NSC-Math-181

#Guided Practice 1.4: The derivative function

Overview

In this section we generalize the idea of the derivative at a point from previous work to find a formula for the derivative of a function at any point. In other words, given a function \(f\), we will learn how to find a function \(f’\) for its derivative. This will involve generalizing the limit-based definition of the derivative we learned earlier. We will learn how to calculate \(f’\) from \(f\) if \(f\) is given as a (simple!) formula and if \(f\) is given as a graph.

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 definition of the derivative of a function \(f\) with respect to \(x\) at the value \(x\).
• Use the limit definition of the derivative of \(f\) to find a formula for \(f’\) if \(f\) is a constant, linear, or quadratic function.
• Given a graph of a function \(f\), identify the graph of its derivative \(f’\) from a list, and describe aspects of the behavior of the graph of \(f’\) .

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.

• Give an example of a function \(f\) for which \(f’\) is not defined at one or more points.
• Given a graph of a function \(f\), sketch the graph of its derivative \(f’\).
• Given any function \(f\) (not just a simple one), set up a limit expression that would compute a formula for \(f’\).
• Use the limit definition of the derivative of \(f\) to find a formula for \(f’\) for functions that are not just constant, linear, or quadratic.

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 1.4. 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 derivative function (2:40)
• Limit definition of derivative (5:38)
• Sketching a derivative graph (6:14)

  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.