The Fundamental Theorems of Calculus



The Fundamental Theorems of Calculus

Overview: In this lesson, students will learn the connection between differential and integral calculus. They will also learn the two fundamental theorems of calculus and how to apply them.

Grade Level/Subject: This lesson is for 12th graders in AP Calculus.

Time: 1-50 minute class period

Purpose: This lesson will finally link the two areas of calculus, differentiation and integrals, together. The students will learn two of the most important theorems in calculus and how to make their calculus lives much simpler!

Prerequisite Knowledge:

Students should:

- Understand both definite and indefinite integrals

- Know derivatives very well

Objectives:

1. Students will learn the connection between derivatives and integrals.

2. Students will learn the two fundamental theorems of calculus and become very comfortable in the application of them.

Standards:

1. Connections: Students will make connections between derivatives and integrals. They will also be asked to consider a little bit of history in their homework by writing a short paragraph about the calculus controversy.

Resources/Materials Needed:

1. Calculus Book

2. Dry Erase Board and Dry Erase Markers

Activities/Procedures:

1. Explain the inverse relationship between derivatives and definite integrals that was discovered by Newton and Leibniz through the illustration of a car’s motion. Calculating the area under the velocity graph using a definite integral gives us the distance traveled by the car. Or, by plotting running totals of the distance traveled, we can antidifferentiate the velocity function to find the original position function.

2. Explain how to formulate an area function using the definite integral. Have the students use their calculator to graph [pic] for [pic]. Use the calculator’s definite integration capabilities to compute A(x) for several values of x and plot them.

(i.e. x = 1.0, 1.1, 1.2, … , 2.9, 3.0)

3. Now have the students find the rate of change of the area function. They should come up with a graph that looks similar to y = 1/x.

4. First Fundamental Theorem of Calculus:

Hypothesis: Suppose that f is a continuous function such that [pic] exists for every real number [pic].

Conclusion: If a < x < b, then A’(x) = f(x).

5. Prove this Theorem.

6. Go through a few examples with the class.

a. [pic]

b. [pic]

7. The Second Fundamental Theorem of Calculus:

Hypothesis: F is any antiderivative of a continuous function f.

Conclusion: [pic]

8. Go through several examples of this.

a. [pic]

b. [pic]

9. Have the students read about the “calculus controversy.” (See Attached). Have them discuss it within their group and write a short paragraph answering the following questions.

a. If you were a judge for the plagiarism case between Newton and Leibniz, what would your ruling be on the case?

b. Why do you think that Newton and Leibniz get the credit instead of Archimedes and the ancient Greeks? Is this fair?

c. What do you think Newton means by his statement, “If I have seen farther than others, it is because I have stood on the shoulders of giants.”

Homework: Read Section 6.6 and take notes. Complete the attached worksheet.

[pic]

Use the fundamental theorems of calculus to find an antiderivative F of the given function f having the specified initial value.

1) [pic] F(1) = 0

2)[pic]; F([pic]) = 0

3)[pic] F([pic]) = 0

Suppose that G is the area function with formula G(x) =[pic]

Find [pic]

4) y = G(x)

5) y = x2G(x)

6) y = (G(x))2

Use the fundamental theorems of calculus to calculate each of the definite integrals given.

7) [pic]

8)[pic]

9)[pic]

10)[pic]

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