Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. (Enter your answers as a comma-separated list.)
f(x) = x^4, [0, 4]
Step 1: multiply one over b-a by the integral of f(x) from [a,b]
1 b
--------- ∫ f(x) dx
b - a a
1 4
--------- ∫ x^4 dx
4 - 0 0
1 [ x^5 ] 4
----- [ -------- ]
4 [ 5 ] 0
1 [ (4)^5 (0)^5 ]
----- [ --------- -- ---------- ]
4 [ 5 5 ]
1 [ 1024 ] 256
----- [ -------- ] = --------
4 [ 5 ] 5
To find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function f(x) = x^4 over the interval [0, 4], we need to compute the average value of the function and find the value(s) of c such that f(c) equals the average value.
The average value of a function f(x) over an interval [a, b] is given by the formula:
average value = (1 / (b - a)) * ∫[a, b] f(x) dx
So, for the function f(x) = x^4 over the interval [0, 4], the average value is:
average value = (1 / (4 - 0)) * ∫[0, 4] x^4 dx
To find the integral, we can apply the power rule:
∫ x^n dx = (1 / (n + 1)) * x^(n + 1)
Using this rule, we can evaluate the integral:
average value = (1 / 4) * [x^5 / 5] evaluated from 0 to 4
average value = (1 / 4) * [(4^5 / 5) - (0^5 / 5)]
average value = (1 / 4) * (1024 / 5)
average value = 256 / 5
To find the value(s) of c guaranteed by the Mean Value Theorem for Integrals, we need to find when f(c) equals the average value. In this case, we need to solve the equation:
c^4 = 256 / 5
To solve this equation, we can take the fourth root of both sides:
c = (256 / 5)^(1/4)
Calculating this, we get:
c ≈ 2.148
So, the value of c guaranteed by the Mean Value Theorem for Integrals for the function f(x) = x^4 over the interval [0, 4] is approximately 2.148.
To find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function f(x) = x^4 over the interval [0, 4], we need to follow these steps:
1. Calculate the definite integral of f(x) over the interval [0, 4]. The definite integral of f(x) represents the area under the curve of the function over that interval.
∫[0, 4] x^4 dx
To integrate x^4, we can use the power rule of integration:
∫ x^n dx = (x^(n+1))/(n+1) + C,
where (n+1) is the power of the variable.
Applying this rule, we have:
∫[0, 4] x^4 dx = (x^(4+1))/(4+1) |[0, 4]
= (x^5)/5 |[0, 4]
Now, evaluate the integral at the upper and lower limits:
[(4^5)/5] - [(0^5)/5]
= (1024/5) - 0
= 1024/5
2. Calculate the average value of f(x) over the interval [0, 4]. The average value is given by:
Average value = (1/(b-a)) * ∫[a, b] f(x) dx
In this case, a = 0 and b = 4. Substituting these values into the formula, we get:
Average value = (1/(4-0)) * (1024/5)
= (1/4) * (1024/5)
= 256/5
3. According to the Mean Value Theorem for Integrals, there exists at least one value c in the interval [0, 4] such that the function f(x) takes on its average value at c.
Therefore, the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function f(x) = x^4 over the interval [0, 4] is/are:
c = any value in the interval [0, 4].
Step 2: plug in for f(c)
f(c) = x^4
256/5 to the 4th root
The answer is 4 over the 4th root of 5