Day 41 - Optimization - 03.02.16

 Update Calculus Equations/Theorems Bell RingerFind all of the critical numbers for .–3, 0, 333, –30none of the aboveLocate the absolute extrema of the function on the closed interval [–6, 6]No absolute max, Absolute min:  Absolute max: , Absolute min:  Absolute max: , No absolute minAbsolute max: , Absolute min: No absolute max or minDetermine whether Rolle's Theorem can be applied to the function 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 .Rolle’s Theorem applies;  c = –2Rolle’s Theorem applies;  c = 0.5Rolle’s Theorem does not applyRolle’s Theorem applies;  c = 2both a and dDetermine whether the Mean Value Theorem can be applied to the function 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 .MVT applies;  c = 4MVT applies;  c = MVT applies;  c = 8MVT applies;  c = MVT does not applyFind all intervals on which    is concave upward.none of these ReviewPre-calculusExtremaMinimaMaximaAbsolute and Relative Extrema (video)Interval NotationExtrema (video)/Critical Numbers ((checkpoints)How can extrema be defined for a function?How can critical numbers be calculated using derivatives?How are critical numbers related to extrema?How does Extreme Value Theorem work?Increasing/Decreasing Functions (video) (checkpoints)How can derivatives be used to find intervals of increasing and decreasing?How can derivatives be used to find relative extrema?Rolle's Theorem (video)/Mean Value Theorem (video) (Desmos Demonstration) (checkpoints)Concavity and Inflection Points (video) (checkpoints)How are derivative related to intervals of concave upwards and downwards?LessonOptimizationApplications of OptimizationHow can differentiation be used to find optimal conditions? Exit Ticket Posted on the board at the end of the block.HomeworkStudy! Help Request List Standard(s) APC.7Analyze the derivative of a function as a function in itself.Includes:comparing corresponding characteristics of the graphs of f, f', and f''​defining the relationship between the increasing and decreasing behavior of f and the sign of f'​translating verbal descriptions into equations involving derivatives and vice versaanalyzing the geometric consequences of the Mean Value Theorem;defining the relationship between the concavity of f and the sign of f"; and ​identifying points of inflection as places where concavity changes and finding points of inflection.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.**AP Calculus BC will also apply the derivative to solve problems.Includes:​analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors;​numerical solution of differential equations, using Euler’s method;​l’Hopital’s Rule to test the convergence of improper integrals and series; and​geometric interpretation of differential equations via slope fields and the relationship between slope fields and the solution curves for the differential equations.