Concavity Lesson

Prof: Ana Maria Lopez

Lesson purpose

!   Learn about the different concavities of a graph and relating it to rate of change.

!   Explore a real life situation

General idea

!   Because the rate of change changes when X increases or decreases, the slope of a graph changes as well as X increases/decreases, so the graph bends upward if it increases or upward decreases.

All of the information in this slide is taken from the class math textbook cited at the end of this PP presentation.

Four possible concavities

!   Increasing and concave down

!   Increasing and concave up

!   Decreasing and concave down

!   Decreasing and concave up

All of the information in this slide is taken from the class math textbook cited at the end of this PP presentation.

Increasing and concave down Increasing and concave up

Decreasing and concave down

Decreasing and concave up

All of the information in this slide is taking from the class math textbook cited at the end of this PP presentation.

Important!

!   “If F is a function whose rate of change increases (Gets less negative or more positive as we move from left to right), then the graph of F is concave up. In this case, the graph bends upward”

!   “If F is a function whose rate of change decreases (Gets less positive or more negative as we move from left to right), then the graph of F is concave down. In this case, the graph bends downwards”.

All of the information in this slide is taken from the class math textbook cited at the end of this PP presentation.

Real life example!

!   When you eventually work and get a salary that increases, this function would create an increasing graph, meaning it would be concave up.

All of the information in this slide is taken from the class math textbook cited at the end of this PP presentation.

T ( Time) S: Salary ($100s) Rate of change

0 40

10 72

20 128

30 230

40 411

5.6

3.2

10.2

18.1

Table 1 à Salary: increasing rate of change

All of the information in this slide is taken from the class math textbook cited at the end of this PP presentation.

Graph 1: Graph of salary function

All of the information in this slide is taken from the class math textbook cited at the end of this PP presentation.

This is a graph of the point presented on Table #1. We can see that from years 10 to 20 there is a smaller increase in the salary and therefore the graph is less steep. This can be because you are still inexperienced and haven’t been part of the company for very long. Yet, from years 30 to 40 there is a larger increase in the salary and therefore the graph is more steep. This graph can be classified as increasing and concave up.

Sources

!   Hughes-Hallett, Deborah, Andrew M. Gleason, and Et. Al. "Ch 2.5: Concavity." Functions Modeling Change: A Preparation for Calculus. By Eric Connally. 4th ed. N.p.: John Wiley & Sons, 2011,2006,2003. 93-97. Print.