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AP Calculus BC Mr. Joyce Course Expectations

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					                                       AP Calculus BC
                                          Mr. Joyce
                                      Course Expectations

1)   Text/Outline: The text we will be using is CALCULUS, Concepts and Applications, Paul A.
     Foerster; 2005 edition. The text is available online by accessing www.keymath.com and entering
     our “class pass”. The course outline for the year parallels the AP course requirements, as well as
     delving further into application and concepts.


2)   Web-site: www.westonk12-ct.org/page.cfm?p=1101
     e-mail: kevinjoyce@westonps.org
     Web Resource: www.interactmath.com

     The web-site should be utilized for current assignments and handouts, as well as useful links. If
     you miss a class, the web-site should be referenced for the current assignment, and the appropriate
     web links or text book should be used for examples and additional practice.


3)   Calculators: The course is designed to integrate the use of the graphing calculator, and one of the
     goals of the course is the development of the techniques for using such a calculator. Test and
     quizzes may include a calculator as well as non-calculator sections (similar to the AP exam).
     While calculators will be provided in the classroom (TI-83), it would be beneficial for and
     recommended that you own a graphing calculator. Classroom instruction will be based on the
     using the TI-83 (84) calculator. Graphing calculators will be permitted and expected on a major
     portion of the AP exam; although there are two of four sections where calculators are not
     permitted.


4)   Notebooks: Your notebook will be crucial to your success in this course. A three-ring binder is
     the best choice, as I will be using supplemental materials throughout the course. All lecture notes,
     activities, homework, supplemental materials, graded assignments and tests should be in this
     notebook.
5)   Homework: The problem sets in this course will be developed from a variety of sources, in
     addition to those in the text. I will go over conceptual questions you have about the homework
     during class, but I expect that you will have given every problem your best effort. All work is
     each problem must be shown; a list of answers is not acceptable. You can expect to spend an
     equal (or greater) amount of time outside of a class to that spent in class – that is – you should
     plan on a minimum of 55 minutes of homework each day. NOTE: Due to the amount of material
     to be covered prior to the AP examination in early May, class time will not always be used to
     check homework answers.



6)   Grades:        Daily Homework and Participation – 5%
                    Tests, quizzes, AP problem sets – 95%
     The number and types of these assignments will vary, depending on the quarter, schedule, etc.
     You will be told the comparative weight of each as we progress through the course.

     Each quarter grade counts 20% of the final course grade. The mid-term and final examinations
     count 10% each. In keeping with the design of the course, all examination will be cumulative.
     Examinations will be designed similar to the AP Examination – part Multiple Choice and part
     Free Response (half without a calculator, half with).



7)   Attendance: Your prompt attendance to class is critical to your success as a student of calculus
     and to the success of the class as a group. Unexcused tardiness or absence may result in lost credit
     for the day’s assignment, and may also reduce the credit earned in the course. Even excused
     absences can have a negative effect on your mastery of this material. Please try to arrange
     appointments after school.



8)   Getting Help: Calculus is the crowning jewel of any high school mathematics curriculum, and all
     of you are very well prepared for your study of calculus. But there will be times this year
     especially in the beginning, when you may find yourself a bit overwhelmed by this material.
     When this happens, please ask for help. Also, any and every member of the Mathematics
     Department will be delighted to discuss a calculus problem with you. We strongly suggest that
     you “preview” your homework before you go home in the afternoon, so that you can get help
     while you are in school and have teacher available.




