Guided Practice for 4.4: The Fundamental Theorem of Calculus

Overview

In our final section of Chapter 4, we examine the central mathematical concept of the course: The Fundamental Theorem of Calculus (“FTC”). The FTC unifies the concept of the definite integral with the concept of the derivative by giving a way to calculate a definite integral in terms of an antiderivative, instead of using geometry or a Riemann sum. In addition to learning to use the FTC and antiderivatives to calculate definite integrals, we will look at the Total Change Theorem (sometimes referred to as the “Net” Change Theorem) that gives a platform for applying definite integrals to a wide variety of real-world problems.

Learning objectives

Basic objectives: Each student is responsible for gaining proficiency with each of these tasks prior to engaging in class discussions, through the use of the learning resources (below) and through the working of exercises (also below).

• State the Fundamental Theorem of Calculus and explain in basic terms what it tells you.
• Explain how the FTC allows one to evaluate definite integrals by first calculating antiderivatives.
• Calculate a definite integral using the FTC and antiderivatives, for an integral in which the integrand is simple to antidifferentiate.
• State the Total Change Theorem.

Advanced objectives: The following objectives are the subject of class discussion and further work; they should be mastered by each student during and following class discussions.

• Use the Total Change Theorem to find the amount of change in a quantity, given the rate at which that quantity changes.

Learning resources

To gain proficiency in the learning objectives, use the following resources. You may include other resources if you wish, in addition to or in replacement of the following.

Textbook: In Active Calculus, read Section 4.4. Make sure to read actively, working through examples and activities as you go.

Video: Watch the following videos at the MTH 201 YouTube playlist (http://bit.ly/GVSUCalculus).

• Quick review: The Fundamental Theorem of Calculus (3:15)
• Fundamental Theorem of Calculus with power functions (7:33)
• Fundamental Theorem of Calculus with exponential functions (7:57)
• Application of the Total Change Theorem (6:41)

Activities

The following activity is to be done during and following your reading and viewing of the resources. Go to student.desmos.com and join using your name in the format Last, First. For example, I would enter my name as Ballif, Serge. Complete each part of the activity. Some of these problems will require you to work them out on paper before entering your answer. Practice producing high quality work so that your work is readable and meaningful. You will receive a mark of Pass if each item response shows a good-faith effort to be right and is submitted prior to the deadline. Remember to use the Piazza discussion board to ask about any questions you have.