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

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Presentation transcript:

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

 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.

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?

So we are considering questions of...  Change  Rates of change  Using rates of change to predict actual change.

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.

We don’t find average speed by averaging speeds?

 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:

 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:

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

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

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