Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval.
f(x) = 4√x
[4, 9]
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To find the value(s) of c guaranteed by the Mean Value Theorem for Integrals, we first need to check if the conditions of the theorem are met.
The Mean Value Theorem for Integrals states that for a continuous function f(x) on the closed interval [a, b], there exists at least one number c in the interval (a, b) such that the average value of f(x) over [a, b] is equal to f(c).
In this case, we are given the function f(x) = 4√x on the interval [4, 9]. We can start by calculating the average value of f(x) over this interval.
The average value of a function f(x) over an interval [a, b] is given by:
(1/(b-a)) ∫[a, b] f(x) dx.
In our case, we have:
(1/(9-4)) ∫[4, 9] 4√x dx.
Simplifying this expression, we have:
(1/5) ∫[4, 9] 4√x dx.
Next, we need to evaluate the integral.
The integral of 4√x can be found by using the power rule for integration.
The power rule states that the integral of x^n with respect to x is (1/(n+1))x^(n+1) + C, where C is the constant of integration.
Applying the power rule to our function, we can find that:
∫ 4√x dx = (4/3) x^(3/2) + C.
Now we need to evaluate this expression over the interval [4, 9]:
(1/5) [(4/3)(9^(3/2)) - (4/3)(4^(3/2))].
Simplifying further, we get:
(1/5) [(4/3)(27) - (4/3)(8)].
This simplifies to:
(1/5) [108/3 - 32/3].
Further simplifying, we have:
(1/5) (76/3).
Finally, calculating this expression, we find:
76/15 = 5.0667.
Therefore, the average value of f(x) over the interval [4, 9] is approximately 5.0667.
According to the Mean Value Theorem for Integrals, there exists at least one value c in the interval (4, 9) such that f(c) = 5.0667.
In this case, c represents the x-coordinate of a point on the graph of f(x) where the tangent line is parallel to the secant line connecting the endpoints [4, f(4)] and [9, f(9)].
To find the specific value of c, we need to solve the equation 4√c = 5.0667.
Dividing both sides of the equation by 4, we get:
√c = 1.2667.
Squaring both sides of the equation, we obtain:
c = 1.6035.
Therefore, the value of c guaranteed by the Mean Value Theorem for Integrals for the function f(x) = 4√x over the interval [4, 9] is approximately 1.6035.