The student will analyze the derivative of a function as a function in itself. This will include a) comparing corresponding characteristics of the graphs of f, f ', and f "; b) defining the relationship between the increasing and decreasing behavior of f and the sign of f '; c) translating verbal descriptions into equations involving derivatives and vice versa; d) analyzing the geometric consequences of the Mean Value Theorem; e) defining the relationship between the concavity of f and the sign of f "; and f) identifying points of inflection as places where concavity changes and finding points of inflection. APC.8 The student will apply the derivative to solve problems. This will include a) analysis of curves and the ideas of concavity and monotonicity; b) optimization involving global and local extrema; c) modeling of rates of change and related rates; d) use of implicit differentiation to find the derivative of an inverse function; e) interpretation of the derivative as a rate of change in applied contexts, including velocity, speed, and acceleration; and f) differentiation of nonlogarithmic functions, using the technique of logarithmic differentiation. * * AP Calculus BC will also apply the derivative to solve problems. This will include a) analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors; b) numerical solution of differential equations, using Euler's method; c) l'Hopital's Rule to test the convergence of improper integrals and series; and d) geometric interpretation of differential equations via slope fields and the relationship between slope fields and the solution curves for the differential equations.