Guided Practice for 5.2: The Second Fundamental Theorem of Calculus

Overview

In this section we learn how to find the derivative of an area function, $A(x)=\int_c^x f(t)\,dt$. The remarkable answer is that $A’(x)=f(x)$. Here we investigate the relationship between $A$ and $f$ and show in what sense integration and differentiation are inverse operations.

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).

• Be proficient at using the First Fundamental Theorem of Calculus to evaluate an integral.
• Know how to find the derivative of a net-signed area function $A(x)=\int_a^x f(t)\,dt$ such as $A(x)=\int_1^x \sqrt{t^2+3}\,dt$.

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.

• Be able to explain how the First and Second Fundamental Theorems of Calculus enable us to formally see how differentiation and itegration are almost inverse processes.
• Use the chain rule to differentiate an area function that has been composed with it such as $A(x)=\int_1^{x^4} \sqrt{t^2+3}\,dt$.

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 5.2. Make sure to read actively, working through examples and activities as you go.

Video: Watch the following videos.

• Quick Recap - Second Fundamental Theorem (2:47)
• Sketching the graph of an integral function (8:16)
• Differentiating an Integral Function (7:04)

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.