### Day 42 - Concavity and Second Derivative Test - 10.16.14

 UpdatesQuiz 7 on Friday!Covers the entire new unit so farExtremaRolle's TheoremMean Value TheoremIncreasing and Decreasing FunctionsFirst Derivative TestConcavitySecond Derivative TestToday's ScheduleBlock 1 - 8:25-9:55Block 2 - 10:02-11:06Block 3 - 11:13-1:19LunchA - 11:06-11:38B - 12:04-12:35C - 12:50-1:19Block 4 - 1:26-2:30Bell RingerIncreasing/Decreasing FunctionsFind the open intervals where the function f(x) = sin(x) + cos(x) is increasing or decreasing on the closed interval [0, 2π]Increasing: (0, π/4) and (5π/4, 2π) | Decreasing: (π/4, 5π/4)Decreasing: (0, π/4) and (5π/4, 2π) | Increasing: (π/4, 5π/4)Increasing: (0, π/4) | Decreasing: (π/4, 5π/4) and (5π/4, 2π)Decreasing: (0, π/4) | Increasing: (π/4, 5π/4) and (5π/4, 2π)none of the aboveIdentify the open intervals where the function f(x) = -4x2 - 6x - 4 is increasing or decreasing.Increasing: (-∞, -¾) | Decreasing: (-¾, ∞)Decreasing: (-∞, -¾) | Increasing: (-¾, ∞)Increasing on (-∞, ∞)Decreasing on (-∞, ∞)none of the aboveReviewPrerequisitesInterval NotationMaxima/MinimaZero Product PropertyFinding Minima/Maxima GraphicallyExtremaRolle's TheoremMean Value TheoremIncreasing and Decreasing FunctionsFirst Derivative TestLessonConcavity and Inflection PointsSecond Derivative TestHow is the concavity of a function related to its first derivative?How is the concavity of a function related to its second derivative?Checkpoints (page 195)P - #6Q - #16R - #18S - #20T - #24U - #32V - #38W - #52Exit TicketPosted on the board at the end of the class. Lesson Objective(s)How can derivatives be used to find concavity?How can derivatives be used to find points of inflection? 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 - #48