MATHEMATICS COLLEGE

Find the exact values of the numbers c that satisfy the conclusion of the Mean Value Theorem for the interval [−2, 2]. (Enter your answers as a comma-separated list.)

Answers

Answer 1

Answer:

Answer:

The answer is " $\bold{c= \pm (2)/(√(3))}$ "

Step-by-step explanation:

If the function is:

$\to f'(x) = 3x^2-2 \n\n\to f'(c) = 3c^2-2$

points are:

$\to -2 \leq x \leq2$

use the mean value theorem:

$\to f'(c) = ( f(b)- f(a))/(b-a)$

$= ( f(2)- f(-2))/(2-(-2))\n\n= (4-(-4) )/(4)\n\n= (8)/(4)\n\n= 2$

$\to 3c^2-2=2 \n\n\to 3c^2=4 \n\n\to c^2=(4)/(3) \n\nc= \pm (2)/(√(3))$

Answer 2

Answer:

Final answer:

The Mean Value Theorem states that for a continuous and differentiable function on a closed interval, there exists at least one 'c' within that interval where the average change rate equals the instantaneous rate at 'c'. In the given case of interval [-2,2], to find 'c', first calculate the average slope between the points (f(2)-f(-2))/4. Then equate this average slope to the derivative 'f'(c). The solution(s) to this equation are the c values for this problem.

Explanation:

The subject of this question pertains to the Mean Value Theorem in Calculus. According to this theorem, if a function f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the open interval (a, b) such that the average rate of change over the interval equals the instantaneous rate of change at c.

In the given case, we're trying to find the 'c' value for the interval [-2,2]. First, we need to find the average slope between the two points. Assuming f is your function, that would be (f(2)-f(-2))/ (2 - -2). Subtract the function values of the two points and divide by the total interval length. Next, we need to see where this average slope equals the instantaneous slope 'f'(c), this entails solving the equation 'f'(c) = (f(2)-f(-2))/4. The solution to this equation will be the c values that satisfy the Mean value theorem within the provided interval.

Learn more about Mean Value Theorem here:

brainly.com/question/35411439

#SPJ3

Related Questions

uppose a small cannonball weighing 16 pounds is shot vertically upward, with an initial velocity v0 = 290 ft/s. The answer to the question "How high does the cannonball go?" depends on whether we take air resistance into account. If air resistance is ignored and the positive direction is upward, then a model for the state of the cannonball is given by d2s/dt2 = −g (equation (12) of Section 1.3). Since ds/dt = v(t) the last differential equation is the same as dv/dt = −g, where we take g = 32 ft/s2. If air resistance is incorporated into the model, it stands to reason that the maximum height attained by the cannonball must be less than if air resistance is ignored. (a) Assume air resistance is proportional to instantaneous velocity. If the positive direction is upward, a model for the state of the cannonball is given by m dv dt = −mg − kv, where m is the mass of the cannonball and k > 0 is a constant of proportionality. Suppose k = 0.0025 and find the velocity v(t) of the cannonball at time t.
Given: ∠T ≅ ∠V; ST || UVProve: TU || VW4 connected lines are shown. A line from point S goes slightly down and to the left to point T to form S T. A line from point T goes slightly down and to the right to point U to form T U. A line from point U goes slightly down and to the left to point V to form U T. A line goes slightly down and to the right to point W to form point W.Complete the two-column proof.PLEASE HURRYY
A 12 SIDED DIE WITH SIIDES LABLELED 1-12 ARE ROLLED ONCE. EACH NUMBER IS LIKELY TO BE ROLLED .WHAT IS THE PROBABILITY OF ROLLING A NUMBER GREATER THAN 10
Please answer this, I am confused (5^2)(7^2)(3^2)
Write the expanded form 0.45 in this way using fractions and using decimals

hugo sells handmade key chains online. Yesterday he made 6 keychains in 30minutes. hugo wants to make 9 keychains today . if hugo works at the same rate today how long will it take him to make the keychains

Answers

Answer:

45 minutes, hope this helps

Step-by-step explanation:

Answer:

45 minutes.

Step-by-step explanation:

Set up a ratio.

6 keychains: 30 minutes

divide both sides of the ratio by 6

1 keychain: 5 minutes

multiply both sides of the ratio by 9

9 keychains: 45 minutes

It will take him 45 minutes.

15 + x ≤ –45 solve using inverse opertation

Answers

I believe the answer is x ≤ -60.

Sin2x/1+cos2x=tanx
How do I prove this with the double angle law

Answers

$sin(2x) = 2 sin x cos x \n \n cos(2x) = cos^2 x - sin^2 x$
After Substituting:
$(2 sin x cos x)/(1+cos^2 x - sin^2 x)$
Use pythagorean thm:
$1 - sin^2 x = cos^2 x$
......................
$(2 sin x cos x)/(2cos^2 x) \n \n = (sinx )/(cos x) \n \n = tan x$