Syllabus

BC Calculus Syllabus

Course Description:

AP Calculus BC is a demanding course that requires students to investigate ideas and use a variety of skills and concepts to solve problems. Analytic, graphic and numeric techniques are explored and students learn to recognize which method or methods are viable and should be 

used to find the solution. Explaining results and methods of solution is vital to demonstrating understanding and an answer could require a diagram, an algebraic-type proof, or a written explanation. Recognizing answers in a variety of forms is also essential to success in AP calculus. Graphing calculators will provide a powerful tool with which we will explore relationships.

Contact Information:

      J. Bixler     Room 302       jbixler@eanesisd.net    512-732-9280  ext 33412

Grading:

10% HW / 20% Quizzes / 70% Tests. Homework will be graded for completion/effort (legitimately trying to work each problem). Assignments 

are due at the beginning of each class.

Late Homework:

Students can turn in no more than 2 late homework assignments each grading period for a grade of 90, but they must be complete or they will remain a ONE in the grade book. Late assignments must be turned in the day BEFORE their accompanying test date.

Makeup Policy:

 YOU are responsible for knowing what you have missed and getting everything you need from Google Classroom. We will follow the standard school makeup policy – one day

 for each excused day absent. If your first day absent is on a test/quiz day, you are expected to make it up the first day you are back on campus. Assessments taken late will receive NO partial credit as you have had longer to study. If absent more than one day you are expected to do the following:

(1) Watch any lesson videos you miss on Google Classroom and do any accompanying worksheets that will also be on GC,

(2) Turn absent work in as soon as you return.

(3) Utilize homework help videos and email me if you’re having trouble getting something or understanding something. 

Dropped Grades:

At the end of each grading period, I will drop the lowest quiz grade and two homework grades if there are NO missing assignments. Late homework will go into the grade book

as a 90. As mentioned previously, you may have only two 90’s. If you have more than two at the end of 

the grading period, the others will be turned back into a 1 but will count as “turned in” for the optional replacement test.

Optional Replacement Test at End of Each Grading Period:

If the student has no more than 3 tardies, all absences have been cleared, and ALL assignments have been turned in, they will have the option of taking a comprehensive quarterly replacement test that can replace one eligible low test grade.

Test Security: 

The Westlake Math Department policy is not to release tests, and students may not make copies, written or digital, of any exam material. Students will have the opportunity to review their completed exams during class or in other teacher-supervised settings. 

Tutorials:

Daily before school or by appointment.

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Course Outline:

First Semester

Chapter P: Preparation for Calculus 

·  Pre-Calculus review

Chapter 1: Limits and Their Properties 

·       An introduction to limits, including an intuitive understanding of the limit process and the formal definition for limits of functions

·       Using graphs and tables of data to determine limits of functions and sequences

·       Properties of limits

·       Algebraic techniques for evaluating limits of functions

·       Comparing relative magnitudes of functions and their rates of change (comparing exponential growth, polynomial growth 

         and logarithmic growth)

·       Continuity and one-sided limits

·       Geometric understanding of the graphs of continuous functions

·       Intermediate Value Theorem

·       Infinite limits

·       Understanding asymptotes in terms of graphical behavior

·       Using limits to find the asymptotes of a function, vertical and horizontal

Chapter 2: Differentiation 

·       Tangent line to a curve and local linearity approximation

·       Understanding of the derivative: graphically, numerically and analytically

·       Approximating rates of change from graph and table of data

·       The derivative as: the limit of the difference quotient, the slope of a curve at a point and interpreted as an instantaneous rate of change

·       The meaning of the derivative—translating verbal descriptions into equations and vice versa

·       The relationship between differentiability and continuity

·       Functions that have a vertical tangent at a point and points at which there are no tangents

·       Instantaneous rate of change as the limit of average rate of change

·       Approximate rate of change from graphs and table of values

·       Differentiation rules for basic functions, including power functions and trigonometric functions

·       Rules of differentiation for sums, differences, products and quotients

·       The Chain rule

·       Implicit differentiation

·       Related rates, modeling rates of change

Chapter 3: Applications of Differentiation 

·  Extrema on an interval and the Extreme Value Theorem

·  Rolle’s Theorem and the Mean Value Theorem and their geometric consequences

·  Increasing and decreasing functions and the First Derivative Test

·  Concavity and points of inflections and the relationship with the 2nd derivative