9)   The AP Examination: In the first or second week of May, the College Board offers an
     examination in calculus, success on which can earn a student advances placement or college
     credit. A major goal is your preparation for this exam. We will concentrate on the syllabus of the
     exam at the BC level.
AP Calculus BC Course Outline
Unit 1: Limits and Continuity (1-2 weeks)

Overview: Students will evaluate limits numerically, graphically, and analytically for both one sided and
two sided limits. The strict definition of a Limit will be explored and illustrated using the TI-83 and
adjusting the window in terms of delta and epsilon, as well as using the table feature. (Delta-epsilon
proofs will be revisited post-AP exam.) Analytic methods will include substitution and algebraic
techniques. Properties of limits and theorems, as well as unbounded behaviors, will be used in evaluating
limits. Continuity will be defined and explored using domain and limits. The properties of Continuous
Functions, as well as associated Theorems (IVT & EVT), will be used to determine continuity. There
should be both an intuitive and systematic approach in evaluating limits and determining continuity.
Applications will include various types of functions (polynomial, rational, exponential, logarithmic,
trigonometric, inverse, composition, piecewise, etc.), rates of change, and tangent lines.
    • Average and instantaneous speed
    • Definition of a limit (informal and delta-epsilon)
                                                             x
    • One-sided and two-sided limits (and a look at y = )
                                                             x
    • Computing limits as x → a ; numerically, graphically, and algebraically
    • Value of a limit, indeterminate form, and “the limit does not exist”
    • Properties of limits as x → a
                                                    sin x
    • The Squeeze (Sandwich) Theorem and lim
                                               x →0   x
    • Horizontal Asymptotes, vertical asymptotes, and infinite limits
    • Properties of Limits as x → ∞
    • Relative rates of growth and dominant functions (i.e. 2 x vs. x 2 as x → ∞ )
    • Definition of continuity
    • Types of discontinuities (hole, jump, asymptote) - removable
    • Continuous functions and intervals
    • Properties of continuous functions and related theorems
    • Intermediate Value Theorem and it’s consequences
    • Average rate of change and secant slopes
    • Slope of a curve at a point and tangent (and normal) lines
    • Free fall investigation

Unit 2: The Derivative (2-3 weeks)

Overview: Students will develop a working conceptual and procedural knowledge of derivatives and rules
associated with derivatives. The various notations for derivatives will be presented with explicit meaning
for each. Emphasis will be placed on relating graphs, equations, tables, and real world applications. The
question of differentiability will be explored and related to local linearity – zooming in using the TI-83 –
                                                     1
comparing corners, cusps, vertical tangents (i.e. x 3 at x = 0) and discontinuities (jumps, holes, and
vertical asymptotes) both graphically and through the definition of the derivative. The graph of y = x
and its derivative will be thoroughly examined numerically, analytically, and graphically. Students will
also rewrite absolute value functions as piecewise functions in their explorations. Question 4c from the
1986 AB exam is explored graphically by using a list of values from L1 and L2 to satisfy the conditions
of continuity: while also exploring the graphical solution (zooming in at x = 1) that meets the analytic
solution for differentiability. Both the first and second derivatives will be examined in terms of the
characteristics of a function – increasing, decreasing, and concavity (as the rate of the rate of growth).
Curve sketching will include increasing & decreasing intervals, local and absolute maximums &
minimums, points of inflection, discontinuities, and concavity, as well as any initial conditions (including
properties of functions – such as odd, even, etc.). Collaborative and cooperative group work will focus on
application problems (simple harmonic motion, economics) and presentations should include strategies
for the initial set-up as well as corresponding solutions.
    • Definition of the derivative for a function and a point
    • Applications of the derivative definition and approximations
    • Differentiability and local linearity – basic tangent line problems
    • Differentiability and continuity
    • When a function is not differentiable at a point – corners, cusps, and discontinuities
    • One-sided derivatives and continuous functions on a defined interval
    • Interpretations of the derivative and notations, and verbal translations (equation ↔ verbal)
    • The second derivative and basic curve sketching
    • Curve sketching from graphs, equations, tables, characteristics and justification
    • Rules for differentiation – constant, power, constant multiple, sum and difference - (polynomials)
    • Exponential function and its derivative
    • Product and quotient rules
    • Chain Rule
    • Trigonometric Functions and their derivatives
                                                    d
    • Applications of the chain rule – deriving        (arctan x) and more
                                                    dx
    • Implicit differentiation
    • All the rules together


   BC additional topics
   • Simple harmonic motion
   • Derivatives in economics – revenue, demand, cost, profit, marginals




Unit 3 – Applications of the Derivative (2-3 weeks)

Overview: Students will apply their knowledge of the derivative and differentiable functions to further
analyze curves and make approximations. A considerable amount of time is to be spent solving various
application problems, in particular, optimization and related rates. These problems will require
recollection of the accumulated theorems and properties from prior units. Connections should be well
established among all topics before moving on to Integral Calculus. Group work will include
presentations of various parametric problem sets with discussions on the initial set-up, alternative
approaches, and corresponding reasoning.

