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Average rates of change

Average rates of change. Calculus is the branch of mathematics that we use to describe change. Changing Quantities. The value of a stock over time. The weight of a load of concrete as a function of its volume. The temperature of a room over the course of a day.

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Average rates of change

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  1. Average rates of change

  2. Calculus is the branch of mathematics that we use to describe change.

  3. Changing Quantities • The value of a stock over time. • The weight of a load of concrete as a function of its volume. • The temperature of a room over the course of a day. • The gravitational pull of the earth as a satellite rises.

  4. Changing Quantities Given a changing quantity, there are some natural questions: • How much does it change? • How fast is it changing? • How fast is it changing on average? • If we know how fast it is changing, can we predict how much it will change?

  5. Important ideas So we are considering questions of . . . • Change • Rates of change • Using rates of change to predict actual change.

  6. Question 1 You leave at 1 p.m. to drive to Cincinnati. You arrive at 3:45 p.m. and note that the distance was 160 miles. What was your average speed? Answer: Average speed is computed by dividing total distance traveled by total time elapsed.

  7. Average speed We don’t find average speed by averaging speeds?

  8. Average rates of change • Speed is the rate at which distance is changing over time. • In general, the average rate of change of a function over an interval of time is:

  9. Average rates of change • Speed is the rate at which distance is changing over time. • Or, more generally, the average rate of change of a function over an interval is:

  10. Question 2: • Find the average rate of change for each of the following functions over the given interval: • f (x )=x 2+1 on [0,1] • f (x )=x 2+1 on [-1,1] • f (x )=2x +4 on [2,5] For a straight line: the average rate of change is the slope of the line

  11. Question 3: • The graph below shows the fluctuations in the price of a stock over the course of the day. • Find the average rate of change of the stock’s price between 8 a.m. and 2 p.m. • Find the average rate of change of the stock’s price between 9 a.m. and 1 p.m.

  12. Question 4: • Suppose you know that over the last week, the price of gas has been decreasing by $.03 per day on average. • How much has it decreased since last Monday? • Suppose that you see that the price of regular is $3.689 per gallon today. Estimate its price two days from now.

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