consequences of a limiting case Therefore the idea of limits is essential for discovering and developing important ideas definitions formulas and theorems in calculus Students must have a solid intuitive understanding of limits and be able to compute various limits including one sided limits limits at infinity the limit of a sequence and infinite limits They should be able to work with tables and graphs in order to estimate the limit of a function at a point Students should know the algebraic properties of limits and techniques for finding limits of indeterminate forms and they should be able to apply limits to understand the behavior of a function near a point Students must also understand how limits are used to determine continuity a fundamental property of functions BIG IDEA 2 DERIVATIVES Using derivatives to describe the rate of change of one variable with respect to another variable allows students to understand change in a variety of contexts In AP Calculus students build the derivative using the concept of limits and use the derivative primarily to compute the instantaneous rate of change of a function Applications of the derivative include finding the slope of a tangent line to a graph at a point analyzing the graph of a function for example determining whether a function is increasing or decreasing and finding concavity and extreme values and solving problems involving rectilinear motion Students should be able to use different definitions of the derivative estimate derivatives from tables and graphs and apply various derivative rules and properties In addition students should be able to solve separable differential equations understand and be able to apply the Mean Value Theorem and be familiar with a variety of real world applications including related rates optimization and growth and decay models BIG IDEA 3 INTEGRALS AND THE FUNDAMENTAL THEOREM OF CALCULUS Integrals are used in a wide variety of practical and theoretical applications AP Calculus students should understand the definition of a definite integral involving a Riemann sum be able to approximate a definite integral using different methods and be able to compute definite integrals using geometry They should be familiar with basic techniques of integration and properties of integrals The interpretation of a definite integral is an important skill and students should be familiar with area volume and motion applications as well as with the use of the definite integral as an accumulation function It is critical that students grasp the relationship between integration and differentiation as expressed in the Fundamental Theorem of Calculus a central idea in AP Calculus Students should be able to work with and analyze functions defined by an integral Your Assignment For each of these Big Ideas do the following 27 points total 1 Rate yourself 1 10 on how confident you are in your understanding of the Big Idea Pay particular attention to the statements about what students should be able to do 3 points for each Big Idea