### Day 46 - Optimization - 10.22.14

 UpdatesUnit 3 Test this Friday, 10/24!Bell RingerOptimizationA farmer has 160 feet of fencing to enclose 2 adjacent rectangular pig pens. The farmer is looking to maximize the area of the pen. Which formula should be differentiated?PerimeterAreaSurface AreaVolumenone of the aboveWhat overall dimensions should be used to maximize the enclosed area?20 ft x 20 ft53.3 ft x 20 ft26.7 ft x 20 ft53.3 ft x 40 ftnone of the aboveWhat is the maximum area possible?400 ft2533.3 ft2800 ft21066.7 ft2none of the aboveReviewPrerequisitesInterval NotationMaxima/MinimaZero Product PropertyFinding Minima/Maxima GraphicallyExtremaRolle's TheoremMean Value TheoremIncreasing and Decreasing FunctionsFirst Derivative TestConcavity and Inflection PointsSecond Derivative TestLimits at InfinityHorizontal AsymptotesLessonOptimizationFind two positive numbers that have a product of 100 and the sum is a minimum.Find the length and width of a rectangle that has a perimeter of 100 feet and a maximum area.CheckpointsExit TicketPosted at the end of the block. Lesson Objective(s)How can derivatives be used to find optimum conditions?Standard(s)APC.8Apply the derivative to solve problems.Includes:​analysis of curves and the ideas of concavity and monotonicityoptimization involving global and local extrema;modeling of rates of change and related rates;use of implicit differentiation to find the derivative of an inverse function;interpretation of the derivative as a rate of change in applied contexts, including velocity, speed, and acceleration; anddifferentiation of nonlogarithmic functions, using the technique of logarithmic differentiation.*Mathematical Practice(s)#1 - Make sense of problems and persevere in solving them#2 - Reason abstractly and quantitatively#5 - Use appropriate tools strategically#6 - Attend to precision#8 - Look for and express regularity in repeated reasoningPast CheckpointsExtrema (page 169)A - #4B - #6C - #18D - #22E - #34Rolle's Theorem (page 176)F - #4G - #10H - #26Mean Value Theorem (page 176-177)I - #34J - #42K - #44Increasing/Decreasing Functions (page 186)L - #6M - #20N - #44O - #48Concavity and Points of Inflection (page 195)P - #6Q - #16R - #18S - #20T - #24U - #32V - #38W - #52Limits at InfinityCheckpoints