   •   Linear approximation and the derivative – tangent line approximations and error
                                                                             sin x
   •   Local linearity to find limits – L’Hopital’s rule – revisit lim             and other cases of the
                                                                        x →0   x
       indeterminate form
   •   L’Hopital’s Rule for indeterminate cases
   •   Newton’s Method
   •   The Mean Value Theorem and Rolle’s Theorem
   •   Return to curve sketching – using the first and second derivatives to interpret and solve application
       problems
   •   The Extreme Value Theorem
   •   Particle problems (motion along a line) and other applications
   •   Optimization and modeling
   •   Related Rates

   BC additional topics
   • One dimensional parametric equations, graphic interpretation, and vectors
   • Two dimensional motion – position, velocity & speed, and acceleration – vector notation
   • Slopes of parameterized curves and chain rule
   • Plots of particle motion and path characteristics

Unit 4 – The Definite Integral (2-3 weeks)

Overview: The student will develop conceptual understanding and make a well founded connection
between area and the Fundamental Theorem of Calculus. They will first examine the connection between
the area under the curve of a velocity graph and displacement, as well as distance traveled. Units should
be included and referred to in all arguments. Approximations will include left hand, right hand, and
midpoint Riemann sums, as well as the trapezoid method. Error should be explained using both area and
formulas. Sigma notation will be used in defining the definite integral. Geometric techniques
(recognizing semi-circles, etc) will be examined and the various integral properties involving sign
examined. The Fundamental Theorem of Calculus will also be thoroughly explored as an “accumulator
                          b
function”, F (b) = F (a ) + ∫ f ( x)dx , with the “idea” of where you are now is a result of where you were
                          a
plus how much you have accumulated. The Mean Value Theorem for Integrals will also be modeled
graphically with emphasis on units (average value vs. average rate of change) to aid in distinguishing how
it differs from the Mean Value Theorem introduced with derivatives. Applications must refer to a breadth
of examples that require use of tables, graphs, equations, and verbal explanations, as well as connections
among them.
     • Measuring distance traveled – velocity graphs and equations
     • Approximation techniques – Riemann sums (LH, RH, Mdpt.) and Trapezoid method (include
         Simpson’s if time permits – if not, revisit later in the year)
     • The Definite Integral
     • Interpretations of the definite integral
     • Geometric interpretations of the definite integral (semi-circles, triangles, etc.)
     • Properties of Limits of Integration – the role of sign
     • Definite integrals and antiderivatives – a connection to differential calculus
     • Comparison of definite integrals
     • The Mean Value Theorem for Integrals and applications/models
     • The Fundamental Theorem of Calculus and the definite integral as an accumulator
                                                                           x
   •   The Fundamental Theorem of Calculus and graphing                ∫a
                                                                               f (t )dt
   •   Using tables, graphs, and equations as they relate to the definite integral and applications

Unit 5 – Differential Equations and Mathematic Modeling (3-4 weeks)

Overview: The student will learn various methods to integrate for both indefinite and definite integrals.
They will construct and solve antiderivatives using various techniques including: “sight recognition”,
change of variable (u-substitution), integration by parts, and trigonometric substitutions. The student will
solve differential equations graphically, numerically, and analytically. Differential equations will be used
to model radioactive material, population growth, and other applications. Slope fields will be used to
model curves for both separable and non-separable differential equations. The importance of the initial
value and its effect on the curve will be modeled. The technique for separation of variables will be
derived and applied to various applications. An emphasis will be placed on population models, including
exponentials and logistics. Domain restrictions and uniqueness of solutions will require various examples
and a return to the phrase, “a function y that satisfies y = f (x) ”. Students will work in groups to solve
various logistics problems (including method of partial fractions) and present their results and any
shortcuts discovered. They will also work in groups to create an assessment (both problems and
solutions) that covers all material from this unit – they will defend their choices.

