, r, space, o, f, space, p, a, n, t, s, end text, end color #6495ed, and \pink{\text{1 pink pair of pants}}1 pink pair of pantsstart color #ff00af, start text, 1, space, p, i, n, k, space, p, a, i, r, space, o, f, space, p, a, n, t, s, end text, end color #ff00af.
Iman selects one of these garments at random. Let AAA be the event that she selects a pink garment and BBB be the event that she chooses a shirt.
What is P(A\text{ or }B)P(A or B)P, left parenthesis, A, start text, space, o, r, space, end text, B, right parenthesis, the probability that the garment Iman chooses is pink or a shirt?

The area A of a circle in square feet with a radius r in feet is given by the formula, A=πr2. What is the area of a circle with a radius of 3 feet? Round your answer to the nearest thousandths.

m^2+5=14
[tex] {x}^{2} [/tex]
x2+5=14
​

Brandon has a jar of quarters and dimes with a total value of $7.85. The number of quarters is 7 less than twice the number of dimes. How many quarters and how many dimes does Brandon have?

Could someone answer and explain. Will mark the brainliest

Can anyone help me pls

HELP!! will give brainliest!
ху
7 21
8 24
9 27
Function B
Function B
6
4
2
0
2
4
6
8
10
Which statement best compares the rate of change of the two functions?
The rate of change of both functions is 2.
The rate of change of both functions is 3.
The rate of change of function A is greater than the rate of change of function B.
The rate of change of function B is greater than the rate of change of function A.

which expression is equivalent to (2m+3)+(m-1)?
A.5m
B.3m + 2
C.2m+2
D.3m-2

The total profit made by an engineering firm is given by the function p=x^2 - 24x +5000 where x is the number of clients the firm has and p is the profit.


1. Find the maximum profit made by the company.

2. How many clients are necessary to reach the profit outlined in part one?

The sum of three consecutive even integers is 486. find all three integers​

Factor the trinomial below COMPLETELY

5x^2 - 20x + 60​

Paul is a newspaper boy who received a total piecework
paycheck of $265.72. He receives 73 cents for every
newspaper he delivers. How many newspapers did he
deliver?

Silvia has a coupon for 25% off an item that regularly costs $8. How much will Silvia pay if she uses the coupon?

hich expression is equivalent to -4(7 - 2m) + 11m?

Alaina baked a cake that weighs 28 ounces. Her family has eaten 49% of it. About how many ounces of cake have they eaten?

of the last 60 people who went to the cash register at the department store, 16 had blomd hair, 20 had black hair, and 3 had red hair. determine the empirical probability the next person will have brown hair​

10g of Copper reacts with 5g of Sulfur

Write the ratio of Copper to Sulphur

​