Week of 1/6-10/20 Mean Value Theorem

POW MVT  Due Friday 1/10

This POW is designed to get you thinking about a theorem that we need soon, although why we will need will need it isn't initially clear. For each case, I want you to:

• Draw an illustration and annotate it. 
• Articulate the idea in precise mathematical language. 
• Generate a logical justification for it. 
• State clearly why all the conditions are needed. 
• Ideally, you will be able to formally prove it as well. 

Prove the following statements using the criteria above:

1. If a function on a closed interval is continuous and differentiable and it has the same value on each end of the interval, there is at least one point in that interval where the function derivative is zero. 
2. If a function is continuous on a closed interval is continuous and differentiable and it has different values on each end of the interval, then the function must have point where the derivative is equal to the slope of a straight line that goes to one end point to another.