Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
f(x) = sqrt(x), [0,16]
f(0) = 0
f(16) = 4
slope of secant: 1/4
f' = 1/2√x
1/2√x = 1/4
2√x = 4
x = 4
which is in [0,16]
verify at
http://www.wolframalpha.com/input/?i=plot+y%3D%E2%88%9Ax%2C+y+%3D+%281%2F4%29%28x-4%29+%2B+2+for+0+%3C%3D+x+%3C%3D+16
Well, I don't know about the Mean Value Theorem, but I do know a thing or two about being mean. As an expert in clowning around, I'll do my best to give you an answer that will hopefully bring a smile to your face.
Now, to find the number c that satisfies the conclusion of the Mean Value Theorem, we need to find a point on the interval [0,16] where the slope of the tangent line is equal to the average rate of change between the endpoints.
I'm not a mathematician, but I can make a wild guess and say that the number c is somewhere between 0 and 16. You might think that's not a very helpful answer, but hey, at least I'm honest about my limitations.
So, to sum it up, the number c that satisfies the conclusion of the Mean Value Theorem on the interval [0,16] is somewhere between 0 and 16. But always remember, laughter is the best solution to any math problem!
To find the number c that satisfies the conclusion of the Mean Value Theorem on the interval [0, 16] for the function f(x) = √x, we need to find the average rate of change of the function over the interval and then find the value of c at which the instant rate of change is equal to the average rate of change.
First, let's find the average rate of change of f(x) over the interval [0, 16]. Average rate of change is given by:
f'(c) = (f(16) - f(0)) / (16 - 0)
Now, let's substitute the values into the formula:
√16 - √0 = (√16 - √0) / 16
√16 = (√16 - √0) / 16
4 = (4 - 0) / 16
4 = 4 / 16
4 = 1/4
Now, let's find the value of c at which the instant rate of change is equal to 1/4. We can find the derivative of f(x) and set it equal to 1/4:
f'(x) = (1/2) * (x^(-1/2))
Setting f'(x) = 1/4:
(1/2) * (x^(-1/2)) = 1/4
Now, let's solve for x:
(x^(-1/2)) = 1/2
1/sqrt(x) = 1/2
Multiply both sides by 2:
2/sqrt(x) = 1
Multiply both sides by sqrt(x):
2 = sqrt(x)
Square both sides:
4 = x
Therefore, the number c that satisfies the conclusion of the Mean Value Theorem on the interval [0, 16] is x = 4.
To find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval [0, 16], we need to check two conditions:
1. The function f(x) = sqrt(x) must be continuous on the interval [0, 16].
2. The function f(x) = sqrt(x) must be differentiable on the open interval (0, 16).
Let's check each condition:
1. Continuity:
The function f(x) = sqrt(x) is continuous on the interval [0, 16] because it is defined and finite for all values of x in that interval.
2. Differentiability:
The derivative of f(x) = sqrt(x) is f'(x) = (1/2) * (1/sqrt(x)) = 1 / (2 * sqrt(x)). It is defined and finite on the open interval (0, 16) since the denominator is never zero.
Since both conditions are satisfied, we can conclude that there exists a number c in the open interval (0, 16) that satisfies the conclusion of the Mean Value Theorem.
To find the specific number c, we need to use the formula from the Mean Value Theorem:
f'(c) = (f(b) - f(a)) / (b - a)
In this case:
a = 0
b = 16
f(a) = f(0) = sqrt(0) = 0
f(b) = f(16) = sqrt(16) = 4
Using the Mean Value Theorem formula:
f'(c) = (4 - 0) / (16 - 0)
f'(c) = 4 / 16
f'(c) = 1/4
Now, to find c, we need to solve the equation f'(c) = 1/4:
1 / (2 * sqrt(c)) = 1/4
To solve this equation, we can cross multiply:
4 * 2 * sqrt(c) = 1
8 * sqrt(c) = 1
Divide both sides by 8:
sqrt(c) = 1/8
Square both sides:
c = (1/8)^2
Simplifying the right-hand side:
c = 1/64
Therefore, the number c that satisfies the conclusion of the Mean Value Theorem on the interval [0, 16] is c = 1/64.