From previous sections

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Presentation transcript:

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.

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

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

Example 1 x y Explore how AROC changes as 3 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 =

Example 2 x y Explore how AROC changes as 1 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 =

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