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Math 170 Functions, Data, and Models

Math 170 Functions, Data, and Models. 11 Concavity Section 2.5. Basic Concepts. If is a function whose rate of change is constant, then the graph of is a line (straight). If is a function whose rate of change is increasing, then the graph of is concave up (bends upward).

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Math 170 Functions, Data, and Models

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  1. Math 170 Functions, Data, and Models 11 Concavity Section 2.5

  2. Basic Concepts • If is a function whose rate of change is constant, then the graph of is a line (straight). • If is a function whose rate of change is increasing, then the graph of is concave up (bends upward). • If is a function whose rate of change is decreasing, then the graph of is concave down (bends downward).

  3. Formula Examples • Sketch and describe the graphs of the following functions:

  4. Cycling Example • Let , , and be Ann’s, Bob’s, and Ono’s distances (in miles), respectively, cycled hours after starting. • Evaluate and interpret . • Find and interpret the average rate of change of each function over the entire domain. • Find the average rate of change of each function over each one hour interval. Give a qualitative description for each set of numbers. • Graph the three functions. Which graph is straight? concave up? concave down?

  5. Graph Example • A man drives from his home to a store and back. The entire trip takes 30 minutes. The graph gives his velocity (in mph) as a function of the time (in minutes) since he left home. A negative velocity indicates that he is traveling away from the store back to his home. • Where is increasing? decreasing? constant? • Where is the rate of change of increasing? decreasing? constant?

  6. Word Examples • Sketch and describe the graphs of the following functions: • US population as a function of time in recent years. ‘U.S. population’ • The amount of carbon dioxide in the Earth’s atmosphere as a function of time over the last 200 years. ’global carbon dioxide concentration’ • The amount of drug in a person’s body as a function of time since an injection.

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