   •   Antiderivatives and slope fields – graphically and numerically
   •   Constructing antiderivatives analytically (a family of curves – don’t forget C)
   •   Model the general antiderivatives using slope field program (return to slope fields in Unit 7)
                            b
   •   Derivation for   ∫
                        a
                                x 2 dx and the rule for integrating x n , n ≠ −1
                                                                                     1
   •   “Sight recognition” for antiderivatives with a closer look at               ∫ x dx
   •   Properties of antiderivatives: sums and constant multiples
   •   Antiderivatives and differential equations (basic – applications to motion and initial value)
   •   Integration by change of variable – u-substitution – derive method and identify key components to
       look for when applying this technique – consider the change of limits u(a) & u(b)
   •   Integration by parts - derive method and identify key components to look for when applying this
       technique – LIPET for choosing “u”
   •   Tabular integration
   •   Brief look at basic trigonometric substitutions
   •   Another look at approximation techniques in context
   •   Differential equations to model change
   •   Constructing and using slope fields
   •   Uniqueness of solution for differential equation
   •   Defining the domain (consider function that satisfies hyperbola equation)
   •   Separation of variables derived and applied
   •   Growth and decay models
   •   More applications (Newton’s law of cooling) and modeling
   •   Logistics models and basic partial fraction
   •   Euler’s method for approximation

   BC additional (and/or “ deeper”) topics
   • Slope fields
   •   Separable differential equations
   •   Partial Fractions
   •   Integral Tables
   •   Trigonometric substitutions
   •   Exponential growth and decay – Newton’s Law of cooling
   •   Population modeling – logistics growth model, logistics regression
   •   Euler’s Method

Unit 6 – Applications of Integrations (4 weeks)

Overview: The student will solve integration problems involving the area between curves – in terms of x
or y. The will build upon this idea and “slicing” to solve volume problems. They will develop techniques
for solving general volume problems and then derive and model the methods for disks and washers.
Again, this should be done in both terms of x and y with justification for the choice of method. The shell
method will also be derived and modeled in terms of x and y. Students will also derive various formulas
for volumes of regions of known cross-sections. Browser based demos will be utilized as well as physical
models for both. Students should have the opportunity to create a model(s) for any of the aforementioned.
A considerable amount of time is to be spent exploring improper integrals and the implications of infinite
discontinuities. Students should develop a “feel” for convergence and divergence for both infinite limits
and infinite discontinuities. Naturally, analytic methods and the application of limits (and L’Hopital’s
Rule) will serve as initial justification. Students will work in groups and make presentations on their
findings from exploring p-series, and the methods of Direct and Limit Comparison. Group work will also
include applications for infinite solids and applications of derived techniques.

   •   Area between curves – in terms of x or y
   •   Finding volume – a general approach – slices that are ∆x or ∆y thick
   •   Discs and washer method derived and applied – in terms of x or y
   •   Shell method derived and applied – in terms of x or y
   •   Both methods – the better choice with justification – be aware of limits
   •   Volume of known cross sections – square, right isosceles, equilateral, semi-circle, any
   •   Use of table to approximate volume (MRI of tumor, etc.)
   •   “Superset” – area between two curves used to create/model each method

   BC additional (and/or “ deeper”) topics
   • Lengths of curves – sine wave, smooth curves, vertical tangents, cusps, & corners
   • Science application – Work, fluid force & pressure, normal probabilities
   • Revisit L’Hopital’s Rule and the indeterminate forms
   • Improper Integrals – infinite limits and infinite discontinuities
                            ∞ 1            1 1
   • P-series; examining ∫ p dx and ∫ p dx ; a closer look using calculator features for fnInt, tables,
                           1 x            0 x

      and graphs, and of course - limits
   • Direct Comparison
   • Limit Comparison
   • Volumes of infinite solids and other applications

