Guided Practice 1.6: The second derivative

Overview

In this section we study the second derivative of a function, which is just the derivative of the first derivative. That is – “taking a derivative” is something we do to a function, and since the derivative $f’$ is a function, we can take its derivative too. The second derivative is an important ingredient for understanding the subtle behaviors of a function, and in particular the concept of concavity will distinguish between a function that is increasing at an increasing pace and a function that is increasing at a decreasing pace. Our main highlight for this section is to have a clear understanding of the relationships between the sign of $f’$, the sign of $f’’$ (the second derivative), the increasing/decreasing behavior of $f$, and the concavity of $f$.

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).

• Define what it means for a function to be increasing or decreasing on an interval, and identify intervals on which a function is increasing or decreasing given a graph of the function.
• Define what is meant by the second derivative of a function and correctly use the double-prime notation to denote a second derivative.
• Define what it means for a function to be concave up or concave down on an interval.
• Visually identify intervals on which a function is concave up or concave down, given a graph of the function.
• Given the sign (positive, negative, zero) of $f’$ at a point, tell whether the original function $f$ is increasing or decreasing at that point.
• Given the sign (positive, negative, zero) of $f’’$ at a point, tell whether the original function $f$ is concave up or concave down at that point.

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.

• Given the graph of a function, sketch the graph of its second derivative.
• Explain what the six different combinations of increasing/decreasing and concave up/concave down/linear mean in real-life terms (such as “increasing at a decreasing rate”).
• Given a table of values for a function, construct a table of approximate values for its second derivative.

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 1.6. 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 second derivative (3:07)
• Limit definition of the second derivative (9:49)
• Determining concavity from a graph (7:43)

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.