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