### Day 21 - Product and Quotient Rule - 09.16.14

 Bell RingerAll tests need to be remediated by Friday! Find the point at which the tangent line is horizontal. Prove using two different methods. f(x) = (x + 2)2(0, 2)(2, 0)(-2, 0)does not existnone of the aboveFind the slope of the tangent line at x = 3. Prove using two different methods. f(x) = |x - 2|01-1does not existnone of the aboveFind the point at which the tangent line is vertical. Prove using two different methods. f(x) = (3/2)x2/3 + 2x + 1(0, 0)(-1, 0)(0, 1)(-1, 1)none of the aboveFind the derivative of the following function: f(x) = sin x - 2xdy/dx = cos x - 2dy/dx = cos xdy/dx = -cos x -2dy/dx = 2cos xnone of the aboveAt what point does the following function have a horizontal tangent line: y = sin x - 2x, 0 ≤ x ≤ 2π(0, 0)(π, 0)(2π, 0)nevernone of the above Review Secant vs. Tangent Line Slope Definition of Derivative Importance of Derivative What does it allow us to do? Drawing a Tangent Line on a Graph Basic Differentiation Rules Constant Rule Power Rule Sum and Difference RuleSine and Cosine DerivativesDerivative NotationDifferentiationRates of ChangePosition FunctionAverage VelocityInstantaneous VelocityFree Fall Problems Lesson Product RuleQuotient RulePractice (Section 2.3) Checkpoint #1 #2, 4, 6 Checkpoint #2 #8, 10, 12 Checkpoint #3 #30, 34, 36, 38 Checkpoint #4 #50, 52, 54Checkpoint #5#64, 68 Exit Ticket Exit TIcket will be posted on the board in class. Lesson Objective(s)How can derivatives be used to create a rule for finding a product and quotient rule for derivatives?Standard(s)APC.5Investigate derivatives presented in graphic, numerical, and analytic contexts and the relationship between continuity and differentiability.The derivative will be defined as the limit of the difference quotient and interpreted as an instantaneous rate of change.APC.6​The student will investigate the derivative at a point on a curve.Includes:finding the slope of a curve at a point, including points at which the tangent is vertical and points at which there are no tangentsusing local linear approximation to find the slope of a tangent line to a curve at the point​defining instantaneous rate of change as the limit of average rate of changeapproximating rate of change from graphs and tables of values.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 versaAPC.8Apply the derivative to solve problems.Includes:optimization 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; andAPC.9Apply formulas to find derivatives.Includes:derivatives of algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functionsderivations of sums, products, quotients, inverses, and composites (chain rule) of elementary functionsderivatives of implicitly defined functionsMathematical Practice(s)#1 - Make sense of problems and persevere in solving them#2 - Reason abstractly and quantitatively#3 - Construct viable arguments and critique the reasoning of others#4 - Model with mathematics