The focus of this Master Class is to explore the pivotal theorems of Calculus. We will discuss Mean Value Theorem, Squeeze Theorem, and the Fundamental Theorem of Calculus. From how to prove these ideas to your students to examples and applications, we will build a deeper understanding of Calculus theorems in ourselves and our students.
Mean Value Theorem
Explore the applet below about the Mean Value Theorem.
Use the slider to move c and find the point where the derivative equals the average rate of change.
What happens when you select A and B as zeros of the function?
Explore how the blue function is squeezed between the orange and pink functions as x approaches zero.
Notice that the orange and pink functions don't equal the blue function anywhere except at zero.
Would this setup work for evaluating the limit anywhere but zero?
Fundamental Theorem of Calculus
This applet goes through the main ideas behind the proof of the Fundamental Theorem of Calculus.
Two areas are defined, the area from a to t, and from a to t+h.
As h approaches zero, what happens to t+h?
As h approaches zero, what happens to the areas of the big rectangle and small rectangle?
Click on the box titled "Using the Squeeze Theorem".
You can see how the actual area under the curve is squeezed between the area of the big rectangle and the small rectangle.