### Day 47 - Unit 3 Synthesis - 10.23.14

 UpdatesUnit 3 Test tomorrow, 10/24!RecordingBell RingerReviewLocate the absolute extrema of the function f(x) = 2x2 + 12x - 4 on the closed interval [-6, 6].no absolute max; absolute min: f(6) = 140absolute max: f(-3) = -22; absolute min: f(6) = 140absolute max: f(6) = 140; no absolute minabsolute max: f(6) = 140; absolute min: f(-3) = -22no absolute max or minFind all the relative extrema for the following function: f(x) = sin(x) + cos(x) on the interval [0, 2π].Relative Max: (5π / 4, sqrt(2)); Relative Min: (π / 4, -sqrt(2))Relative Max: (π / 4, sqrt(2)); Relative Min: (-π / 4, -sqrt(2))Relative Max: (π / 4, sqrt(2)); Relative Min: (5π / 4, -sqrt(2))Relative Max: (π / 4, sqrt(2)); Relative Min: (5π / 4, sqrt(2))none of the aboveDetermine whether Mean Value Theorem can be applied to the function f(x) = x3 on the closed interval [0, 16]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (0, 16) such that f’(c) = [f(b) - f(a)] / (b - a).MVT applies; c = 4MVT applies; c = -16sqrt(3) / 3MVT applies; c = 8MVT applies; c = 16sqrt(3) / 3MVT does not applyFind the open intervals where the function f(x) = sin(x) + cos(x) is increasing or decreasing on the closed interval [0, 2π]Increasing: (-∞, π/4) and (5π/4, ∞) | Decreasing: (π/4, 5π/4)Decreasing: (-∞, π/4) and (5π/4, ∞) | Increasing: (π/4, 5π/4)Increasing: (-∞, π/4) | Decreasing: (π/4, 5π/4) and (5π/4, ∞)Decreasing: (-∞, π/4) | Increasing: (π/4, 5π/4) and (5π/4, ∞)none of the aboveFind all intervals on which f(x) = (x + 1) / (x - 3) is concave downward.(1. ∞)(-∞, ∞)(-∞, 3)(3, ∞)none 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 AsymptotesOptimizationLessonReviewExit TicketN/A 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 InfinityCheckpointsOptimizationCheckpoints