District Curriculum Overview

Subject: Mathematics Course: 342 AP Calculus BC Grade Level: 9 – 12

Concepts	Topics/Units	Content/Skills	Essential Activities/Agreements
Understanding graphs of continuous functions and applying the Intermediate Value Theorem and the Extreme Value Theorem	Limits and Continuity	· Limits, Continuous functions, Asymptotes, End behavior · Curve sketching · Intermediate Value Theorem and Extreme Value Theorem
Explaining the concept of derivative of a function, graphically, numerically and algebraically using limits. Applying the Mean Value Theorem and its geometric consequences.	Derivatives I	· Slope of a curve at a point, including points at which there are vertical tangents and at which there are no tangents. · The concept of derivative · Derivatives involving polynomials · Graphing the derivative from data · Mean Value Theorem
Finding the linearization of a function at a given point Using differentials to estimate the relative change in the value of a function	Derivatives II	· Derivatives of functions involving fractional powers & trig functions · Chain rule and Implicit differentiation · Differentials and Linear approximation
Using first and second derivatives to find critical points, intervals of increase/decrease, inflection points, and concavity of graphs of polynomial, trigonometric, rational, and radical functions	Curve Sketching	· Critical points, Concavity · Local (relative) and Absolute (global) extreme values · First and second derivative tests · Rational and radical functions
Using derivatives to solve real world problems based on rates of change and optimization	Applications of the Derivative	· Speed, Velocity, Acceleration, Related Rate Problems · Selected problems from physics, chemistry, industry, economics, medicine and life experiences
Understanding and using the geometric interpretation of differential equations via directional fields and the relationships between slope fields and solution curves for differential curves.	Antiderivatives	· Antiderivatives · Differential Equations and Slope fields (Directional fields) · Solution curves for differential equations · Solving logistic differential equations; using them in modeling · Numerical solution of differential equations using Euler’s method
Using Riemann Sums (using left, right, and midpoint evaluation points) , Trapezoidal Rule, and Simpson’s Rule to approximate a definite integral of a function represented algebraically, graphically and by a table of values.	Integrals I	· Area · Riemann sum · Trapezoidal Rule · Simpson’s Rule · Definite integral as an accumulation function
Estimating, evaluating and calculate values of integrals	Integrals II	· Fundamental Theorem · Indefinite integrals · Substitution and numerical methods
Using integrals to solve real world problems	Applications of definite integrals	· Area under a graph as a net accumulation of rate of change · Volume of a solid by disc, washer, or shells, and by cross sections · Average Value of a function
Determine when logarithmic differentiation must be used and applying it correctly Solving separable differential equations which model exponential growth or decay	Exponential and logarithmic functions	· Derivatives and integrals of exponential and logarithmic functions · Logarithmic differentiation · Exponential growth and decay
Using inverse trig functions to solve real world problems	Inverse trig functions	· Graphs, derivatives, and integrals involving inverse trig functions
Solving integrals by the methods of parts, trig substitution, and partial fractions Using partial fractions to solve separable differential equations and exponential growth problems	Methods of Integration	· Integration by parts · Trigonometric substitution · Partial fractions · Algebraic manipulations · General trig manipulations
Applying L’Hopital’s rule correctly to calculate a limit Evaluating improper integrals using limits	Improper integrals	· L’Hopital’s rule, including its use in determining limits and convergence of improper integrals and series · Indeterminate forms · Improper integrals
Understanding the concept of a series Expressing functions as general Taylor series centered at x=a or as a Maclaurin series. Manipulating a Taylor series for a given function, to obtain a new series from a known series	Polynomial Approximations and Series	· Define a series as the limit of a sequence of partial sums. · Use various tests to determine whether a series converges or diverges · Determine whether alternating series converge or diverge · Calculate error bound of an alternating series approximation · Compare graphs of functions and their Taylor polynomials · Approximate functions using Taylor polynomials · Determine interval and radius of convergence · Find Lagrange error bound for Taylor polynomials · Form new Taylor series by substituting, by differentiating, or by integrating. · Find a Maclaurin series for a function
Using parametric and polar curves to solve real world problems	Parametric and polar curves	· Analysis of planar curves in given parametric form, polar form, and vector form, including velocity and acceleration · Derivatives of parametric, polar, and vector functions · Finding Arc Length of a Parameterized Curve · Finding the Area bounded between two polar equations
	AP Test Preparation	· Review of course	Released AP multiple choice and free response questions
· Use understanding of calculus ideas to do an in-depth study of one application of calculus to a real world situation	Special Projects: Time left after AP exams	Topics selected by students may include · Length of Curve · Calculating Work and Hooke’s Law · Area of a Surface of Revolution · Hydrostatic Force · Moments and Center of Mass