• Course Outline (order of topics and time allowed is approximate)

# Unit 1 Limits and Continuity (10 days)

·         Informal Definition of Limit – one and two-sided, limits that fail to exist

·         Properties of Limits

·         Finding limits graphically, numerically, and analytically

·         Use limits to compare relative magnitudes of functions and their rates of change

·         Use infinite limits to determine the asymptotic behavior of a function, including vertical asymptotes.

·         Determining continuity from graphs at a point and on an open interval

·         Determine one-sided limits and continuity on a closed interval

·         Use properties of continuity

·         Determining continuity analytically using limits

·         Apply the Intermediate Value Theorem

·         Points of discontinuity – removable and non-removable, asymptotic behavior

# Unit 2 Differentiation (13 days)

·         Local linearity

·         Formal definition of the derivative

·         Alternate limit form of the derivative

·         Finding equations of tangent lines

·         Relating the derivative to the slope of the tangent line

·         Exploration of the concept of a derivative- Interpretation of the derivative as the slope of a tangent line, local linearity, and instantaneous and average rate of change

·         Estimating rates of change (derivatives) from tables of data and graphs

·         Graphical interpretation of differentiability / non-differentiability

·         Differentiability implies continuity, but continuity does not necessarily imply differentiability

·         Techniques of differentiation:  Constant, Multiple, Sum and Difference, Power, Product, Quotient, and Chain Rule.

·         Derivatives of the transcendental functions – power, exponential, logarithmic, trigonometric, inverse trigonometric.

·         Higher Order Derivatives

# Unit 3:  Curve Sketching (17 days)

·         Finding relative and absolute extrema of a function on an interval

·         Compute the derivative of a function numerically using a graphing calculator

·         Extreme Value Theorem and Mean Value Theorems and their applications

·         Determine the critical values of a function

·         Increasing and Decreasing Intervals – including Monotonic Functions

·         Finding local extrema using the First and Second Derivative Tests

·         Determine intervals of Concavity and Points of Inflection

·         Find limits at infinity and discuss these limits in relation to the end behavior of a function

·         Analyzing the relationship between f, f’, and f” – graphical and numerical applications

·         Modeling and Optimization  - solving applied minimum and maximum problems

# Unit 4:  Applications of Differentiation  (20 days)

·         Local Linear Approximations - estimate numeric derivatives and find error using concavity

·         Determine the numerical solution of differential equations using Euler’s Method

·         Interpret rates of change including numerical data and introduce slope fields

·         Apply Rolle’s Theorem and the Mean Value Theorem

·         Interpret the derivatives as a rate a change in a variety of applied contexts including velocity, speed, and acceleration

·         Implicit Differentiation

·         Derivatives of Inverses of Functions

·         Apply derivatives to real-world problems involving related rates

·         Determine the derivative of parametric, polar, and vector-valued functions

·         Apply derivatives in the analysis of planar curves given in parametric, polar, vector form, including velocity and acceleration

# Unit 5:  Introduction to Integral Calculus (17 days)

·         Approximating area using  left, right, midpoint, upper, lower, and trapezoidal sums

·         Approximate definite integrals of functions represented algebraically, geometrically, and numerically using Approximating Sums (including distance traveled by a particle along a line )

·         Compute Riemann sums using left, right, midpoint, upper, lower, and evaluation points

·         Find area using the limit definition of a Riemann sum

·         Definition of definite integrals as a limit of a summing process

·         Antidifferentiation – from known derivatives of basic functions including use of the general power rule.

·         Properties of definite integrals

·         Find the constant of integration from initial conditions.

·         First and Second Fundamental Theorems of Calculus

·         Mean Value Theorem for Integrals – average value of a function

·         Evaluate a definite integral using a graphing calculator

·         Graphical Analysis of functions using Fundamental and Mean Value Theorems

·         “U” Substitution – including change of variables and change of limits of definite integrals.

·         Integration of transcendental functions – power, exponential, trigonometric, inverse trigonometric

·         Antiderivatives by parts and simple partial fractions with non-repeating linear factors only

# Unit 6:  Techniques of integration (10 days)

·         Use the Fundamental Theorem of Calculus to represent a particular antiderivative and perform an analytical and graphical analysis of a function so defined

·         Use the Second Fundament Theorem of Calculus to interpret a rate of change as the change of a quantity over an interval.

·         Solving separable differential equations graphically and analytically

·         Use separable differential equation in modeling, including applications with growth and decay.

·         Solve logistic differential equations and use them in modeling

# Unit 7:  Applications of Integration (19 days)

·         Apply integration methods to find the area of a region bounded by planar curves

·         Using integrals of rates of change to find accumulated change – motion revisited

·         Volumes of solids of revolution about a horizontal or vertical axis – disc and washer

·         Volumes of solids with known cross sections

·         Length of a curve including a curve given in parametric form

·         Apply integration methods to find the area of a region bounded by polar curves

·         Apply terms of series as areas of rectangles and relate them to improper integrals

·         Use technology to explore and determine convergence and divergence

·         Apply the concept of a series to examples including decimal expansion

·         Apply geometric series in applications

·         Define a series as a sequence of partial sums and convergence as the limit of the sequence of partial sums

·         Find limits using L’Hopital’s Rule

# Unit 8:  Convergence (14 days)

·         Find improper integrals as limits of definite integrals

·         Use L’Hopital’s Rule to determine convergence and divergence of improper integrals and series

·         Determine the convergence or divergence of geometric, telescoping, p, and harmonic series using the Integral Test

·         Determine convergence or divergence of series using the Ratio Test

·         Determine convergence or divergence and approximate the sum of an alternating series and determine error bound

·         Choose the most efficient test for determining the convergence or divergence of an infinite series by recognizing when to apply which test to a given series.

·         Find an nth degree polynomial approximation of a given function, including Taylor and Maclaurin’s polynomials, with graphical demonstration of convergence

·         Determine the Lagrange error bound for Taylor polynomials

·         Determine the radius and the interval of convergence of a power series

# Unit 9:  Infinite Series (12 days)

·         Represent functions by a power series

·         Determine the general Taylor series centered at x = a

·         Find the Maclaurin Series for functions , sin x, cos x, and  then use them in the formation of the new series

·         Use formal manipulation of Taylor series and shortcuts to compute Taylor series, including differentiation, antidifferentiation and formation of new series from known series

# Course Review: (31 days)

The remainder of the time before the AP Exam is used for review with a focus on the relationships between the various concepts and on calculus as a cohesive whole.  The entire course is divided into several review units that allow time for additional discussion in areas where students are still having difficulties.  The unit packets allow students to work independently or in groups both in and out of class.  The review includes review presentation of topics by students, individual and group work including discussions of past AP problems, review quizzes, and timed practice tests.  After the exam, various additional topics are discussed at time allows.