1. 5023 MAX - Min: Optimization AP Calculus

2. OPEN INTERVALS: Find the 1 st Derivative and the Critical Numbers First Derivative Test for 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 Test for Max / Min –FIND 2 nd Derivative –PLUG IN the critical number –MAX: if the value is negative –MIN: if the value is positive

3. Example 1: Open - 1st Derivative test VA=1 x=3, x=-1 Undefined at 1 f’(x) 13 + -- + -202 VA 4 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 1 st derivative goes from negative to positive at 3. A VA at x=1 because 1 st derivative and the function are undefined at x=1

4. Example 2: Open - 2nd Derivative Test

5. LHE p. 186

6. CLOSED INTERVALS: 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 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 Critical #

7. CLOSED INTERVALS: Find the 1 st Derivative 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

8. Example : Closed Interval Test Consider endpts. Absolute max absolute min

12. LHE p. 169 even numbers

13. OPTIMIZATION PROBLEMS Used to determine Maximum and Minimum Values – i.e. »maximum profit, »least cost, »greatest strength, »least distance

14. 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. Find the Derivative and perform one the tests.

15. Relative Maximum and Minimum FIRST DERIVATIVE TEST FOR MAX/MIN. If f is continuous on [a, b] containing a critical number,c, and differentiable on (a,b) except possibly at c and if:  f / (x) changes signs from (+) to (-) then f(c) is a relative Maximum.  f / (x) changes signs from (-) to (+) then f(c) is a relative Minimum.  f / (x) has no sign change then f(c) is neither. Note: c is the location. f (c) is the Max or Min value. DEFN: Relative Extrema are the highest or lowest points in an interval. Where is xWhat is y

16. Concavity and Max / Min SECOND DERIVATIVE TEST FOR MAX/MIN. If f is continuous on [a, b] and twice differentiable on (a,b) and if f (c) is a critical number, then  if f // (c) > 0 then f (x) is concave up and c is a minimum  if f // (c) < 0 then f (x) is concave down and c is a maximum  if f // (c) = 0 then the test is inconclusive.

17. 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? Statement: Generic formula: Personalized formula: Which Test? Figure: y x Max area A=lw A=xy x+2y=2000 x=2000-2y x=1000 y=500 To maximize the area the lengths of the sides should be 500 meters.

18. Example 2: Design an open box with the MAXIMUM VOLUME that has a square bottom and surface area of 108 square inches. Statement: max volume of rect. prism Formula: x x y Personalized Formula:

19. Example 3: 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. ellipse Major axis y-axis length is 2 Minor axis x-axis length is 1 Max Area: A=lw A=2x*2y A=4xy Secondary equations

20. Example 4: Find the point on closest to the point (0, -1). min (0,-1)

21. Example 5: 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 will cost 1 cent per square inch. Find the dimensions that will minimize the cost. Statement: minimize cost of material Formula: x x y

22. 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.

23. Example 7:AP Type Problem: At noon a sailboat is 20 km south of a freighter. The sailboat is traveling east at 20 km/hr, and the freighter is traveling south at 40 km/hr. If the visibility is 10 km, could the people on the ships see each other?

24. Example 8:AP - Max/min - Related Rates The cross section of a trough has the shape of an inverted isosceles triangle. The lengths of the sides of the cross section are 15 in., and the length of the trough is 120 in. 15in. 120 in. 1) Find the size of the vertex angle that will give the maximum capacity of the trough. 2) If water is being added to the trough at 36 in 3 /min, how fast is the water level rising when the level is 5 in high?

25. Last Update: 12/03/10 Assignment: DWK 4.4