### Day 33 - Related Rates - 10.02.14

 Bell RingerRemindersDifferentiation Test 2 on this Friday, 10.03.14!Summative Exam 1 on next Friday, 10.10.14!Implicit DifferentiationFind the derivative of the following implicitly: x3 + y2 = 5dy/dx = 3x2 / (2y)dy/dx = -3x2 / (2y)dy/dx = -3x2 / ydy/dx = -x2 / ynone of the aboveFind the second derivative of the following implicitly: x3 + y2 = 5dy/dx = (-12xy - 9x4y-1) / (4y2)dy/dx = (12xy - 9x4y-1) / (4y2)dy/dx = (-12xy - 9x4y-1) / y2dy/dx = (-12xy + 9x4y-1) / (4y2)none of the aboveFind the equation of the tangent line at (1, -2) for the following: x3 + y2 = 5y = ¾x + (5/4)y = ¾x - (5/4)y = -¾x + (5/4)y = -¾x - (5/4)none of the aboveRelated RatesGrain is poured into a conical pile at a rate of 20 ft3/min. The diameter of the base of the cone is approximately 4 times the height. At what rate is the height of the pile changing when the pile is 10 ft high?12.5/π0.8/π1/(20π)1/(2π)none 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 ProblemsProduct RuleQuotient RuleChain RuleImplicit Differentiation Lesson Related RatesBrainstorming ActivityDiscussionCheckpointsA - page 154 #2B - page 154 #16C - page 154 #18D - page 154 #20E - page 154 #14 Exit Ticket Exit Ticket will be posted on the board in class. Lesson Objective(s)How can differentiation be used to find the related rates?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