calculus is the branch of mathematics that we use to describe change
TRANSCRIPT
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Average rates of change
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Calculus is the branch of mathematics that we use to describe change.
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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.
Changing Quantities
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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?
Changing Quantities
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So we are considering questions of . . .
Change
Rates of change
Using rates of change to predict actual change.
Important ideas
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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:
Question 1
mph 58hours 752
miles 160
.
Average speed is computed by dividing total distance traveled by total time elapsed.
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Average speed
We don’t find average speed by averaging speeds?
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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:
in time change total
quantityin change total
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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:
t variableindependen in the change total
variabledependent in the change total
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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
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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.
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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?
b) Suppose that you see that the price of regular is $3.689 per gallon today. Estimate its price two days from now.