1. 2.3 Rates of Change b

2. We’ve talked about how a derivative can be used to find the slope of a tangent line.
The derivative can also be used to determine the rate of change of one variable with respect to another. Or the INSTANTANEOUS RATE OF CHANGE
i.e. production rates, population growth rates, water flow rates, velocity, and acceleration.

3. 1.6 Differentiation Techniques: The Product and Quotient Rules
DEFINITION:
If C(x) is the cost of producing x items, then the
average cost of producing x items is
If R(x) is the revenue from the sale of x items, then the
average revenue from selling x items is
If P(x) is the profit from the sale of x items, then the
average profit from selling x items is

4. 1.6 Differentiation Techniques: The Product and Quotient Rules
Example 6: Paulsen’s Greenhouse finds that
the cost, in dollars, of growing x hundred geraniums is
given by If the revenue from the sale of x hundred geraniums is given by
find each of the following.
a) The average cost, the average revenue, and the average profit when x hundred geraniums are grown and sold.
b) The rate at which average profit is changing when 300 geraniums are being grown.

5. 1.6 Differentiation Techniques: The Product and Quotient Rules
Example 6 (continued):
a) We let AC, AR, and AP represent average cost,
average revenue, and average profit.

6. 1.6 Differentiation Techniques: The Product and Quotient Rules
Example 6 (continued):
b) First we must find AP(x). Then we can substitute 3
(hundred) into AP(x).
NOTE: In the textbook, on p. 164, they substitute in 300 (instead of 3).
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

7. 1.6 Differentiation Techniques: The Product and Quotient Rules
Example 6 (concluded):
Thus, at 300 geraniums, Paulsen’s average profit is
increasing by about $11.20 per plant.

8. The temperature T of a person during an illness is given by
where T is the temperature in degrees Fahrenheit, at time t, in hours.
Find the rate of change of the temperature with respect to time.
Find the temperature at t=2hrs
Find the rate of change of the temperature at t=2hrs.

9. The Population P, in thousands, of a small city is given by
Where t is the time, in months
a.) Find the growth rate
b.) Find the population after 12 months
c.) Find the growth rate at t=12 months

10. Position, Acceleration, and Velocity
A common use of rate of change is to describe the motion of an object moving in a straight line. (either horizontally or vertically)
The function s that gives the position (relative to the origin) of an object as a function of time t is called a position function.

11. (position)
g is an object’s acceleration due to earths gravity, you have learned in science that it is -9.8 meters per second per second or -32 feet per second per second, not stuttering (this means that after 1 second its 32 after 2 seconds its 64 its acceleration is in a linear fashion, it is increasing speed all the time, we neglect the idea of air resistance in this scenario, so for a bowling ball this would be accurate but for a feather it would not be)
v0  initial velocity, so if something was being thrown or pushed or launched. (someone jumping on a trampoline, a rock or baseball being thrown, etc)
-s0  initial height or initial position.

12. An object is dropped from a height of 144 feet
An object is dropped from a height of 144 feet. Find its position (height) after 2 seconds.
-its dropped so no initial velocity
-its at a height of 144 so its initial position is 144.
s(t)
=80 ft s t

13. What other formula do we know as being equal to distance/time?
s(c)
s(t)
The derivative of this s t c
What other formula do we know as being equal to distance/time?
So what we can say is that s’(c) or the derivative of the position function is a velocity function.
Velocity is speed with the added property of direction.
s(c) = position function
s’(c) = velocity function
s’’(c) = rate of change of the velocity with respect to time, which is acceleration.

14. At a time t seconds after it is thrown up in the air, a tomato is at a height of meters.
a.) what is the average velocity of the tomato during the first 2 seconds? 15.2m/sec
b.) Find the instantaneous velocity of the tomato at t= m/sec
c.) what is the acceleration at t=2? -9.8m/sec
d.) How high does the tomato go? 34.9
e.) How long is the tomato in the air? 5.2 sec

15. At time t=0 a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by
where s is measured in feet and t is in seconds.
a.) when does the diver hit the water?
b.) what is the diver’s velocity as he enters the water?