5023 MAX - Min: Optimization AP Calculus
First Derivative Testfor Max / Min TEST POINTS on either side of the critical numbers MAX:if the value changes from + to – MIN: if the value changes from – to + Second Derivative Testfor Max / Min FIND 2nd Derivative PLUG IN the critical number MAX: if the value is negative MIN: if the value is positive OPEN INTERVALS:Find the 1stDerivative and the Critical Numbers
Example 1: Open - 1st Derivative test VA=1 x=3, x=-1 Undefined at 1 f’(x) + + - - -1 1 3 VA 4 -2 0 2 max at -1 min at 3 and a VA at 1 why? Max of -2 at x=-1 because the first derivative goes from positive to negative at -1. Min of 6 at x=3 because 1st derivative goes from negative to positive at 3. A VA at x=1 because 1st derivative and the function are undefined at x=1
Critical # CLOSED INTERVALS: EXTREME VALUE THEOREM: If f is continuous on a closed interval [a,b], then f attains an absolute maximum f(c) and an absolute minimum f(d) at some points c and d in [a,b] local Closed Interval Test Find the 1st Derivative and the Critical Numbers Plug In the Critical Numbers and the End Points into the original equation MAX: if the Largest value MIN: if the Smallest value
CLOSED INTERVALS:Find the 1stDerivative and the Critical Numbers • Closed Interval Test • Plug In the Critical Numbers and the End Points into the original equation • MAX: if the Largest value • MIN: if the Smallest value
Example : Closed Interval Test Consider endpts. and x=2 Absolute max absolute min
OPTIMIZATION PROBLEMS • Used to determine Maximum and Minimum Values – i.e. • maximum profit, • least cost, • greatest strength, • least distance
METHOD: Set-Up Make a sketch. Assign variables to all given and to find quantities. Write a STATEMENT and PRIMARY (generic) equation to be maximized or minimized. PERSONALIZE the equation with the given information. Get the equation as a function of one variable. < This may involve a SECONDARY equation.> Find the Derivative and perform one the tests.
Relative Maximum and Minimum DEFN: Relative Extrema are the highest or lowest points in an interval. Where is x What is y
y-could be 0 y – could be 1000 1 ILLUSTRATION : (with method) A landowner wishes to enclose a rectangular field that borders a river. He had 2000 meters of fencing and does not plan to fence the side adjacent to the river. What should the lengths of the sides be to maximize the area? Figure: Statement: y Max area Generic formula: x A=lw Personalized formula: A=xy x+2y=2000 x=2000-2y x=1000 y=500 Which Test? To maximize the area the lengths of the sides should be 500 meters.
Design an open box with the MAXIMUM VOLUME that has a square bottom and surface area of 108 square inches. Example 2: y Statement: x x max volume of rect. prism Formula: Personalized Formula: Plug in 6 Get a -
Find the dimensions of the largest rectangle that can be inscribed in the ellipse in such a way that the sides are parallel to the axes . Example 3: ellipse Major axis y-axis length is 2 Minor axis x-axis length is 1 Secondary equations Max Area: A=lw A=2x*2y A=4xy CIT: A(0)=0 A A(1)=0 Dimensions: undef: x=1,-1 Zeroes:
min Find the point on closest to the point (0, -1). Example 4: (0,-1) =0
A closed box with a square base is to have a volume of 2000in.3 . the material on the top and bottom will cost 3 cents per square inch and the material on the sides willcost 1cent per square inch. Find the dimensions that will minimize the cost. Example 5: Statement: minimize cost of material y x Formula: x
Example 6: Suppose that P(x),R(x), and C(x) are the profit, revenue, and cost functions, that P(x) = R(x) - C(x), and x represents thousand of units. Find the production level that maximizes the profit.
Last Update: 12/03/10 Assignment: DWK 4.4