Calculus BC Advanced Placement
STATEMENT OF PURPOSE
The purpose of Calculus BC Advanced Placement is to provide a rigorous, well defined curriculum for Advanced Placement Calculus. Calculus BC Advanced Placement is primarily concerned with developing the students understanding of the concepts of calculus and providing experience with its methods and applications. The course emphasizes a multi-representational approach to calculus, with concepts, results, and problems being expressed geometrically, numerically, analytically, and verbally. The connections among these representations are important. Calculus BC is intended to be challenging and demanding. Calculus BC is an extension of Calculus AB rather than an enhancement; common topics require a similar depth of understanding. Technology is used regularly by students and teachers to reinforce the relationships among the multiple representations of functions, to confirm written work, to implement experimentation, and to assist in interpreting results. Each student has their own calculator. The calculator is TI-83 or TI-84. A few students use the TI-89. Assessments are designed to contain problems which are calculator active, as well as other problems for which the calculator is not permitted. Through the use of unifying themes of derivatives, integrals, limits, approximation, and applications and modeling, the course becomes a cohesive whole. The course includes, but is not limited to, college level mathematics for which most colleges grant advanced placement credit according to the results of an Advanced Placement Examination. I cover all the topics that appear in the AP Calculus BC Course Description, plus: area of surface of revolution, and Newton’s Approximation. It is intended that the Calculus BC course enables a student to obtain credit for the first two semesters of college calculus.
MATERIALS Textbook Finney, Ross L., Franklin Demana, Bert Waits, and Daniel Kennedy. Calculus: Graphical, Numerical, Algebraic, AP Edition; Reading, MA: Addison-Wesley, 2007.
Supplemental Resources Best, George, Richard Lux. Preparing for the Calculus (BC) Exam. Venture Publishing. Andover, MA. 1997. Broadwin, Judith, George Lencher, Martin Rudolph. Solutions to the BC Calculus Free Response Questions 1969-2001. Bellmore, NY. Foerster, Paul A. Instructor’s Resource Book. Key Curriculum Press. Berkeley, CA. 1998. Hockett, Shirley. How to Prepare for the Advanced Placement Examination, Calculus. 6th ed. Hughes-Hallet, Deborah, and Andrew M. Gleason, et. al. Calculus Ostebee, Arnold and Zorn, Paul. Calculus from Graphical Numerical Symbolic Points of View Technology: Geometer’s Sketchpad, CBR/CBL
COURSE OBJECTIVES Students will be able to: 1.
Analyze and graph elementary functions including algebraic, trigonometric, exponential and logarithmic
2.
Identify the properties of domain, range and symmetry of a function
3.
Locate real zeros and asymptotes of a function
4.
Compute limits of a function
5.
Derive and apply the definition of the derivative
6.
Determine the conditions for differentiability and continuity of a function
7.
Apply L'Hopital's rule for determining limits of indeterminate forms
8.
Compute the derivative of a product quotient, composite, implicitly defined, inverse logarithmic, exponential and parametric functions
9.
Apply the derivative to determine the slope of a function and the equations of the tangent and the normal
10.
Utilize the derivative to compute linear approximations, identify increasing and decreasing functions, concavity of a function and points of inflection
11.
Solve word problems dealing with min-max and related rates
12.
Determine velocity acceleration and direction of movement-of a function
13.
State and apply the Mean-Value theorem and Rolle's Theorem
14.
Apply the definition of the definite integral as the limit of a Riemann sum
15.
Evaluate and apply the integrals of polynomial, trigonometric, exponential and logarithmic functions
16.
Evaluate the definite and indefinite integral by the methods of substitution, trigonometric substitution, partial fractions, improper integrals and integration by parts
17.
Use the definite integral to find the average value of a function, area under a curve, and volume of a solid of revolution
18.
Employ the definite integral to solve problems of growth and decay
19. Determine the solution of variable separable differential equations of the form 20.
Approximate the solution of differential equations through slopefields and Euler’s Method
21.
