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

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

Example 1 Example 2

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

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

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

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

Product/Quotient Rules Product Rule: Quotient Rule:

Examples 1&2

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

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 .

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

Examples 1&2

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.

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 .

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

Example 2

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.

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.