How do you use the mean value theorem for #2sinx +sin2x# on closed interval of #[4pi, 5pi]#?
(Note: The amount of detail you'll be asked to provide depends, to some extent, on your teacher's goals for the class.)
Theerfore, the Mean Value Theorem allows us to conclude that:
Doing some arithmetic we can rewrite this conclusion as;
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To use the Mean Value Theorem for (2\sin(x) + \sin(2x)) on the closed interval ([4\pi, 5\pi]), follow these steps:

Verify that the function (2\sin(x) + \sin(2x)) is continuous on the interval ([4\pi, 5\pi]) and differentiable on the open interval ((4\pi, 5\pi)).

Calculate the derivative of the function. The derivative of (2\sin(x) + \sin(2x)) is (2\cos(x) + 2\cos(2x)).

Find the average rate of change of the function over the interval ([4\pi, 5\pi]). This is done by evaluating the function at the endpoints of the interval and taking their difference, then dividing by the length of the interval. The average rate of change is given by:
[\frac{f(5\pi)  f(4\pi)}{5\pi  4\pi}]
 Apply the Mean Value Theorem, which states that there exists at least one (c) in the open interval ((4\pi, 5\pi)) such that the derivative of the function at (c) is equal to the average rate of change found in step 3.
So, set the derivative of the function equal to the average rate of change and solve for (c):
[2\cos(c) + 2\cos(2c) = \frac{f(5\pi)  f(4\pi)}{5\pi  4\pi}]
 Solve for (c) to find the value where the derivative equals the average rate of change.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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