Determine the properties of vector functions and their derivatives and integrals
22.
Apply the integral to determine area bounded by polar curves, volumes of solids with know cross sections, length of a path and surface area
23.
Apply the tests of convergence to a series – absolute and conditional
24.
Study Taylor series for remainder and error approximation and for the operations with these series
25.
Communicate mathematics both orally and in well-written sentences
26.
Model a written description of a physical situation with a function, a differential equation, or an integral
27.
Use technology to help solve problems, experiment, interpret results, and verify conclusions
28.
Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement
TIME FRAME Note: Calculus BC meets 7 periods a week. Unit
Periods
I.
Prerequisites for Calculus
3
II.
Limits and Continuity
7
III.
Derivatives
20
IV.
Applications for Derivatives
15
V.
Integration
20
VI.
Application of Definite Integrals
15
VII.
Differential Equations
10
VIII.
Application of Differential Equations
15
IX.
Advanced Integration
12
X.
Sequence and Series of Constants
10
XI.
Infinite Power Series
15
XII.
Polar Coordinates
7
XIII.
Vectors
10
XIV.
Practice for AP Exam
20 179
Total
CURRICULAR REQUIREMENTS [CR 1] The teacher has read the most recent AP Calculus Course Description. [CR 2] The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description. [CR 3] The course provides students with the opportunity to work with functions represented in a variety of ways—graphically, numerically, analytically, and verbally— and emphasizes the connections among these representations. [CR 4] The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences
* Please note, Calculus problems inherently require and practice all the New Jersey Core Curriculum Content Standards for Mathematics. Instead of noting each standard for each section, this curriculum guide lists the criteria of the College Board for Advanced Placement Calculus BC courses.
COURSE OUTLINE
[CR 1], [CR 2]
I.
PREREQUISITES FOR CALCULUS [CR 4], [CR 5] A. Coordinates and Graphs in the Plane B. Slope and Equations for Lines C. Relations, Functions and Their Graphs 1. Support symmetry graphically, numerically and algebraically 2. Properties of functions from properties of graphs 3. Domain and range of a function from a sketch of the graph. D. Geometric Transformations: Shifts, Reflections, Stretches, and Shrinks E. Solving Equations and Inequalities Graphically F. Relations, Functions and Their Inverses G. A Review of Trigonometric Functions
II.
LIMITS AND CONTINUITY [CR 3], [CR 4], [CR 5] A. Limits 1. Find limits algebraically 2. Estimate limits graphically and numerically. Use limit notation correctly. B. Continuous Functions 1. Intermediate Value Theorem 2. Extreme Value Theorem C. The Sandwich Theorem and (sin Ø)/Ø D. Limits Involving Infinity 1. Find horizontal and vertical asymptotes using limits. 2. Describe asymptotes in terms of graphs and limits. E. Controlling Function Outputs: Target Values
III.
DERIVATIVES [CR 3], [CR 4], [CR 5] A. Slopes, Tangent Lines and Derivatives 1. Explain and derive formula for f’(x). 2. Explain relationship between differentiability and continuity. B. Numerical Derivatives 1. Estimate derivatives of f at points from a graph. 2. Estimate derivatives of f at points from a table of values. 3. Use numerical derivative capabilities of a calculator. C. Differentiation Rules D. Velocity, Speed, and Other Rates of Change 1. Explain difference between average and instantaneous rate of change. 2. Approximate rates of change from graphs and tables. 3. Sketch graph of f’(x) given the graph of f(x) E. Derivatives of Logarithmic and Exponential Functions F. Derivatives of Trigonometric Functions G. Derivatives of Inverse Trigonometric Functions
H. I. J. K.
IV.
APPLICATIONS OF DERIVATIVES [CR 3], [CR 4], [CR 5] A. Maxima, Minima, and the Mean Value Theorem 1. Explain how to find critical points and extreme values. 2. State and apply Mean Value Theorem. B. Predicting Hidden Behavior 1. Sketch a graph of f(x), given the characteristics of f ′ ( x ) and f ′′ ( x ) . C. D. E. F. G.