Unit 7 – Infinite Series (5 weeks)
Overview: The student will solve problems involving infinite series – examining converge, divergence,
approximations, and other connections to functions and calculus. A return to geometric series and it’s
graphical implications will serve as the introduction. A connection is drawn from infinite geometric
series and it’s sum – this is done numerically, algebraically, and graphically. Values explored should also
serve as an introduction to radius of convergence. The connection is carried through to the definition
Power Series and it’s center. Transformations, differentiation, and integration techniques are then applied
to these concepts. Linear approximation will serve as the lead into deriving Taylor Polynomials. Various
degrees of Taylor Polynomials will be explored graphically and serves as discussion pieces for radius of
convergence and error. MacLaurin and Taylor Series will be generalized and defined using sigma
notation. The various techniques/tests for determining convergence of a series, radius of converge, and
related topics, are thoroughly explored and compares to each other. Students will work together to
explain their reasoning for choosing their particular approach in determining convergence of a series (i.e.
integral test vs. limit comparison) and develop an algorithm. Beyond this, students will explore
alternating series (and it’s test for convergence), convergence at endpoints, and errors in approximation.
Throughout this unit, students will be required to revisit and explain their reasoning for solutions and
methods chosen – they will be drawn back to the basis for this unit – infinite geometric series &
polynomial approximations with the application of transformations, differentiation, and integration. Due
to the nature of this topic, the connections among numerically, analytically, and graphically supporting
solutions cannot be overemphasized.

   •   Geometric Sequences Series – infinite sums, convergence, and divergence
   •   Representation of a function by a series – getting better with more terms
   •   Applying transformations, differentiation, and integration to find a power series
   •   Identifying series
   •   Deriving and constructing a Taylor Series
                                                             1     1
   •   Representing various functions by a power series:         ,    ,sin x, cos x, e x , ln(1 + x), tan −1 x
                                                           1− x 1+ x
   •   Using a series to find another series
   •   Taylor Theorem with Remainder and the Remainder Estimation Theorem, Lagrange Error
                                 πi
   •   Euler’s Formula and e + 1 = 0
   •   Convergence and the nth term test
   •   The Direct Comparison Test
   •   Absolute Convergence and Convergence – comparing nonnegative series
   •   The Ratio Test - convergence and radius
   •   Endpoint convergence and comparing techniques – ratio, telescoping, and more
   •   The Integral Test and why it works – a return to Riemann Sums
   •   Harmonic series and p-series – recognizing, proving, and referencing
   •   Limit Comparison Tests – cleaning up an “ugly” function
   •   Alternating Series Test – what’s required? – and the estimation theorem
   •   Absolute and conditional Convergence
   •   Finding Intervals of Convergence

Unit 8 – Parametric & Polar Functions (1-2 weeks)

Overview: The student will solve problems involving parametric and polar functions. Students should
already be well versed in these functions (from the Pre-Calculus curriculum as well). Time is spent
examining the calculus involved and it’s affect on curve characteristics as well as applications of
parametric & polar functions. This is a second look at parametric equations as they were covered in Unit
3. The focus of polar equations is on graphing techniques, rewriting equations, differentiation, area, and
limits of integration.
    • Parametric functions and derivatives
                              d2y
    • Parametric formulas -        and length of a smooth curve
                              dx 2
    • Investigating Cycloids
    • Surface Area from a smooth parameterized curve
    • Polar coordinates and graphs
    • Relating polar and Cartesian coordinates
    • Some “familiar” polar graphs
                                    dy
    • Polar functions and slope –
                                    dx ( r ,ϑ )
   •   Deriving a formula for area using polar functions
   •   Area between polar curves
   •   Length of a Polar Curve
   •   Area of a surface of revolution



   Practice Exam

   More AP Problem sets

   AP EXAM


Unit 8 – Post AP topics

Overview: The student will revisit topics from the year and further delve into application problems.
From here they will either move on to topics in multivariable or differential calculus – or they may
complete a project that includes creating a calculus booklet with a presentation.

				
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