### Day 41 - Increasing and Decreasing Functions - 10.15.14

 UpdatesQuiz 7 on Friday!Covers the entire new unit so farExtremaRolle's TheoremMean Value TheoremIncreasing and Decreasing FunctionsFirst Derivative TestBell RingerRolle's Theorem and Mean Value TheoremDetermine whether Rolle’s Theorem can be applied to the function f(x) = x2 - 4x - 5 on the closed interval [-1, 5]. If Rolle’s Theorem can be applied, find all values of c in the open interval (-1, 5) such that f’(c) = 0.Rolle’s Theorem applies; c = -2Rolle’s Theorem applies; c = 0.5Rolle’s Theorem does not apply.Rolle’s Theorem applies; c = 2both a and dDetermine 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 average value of f(x) = 2x3 + 3 on [3, 7]689158-1584none of the aboveReviewPrerequisitesInterval NotationMaxima/MinimaZero Product PropertyFinding Minima/Maxima GraphicallyExtremaLessonIncreasing and Decreasing FunctionsFirst Derivative TestHow is the increasing/decreasing of a function related to its first derivative?Checkpoints (page 186)L - #6M - #20N - #44O - #48Concavity and Inflection PointsExit TicketPosted on the board at the end of the class. Lesson Objective(s)How can the first derivative be used to find relative extrema?How can the first derivative be used to find intervals in which a function is increasing or decreasing?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 - #44