AP CALCULUS BCSyllabus

COURSE DESCRIPTION

AP Calculus BC is a full year course offered to students in grades 11-12. AP Calculus BC emphasizes the concepts of calculus graphically, algebraically, and with tables of values. The course teaches all topics associated with functions, graphs, limits, derivatives, integrals, and polynomial approximations as well as series as outlined in the AP Calculus Course Description.

COURSE MATERIALS

Major Text:Larson,Roland E., Robert P. Hostetler, and Bruce H. Edwards. Calculus. Sixth Edition. Houghton Mifflin Company, 1998.

Supplementary Materials and Additional Resources:Sallee, Tom, Judy Kysh, and Brian Hoey. College Preparatory Mathematics Calculus, First Edition: CPM Educational Program, 2003

apcentral.collegeboard.com

AP Calculus Course Description. The College Board.

AP Calculus Exam free-response questions from 1997 to present.

AP Calculus Exam multiple-choice questions from1993, 1997, 1998, and 2003.

Texas Instrument TI-83, TI-83 Plus, or TI-89 graphing calculator.

COURSE OUTLINE

Chapter 1: Limits (9 days) 1 test

Find limits graphically and numerically Evaluate limits analytically Estimating limits from both graphs and tables Continuity with one-sided limits and graphs Intermediate Value Theorem Infinite limits and asymptotic behavior Limits at infinity and asymptotic behavior

Students will work with limits graphically, numerically, analytically, and verbally.

o Graphically, students will have to determine that the graph approaches the same value from the left and the right.

o Numerically, students will describe the value of the limito Analytically, students will find the value of the limito Verbally, students will discuss both orally and written how the above

methods are related.

Chapter 2: Differentiation (16 days) 2 Quizzes, 1 Test

Examining rates of change for different types of graphs Definition of derivative Power Rule and derivatives of a sum and difference Derivative and Tangent Line (local linearity) Instantaneous rate of change vs. average rate of change Approximating rates of change from graphs and tables Differentiability and continuity Basic differentiation rules and rates of change Product and Quotient Rules Higher Order Derivatives Chain Rule Implicit Differentiation Related Rates Students will work with differentiation graphically, numerically, analytically, and

verbally. o Graphically, be determining two points on a line students can estimate the

derivative by calculating the slope between the two pointso Numerically, students will use their calculators to verify the derivative at

an exact point.o Analytically, students will understand and be able to use the definition of

derivative to calculate the value of the derivative at a pointo Verbally, students will be able to discuss the meaning of the derivative;

specifically that the derivative is the slope at an exact point

Chapter 3: Applications of Differentiation (17 days) 1 Quiz, 1 Test

Analysis of functions Increasing and decreasing functions and their derivatives Examining the relationship graphically between a function and its derivatives Provide written explanations of the relationship between graphs First derivative test

Concavity, points of inflection, and second derivative test Extrema on an interval Rolles Theorem and Mean Value Theorem Word problems using AVT and MVT require students to express, in written form,

the differences and similarities between the two concepts Curve Sketching and curve analysis Interpreting first and second derivatives in context (including position, vel., acc.) Provide written explanations of relationship between velocity and position

functions Optimization problems Inverse functions and implicit differentiation Newtons Method Differentials Slope fields and Eulers Method Students will work with differentiation graphically, numerically, analytically, and

verbally. For example, when learning optimization:o Graphically, students will plot various data points in order to visually

determine the approximate value at which the maximum occurso Numerically, students will calculate the volume of a given boxo Analytically, students will determine the maximum value of the boxo Verbally, students will discuss the use of the derivative in identifying

max/min. Chapter 4: Integration (13 days) 1 Test

Anti-derivatives and indefinite integration Area Students will be able to provide written explanations of the relationship between

the anti-derivative and area of under a curve Riemann sums: left, right, midpoint, trapezoids Definite integrals Average value of a function Fundamental Theorem of Calculus Word problems require students to use written explanations to verify outcomes Integration by substitution Numerical integration Applications of integration Integration as an accumulation function Students will work with integration graphically, numerically, analytically, and

verbally.o Graphically, students will be required to draw rectangles and trapezoids to

estimate the area under a curve.

o Numerically, students will use their calculator to graph and shade the region under a curve, and finally develop the tool to evaluate a definite integral

o Analytically, students will be able to use integration techniques to solve various problems

o Verbally, students will discuss the meaning of the integral and will be able to explain the relationship between Riemann Sums and the value of an integral; i.e. was the estimate above or below the actual value of the integral

Chapter 5, Sections 1-5.7: Logarithmic and Exponential Functions (10 days) 1 Test

Natural logarithm function and differentiation Natural logarithmic function and integration Inverse functions Exponential functions differentiation and integration Bases other than e and applications Differential equations growth and decay Separable differential equations

Chapter 5.8-5.9: Inverse Functions (6 days) 1 Quiz

Inverse trigonometric functions and differentiation Inverse trigonometric functions and integration

Chapter 6: Applications of Integration (12 days) 1 Test

Area of region between two curves Volumes by disc, washers, shells and cross sections Arc Length Students will work with specific applications of integration graphically,

numerically, analytically, and verbally.o Graphically, students will be able to use their calculators to sketch the two

curves, and thus shade the region between the curveso Numerically, students must be able to differentiate the different methods

of finding volume and must determine the most appropriate method to use.o Analytically, students will be able to use already learned integration

techniques to determine the exact area between two curves, as well as the volume of a rotated region.

o Verbally, students will discuss the various methods (discs, washers, etc.) used and the relationship between the methods.

Midterm (Review 2 days)

Chapter 7: Integration Techniques (10 days) 1 Test

Basic integration rules Integration by parts Partial fractions Indeterminate forms and LHopitals rule Improper integrals

Chapter 9: Parametric Equations and Polar Coordinates (10 days ) 1 Test

Analysis of plane curves and parametric equations Derivative of parametric functions Polar coordinates and polar graphs Derivatives of polar functions Area in polar coordinates Analysis and derivatives of vectors Students will work with polar functions graphically, numerically, analytically, and

verbally.o Graphically, students must be able to graph polar curves o Numerically and analytically, students will be able to find the area

between two polar curves.o Verbally, students must be able to interpret results involving motion of a

particle, given information about one or more of the functions.

Chapter 8: Infinite Series ( 20 days) 2 tests

Sequences and partial sums Series and convergence Geometric Series Power Series Representation of functions using power series Taylor and MacLaurin Series

Using the calculator to view and examine the relationship between ex, sin x, cos x, and 1/(1-x) and their respective MacLaurin polynomials Derivatives and anti-derivatives of Taylor and MacLaurin Series

Nth term test Relationship between series and improper integrals Integral and p series test Direct and limit comparison tests Alternating series test

Ratio test Lagrange error for Taylor polynomials Students will work with Taylor series graphically, numerically, analytically, and

verbally.o Numerically, given a function, students must be able to write terms of a

series about 0=xo Analytically, students will be able to estimate the value of an integral

using terms of a serieso Verbally, students must be to explain the differences between estimated

values and actual values of an integral involving error bounds

Logistics (5 days) 1 Quiz

Modeling the spread of disease using the graphing calculator and randomly generated numbers.

Analyzing and solving logistic differential equations Checking to see how the experimental model reflects the theoretical differential

equation

CalculatorStudents will use a graphing calculator for the following:

Explore functions and tangent lines Explore and interpret how transformations, compositions, and inverses affect a

graph and its domain and range Input data and find a graph of best fit Tool to develop derivative functions Find the derivative of a