1: The question wishes for us to use the -intercepts as the endpoints of our interval.
2: Now by Rolle's Theorem, we know that somewhere between these points, the slope will be zero. Where? Easy: Take the derivative.
Thus, at , we have a spot with a slope of zero. We know that (or 1.5) is between 0 and 3. Thus, Rolle's Theorem is true for this (as it is for all cases).
Let . Then and , which is non-zero if . Then simplifies to , which is the Mean Value Theorem.
1: Using the expression from the mean value theorem
insert values. Our chosen interval is . So, we have
2: By the Mean Value Theorem, we know that somewhere in the interval exists a point that has the same slope as that point. Thus, let us take the derivative to find this point .
Now, we know that the slope of the point is 4. So, the derivative at this point is 4. Thus, . So
1: We start with the expression:
(Remember, sin(π) and sin(0) are both 0.)
2: Now that we have the slope of the line, we must find the point x = c that has the same slope. We must now get the derivative!
The cosine function is 0 at (where is an integer). Remember, we are bound by the interval , so is the point that satisfies the Mean Value Theorem.
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