District Curriculum Overview

Subject: Mathematics Course: 341 AP Calculus AB Grade Level: 9 – 12

Concepts	Topics/Units	Content/Skills	Essential Activities/Agreements
Finding Limits of Functions (including one-sided limits)	Functions, Graphs, and Limits	· An intuitive understanding of limits · Calculating limits algebraically. · Estimating limits from graphs or tables of data. · Describing asymptotic behavior in terms of limits involving infinity.
Understanding Continuity as a Property of Functions	Limits and Continuity	· Understanding continuity in terms of limits. · Geometric understanding of graphs of continuous functions ( Intermediate Value Theorem and Extreme Value Theorem)
Understanding the Concept of Derivative	Definition of a Derivative	· Derivative presented graphically, numerically, and analytically. · Derivative interpreted as an instantaneous rate of change. · Derivative defined as the limit of the difference quotient. · Relationship between differentiability and continuity.
Determining the Derivative at a Point	Derivatives I	· Slope of a curve at a point, including points at which there are vertical tangents and at which there are no tangents. · Tangent line to a curve at a point and local linear approximation. · Instantaneous rate of change as the limit of average rate of change. · Approximate rate of change graphically, and numerically.
Understanding the Derivative as a Function	Derivatives I	· Corresponding characteristics of graphs of f and f ’. · Relationship between the increasing and decreasing behavior of f and the sign of f ’. · The Mean Value Theorem
Understanding and Calculating Second Derivatives	Derivatives I	· Corresponding characteristics of the graphs of f, f ’, and f ’’. · Relationship between the concavity of f and the sign of f ’’. · Points of inflection as places where the concavity changes.
Solving Application Problems Involving Derivatives	Curve Sketching Optimizaton Problems Slopefields	· Analysis of curves · Absolute (global) and Relative (local) extrema. · Modeling rates of change and related rate problems. · Speed, Velocity and Acceleration · Interpeting differential equations via slopefields and the relationship between slopefields, and solution curves for differential equations.
Computing Derivatives	Derivatives II	· Compute derivatives of the following basic functions: power, exponential, logarithmic, trigonometric, and inverse trigonometric functions. · Use the product and quotient rules, and find higher order derivatives. · Use the Chain rule and implicit differentiation.
Interpreting and Using Properties of Definite Integrals	Integrals	· Definite integral as a limit of Riemann sums. · Definite Integral as an accumulation function. · Basic Properties of Definite Integrals.
Finding specific antiderivatives, and solving separable differential equations.	Techniques of Integration	· Fundamental Theorem of Calculus · Indefinite vs. Definite integrals · Substitution Method.
Using integrals in a variety of applications to model physical, biological, or economic situations.	Applications of Integration	· Area of a region between two curves. · Volumes of Solids: the disc, the washer, and the shell method. · Volume of a solid with known cross sections. · Average value of a function · The total distance traveled by a particle along a line.
Using numerical approximations to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.	Definite Integrals by Numerical Approximations.	· Using left, right, and midpoint Riemann Sums to compute an approximate value for a definite integral. · Using the Trapezoidal Rule to find area. · Finding the error in the Trapezoidal Rule.
	AP Test Preparation	· Review of course	Previously released AP multiple choice and free response questions