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Math 170 Functions, Data, and Models

Learn about constant rate of change, concavity, and graphing functions with practical examples and detailed explanations. Explore average rate of change, intervals, and qualitative descriptions.

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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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