5.4 Fundamental Theorem of Calculus

ObjectivesSWBAT:

1) apply the fundamental theorem of calculus2) understand the relationship between the derivative

and definite integral as expressed in both parts of the fundamental theorem of calculus

The Fundamental Theorem of Calculus (FTOC)

• The FTOC has two parts. The two parts tie together the concept of integration and differentiation.

• Some people regard the FTOC as the most important computational discovery in the history of mathematics!!!

• How excited are you to learn this?!?!?!

The total distance traveled was the area under the curve (green region)

To find the total distance traveled, you can find the antiderivative of velocity (s), substitute in 10, substitute in 0, and subtract.

The Fundamental Theorem of Calculus (the evaluation part aka part 2)

Integral Rules• These rules are not formally stated until

Chapter 6, but knowing about derivatives, you should be able to make the following connections.

Example 3: Evaluate each expression.

The Fundamental Theorem of Calculus (part 1…the simple version)

where a is a constant.

Wouldn’t it be great if all of Calculus was this simple?

The Fundamental Theorem of Calculus (part 1…the extended version)

Let’s look more closely at what it means to define a function using an integral.

• Let’s now tie everything together (derivatives and integrals).

• You are almost guaranteed to see something like this on the AP exam.

• (distribute handout of example 12)