Calculus/Some Important Theorems/Solutions

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Rolle's ThoeremEdit

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 TheoremEdit

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.