Which value of c satisfies the MVT for f(x) = x*sinx on [1,4]?
My answer is 2.463.
I did not get this.
for my average rate of change I had
= (4sin4 - sin1)/3 = appr -.72858
f'(x) = xcosx + sinx
by MVT, xcosx + sinx = -.72858
This is a nasty equation to solve, so I tried good ol'
Wolfram
http://www.wolframalpha.com/input/?i=solve+xcosx+%2B+sinx+%3D+-.728579665
I picked the solution of x = 2.808 as my choice of c in the given interval.
I don't know what method you used to solve the equation and got your answer
Khan Academy does a good job of introducing the MVT here:
https://www.khanacademy.org/math/ap-calculus-ab/ab-existence-theorems/ab-mvt/v/mean-value-theorem-1
with an example using an actual function here:
https://www.khanacademy.org/math/ap-calculus-ab/ab-existence-theorems/ab-mvt/v/finding-where-the-derivative-is-equal-to-the-average-change
Wolframalpha gives (4sin4 - sin1)/3 = -1.29.
You are correct, I don't know what I did to get the easy part of the problem,
And your final answer is correct, good job.
Oh, you're close! But you know, finding the value of c that satisfies the Mean Value Theorem is no joking matter. In this case, the correct value of c is actually 2.784. So close, yet so far! Keep those calculations going!
To find the value of c that satisfies the Mean Value Theorem (MVT) for the function f(x)=x*sin(x) on the interval [1,4], we first check if the function satisfies the conditions of the MVT.
1. First, we check if the function is continuous on the closed interval [1,4]. The function f(x)=x*sin(x) is the product of two continuous functions (x and sin(x)), so it is continuous on the interval [1,4].
2. Next, we need to check if the function is differentiable on the open interval (1,4). The function f(x)=x*sin(x) is differentiable on the entire interval (1,4) as the product of two differentiable functions (x and sin(x)).
Since the function f(x)=x*sin(x) satisfies both conditions, we can apply the MVT, which states that there exists at least one value c in the open interval (1,4) such that the derivative of the function at c is equal to the average rate of change of the function over the interval [1,4].
The derivative of f(x) = x*sin(x) can be found by applying the product rule:
f'(x) = 1*sin(x) + x*cos(x) = sin(x) + x*cos(x)
Next, we find the average rate of change of the function over the interval [1,4]. The average rate of change is equal to the difference in the function values at the endpoints divided by the difference in the x-values:
Average Rate of Change = (f(4) - f(1))/(4 - 1)
= (4*sin(4) - 1*sin(1))/(4 - 1)
= (4*sin(4) - sin(1))/3
Now we set the derivative equal to the average rate of change and solve for c:
f'(c) = Average Rate of Change
sin(c) + c*cos(c) = (4*sin(4) - sin(1))/3
To find the specific value of c, we can solve this equation numerically using a graphing calculator, numerical methods, or software. The approximate solution for c is indeed around 2.463.
Therefore, the value of c that satisfies the MVT for f(x) = x*sin(x) on the interval [1,4] is approximately 2.463.