V.
The Chain Rule Implicit Differentiation and Fractional Powers Derivatives of Parametric Functions Linear Approximations and Differentials 1. Use tangent lines as a local linear approximation. 2. Illustrate graphically the difference between df and delta f. 3. Use df to approximate delta f
2. Find equation of f(x) given characteristics of f ′ ( x ) and f ′′ ( x ) . Polynomial Functions, Newton’s Methods, and Optimization 1. Develop and apply Newton’s method. 2. Use calculus to solve optimization problems Rational Functions and Economics Applications Radical and Transcendental Functions Related Rates of Change Rolle’s Theorem – Mean Value Theorem
INTEGRATION [CR 3], [CR 4], [CR 5] A. Integral as an Area 1. Rectangular Approximation Methods 2. Trapezoidal Method 3. Riemann Series B. The Fundamental Theorems of Calculus C Integrals 1. Polynomials 2. Antiderivatives 3. U Substitiuiton 4. Integrals of Trigonometric Functions 5. Integrals of Inverse Trigonometric Functions 6. Integrals of Logarithmic and Exponential Functions 7. By Parts/Tabular D. Initial Value Problems, and Mathematical Modeling
VI.
APPLICATION OF DEFINITE INTEGRALS [CR 3], [CR 4], [CR 5] A. Areas Between Curves B. Volumes of Solids of Revolution -- Disks and Washers C. Volumes Cylindrical Shells D. Volume by known cross section E. Rectilinear Motion F. Length of a Curve G. Area of a Surface of Revolution H. Work Problems
VII.
DIFFERENTIAL EQUATIONS [CR 3], [CR 4], [CR 5] A. Solving Variable Separable B Slopefields and Eulers Method
VIII.
APPLICATION OF DIFFERENTIAL EQUATIONS [CR 3], [CR 4], [CR 5] A. Area as an Accumulation Function B. Exponential Change C. Logistic Growth D. Rectilinear Motion
IX.
TECHNIQUES OF ADVANCED INTEGRATION [CR 3], [CR 4], [CR 5] A. Powers of Trigonometric Functions B. Trigonometric Substitutions C. Rational Functions and Partial Fractions D. Improper Integrals
X.
SEQUENCE AND SERIES [CR 3], [CR 4], [CR 5] A. Introduction and Definitions 1. Sequence - arithmetic, geometric 2. Series - arithmetic, geometric B. Limit Theorems C. Infinite Series 1. Geometric Series 2. Telescoping Series 3. Harmonic Series D. Test for Convergence 1. The necessary but not sufficient condition – nth term test. 2. Integral Tests 3. Comparison Test 4. Ratio Test 5. Limit Comparison Test 6. Alternating Series with Error 7. Alternating Series - conditional convergence
XI.
INFINITE POWER SERIES [CR 3], [CR 4], [CR 5] A. Definition of Radius of convergence, interval of convergence
B. C. D. E. F.
Power Series - Expansion of Functions Maclaurin and Taylor Series LaGrange Remainder Theorem Indeterminate Forms L’Hopital’s Rule Other Methods of Obtaining Series 1. Long Division 2. Differentiation 3. Integration 4. Substitution
XII.
POLAR COORDINATES [CR 3], [CR 4], [CR 5] A. Polar Coordinate System B. Graphing Polar Equations C. Areas Bounded by Polar Curves D Derivatives of Polar Functions
XIII.
VECTORS [CR 3], [CR 4], [CR 5] A. Definition of a Vector B. Comparison of Vector and Scalar Speed C Derivatives of Vector Functions D. Velocity and acceleration Vectors E. Using vectors to calculate distance
ASSESSMENT 1.
Quizzes
2.
Unit Tests
3.
Midterm Exam
4.
Final Exam
5.
Laboratory Activities
6.
Projects - Research
7.
Calculator
