Calculus/Some Important Theorems/Solutions

Rolle's Thoerem 1. Show that Rolle's Theorem holds true between the x-intercepts of the function . 1: The question wishes for us to use the -intercepts as the endpoints of our interval. Factor the expression to obtain . and are our two endpoints. We know that and are the same, thus that satisfies the first part of Rolle's theorem (). 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). Mean Value Theorem 2. Show that , where is the function that was defined in the proof of Cauchy's Mean Value Theorem. 3. Show that the Mean Value Theorem follows from Cauchy's Mean Value Theorem. Let . Then and , which is non-zero if . Then simplifies to , which is the Mean Value Theorem. 4. Find the that satisfies the Mean Value Theorem for the function with endpoints and . 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 5. Find the point that satisifies the mean value theorem on the function and the interval . 1: We start with the expression: so, (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.