1. From previous sections
Marginals: slope of linear functions
marginal profit = slope = , it means that profit will increase
for the next unit sold.
Extrema: maximum profit or revenue, minimum cost
For quadratic functions, these happen at
the
GOAL: discuss marginals and extrema for any function.

2. Rates of Change and Derivatives
Average Rate of Change: average velocity, slope of secant line
(Instantaneous) Rate of Change: velocity, derivative, marginal, slope of tangent line

3. Notation “f prime of x” Average R.o.C vs. Instantaneous R.o.C:
interval of values vs. a single value

4. Example 1 x y Explore how AROC changes as3
Explore how AROC changes as
the second input get closer x = 3. a b
AROC 3 4
3.5
3.1
3.01
Slope of tangent line at
x = 3 looks like m =

5. Example 2 x y Explore how AROC changes as1
Explore how AROC changes as
the second input get closer x = 2. a b
AROC 2 1
1.5
1.9
1.99
Slope of tangent line at
x = 2 looks like m =

6. Example 3
Using the definition of derivatives and limits, we could have shown the following:
Find the instantaneous rate of change at
Find the slope of the tangent line at
Find the equation of the tangent line at

7. Derivative Rules
Derivatives = rates of change = marginal = tangent line slopes
Constant Rule:
Power Rule:
Coefficient Rule:
Sum/Difference Rule:

8. Example 1
Example 2

9. Example 3 x y 1
Find the tangent line at x = 2.

10. Example 4 Given the demand function based on price:
Find the rate of change of demand with respect to price and explain the value when
If price
demand is expected to

11. Example 5 Given the cost function based on level of production:
Find the rate of change of average cost and what level of production has a rate of change value of 0.
Average cost is changing

12. Example 6
Given the cost function based on level of water impurity, p percent:
Find the rate of change of cost when impurities account for 10%
It would cost about

13. Product/Quotient Rules
Product Rule:
Quotient Rule:

14. Examples 1&2

15. Example 3
Find the slope of the tangent line at x = 2.

16. Example 4
Given the sales function (in thousands of dollars) based on x, the amount spent on advertising (in thousand of dollars),
find and interpret and
If ad expenses increase .
If ad expenses increase .

17. Chain Rule Now that rules for basic arithmetic (addition, subtraction,
multiplication, and division) are known, look at composition:
Composite Rule for Powers:

18. Examples 1&2

19. Example 3 Body-heat loss due to convection involves a coefficient of
convection, KC ,that depends on wind velocity. Find the rate of change of the coefficient when wind velocity is 12 mph.

20. Example 4
The U.S. gross domestic product (GDP) (in billions of dollars) based on t, years after 2000, can be modeled with:
Find and interpret the rate of change in GDP in 2015.
2015:
2016:
Change:
The U.S. GDP was expected .

21. Using the Derivative Rules
Now mult., div. and chain rule in more complicated functions.
Example 1

22. Example 2

23. Example 3
A 12 inch square piece of cardboard has squares of size x inches cut out of each corner and is then folded to create an open-top box. Find the rate of change of the volume for various values of x.

24. Example 4
The distance an object travels (in feet) is based on t, time in seconds, can be modeled with:
Find and interpret the position, velocity and acceleration of the object at times of 4 seconds and 5 seconds.