·  Points of inflection as places where concavity changes

·  Second Derivative Test

·  Limits at infinity

·  Summary of graphing techniques, analysis of curves, including the notions of monotonicity and concavity

·  Relating the graphs of f, f’, and f’’

·  Optimization including both relative and absolute extrema

·  Differentials, tangent line to a curve, linear approximations and Newton’s Method of approximating zeros

·  Application problems including position, velocity, acceleration, and rectilinear motion

Chapter 4: Integration 

·  Antiderivatives and indefinite integration, including antiderivatives following directly from derivatives of basic functions

·  Basic properties of the definite integral

·  Area under a curve

·  Meaning of the definite integral

·  Definite integral as a limit of Riemann sums

·  Riemann sums, including left, right and midpoint sums

·  Use of the First Fundamental Theorem of Calculus to evaluate definite integrals

·  Use of the Fundamental Theorem of Calculus to represent a particular antiderivative, and the analytical and graphical analysis

         of functions so defined

·  The Second Fundamental Theorem of Calculus and functions defined by integrals

·  Use of substitution of variables to evaluate definite integrals

·  Integration by substitution

·  The Mean Value Theorem for Integrals and the average value of a function

·  Trapezoidal sums

·  Use of Riemann sums and trapezoidal sums to approximate definite integrals of functions that are represented analytically,         graphically and by tables of data

Chapter 5: Log, Exponential, and Other Transcendental Functions 

·  Define the natural logarithmic function as a definite integral

·  The natural logarithmic function: differentiation and integration

·  Inverse functions and the use of implicit differentiation to find the derivative of an inverse function

·  Exponential functions: differentiation and integration

·  Bases other than e and applications

·  Solving separable differential equations

·  Applications of differential equations in modeling, including exponential growth

·  Use of slope fields to interpret a differential equation geometrically

·  Drawing slope fields and solution curves for differential equations

·  Inverse trig functions and differentiation

·  Integrals yielding inverse trig functions

Second Semester

Chapter 6: Applications of Integration 

·  The integral as an accumulator of rates of change

·  Area between 2 curves

·  Volume of solids of revolution by disc, washer and shell method

·  Volume of solids of know cross sections

·  Arc length and surface area

·  Problems from past AP tests involving setting up an approximating Riemann sum and representing its limit as a definite

         integral

·  Average value of a function

·  Applications of integration in problems involving a particle moving along a line, including the use of the definite integral with

        an initial condition and using the definite integral to find the distance traveled by a particle along a line

Chapter 7: Integration Techniques, L’Hopital, and Improper Integrals 

·  Review of basic integration rules

·  Integration by parts

·  Trigonometric integrals

·  Trigonometric substitution

·  L’Hopital’s Rule and its use in determining limits

·  Indeterminate forms :   

·  Relative rates of growth

·  Improper integrals and their convergence and divergence

Chapter 8: Infinite Series

·  Convergence and divergence of sequences

·  Definition of a series as a  sequence of partial sums

·  Convergence of series defined in terms of the limit of the sequence of partial sums of a series

·  Geometric series and its applications

·  The nth Term Test for Divergence

·  The Integral Test to prove the convergence or divergences of the p-series (harmonic series as the p-series)

·  Error approximation for the integral test

·  Comparisons of series using both Direct and Limit Comparison Tests

·  Alternating Series and the Alternating Series Remainder

·  The Ratio and Root Tests

·  Taylor Polynomials and approximations: use the graphing calculator to view different Taylor Polynomials for sin x and cos x

·  Powers series and radius and interval of convergence

·  Representation of functions by power series

·  Taylor and Maclaurin series for a given function

·  Taylor and Maclaurin series for sin x, cos x, , and

·  Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, integration,

         addition of series, multiplication of series by a constant and /or a variable and the formation of new series from known series

·  Taylor’s Theorem with the Lagrange Form of the Remainder (Lagrange Error Bound)

Chapter 9: Plane Curves, Parametric Equation, and Polar Curves; Special topics: Numerical solution of differential equations using Euler’s Method, Limits of Sums as definite integrals 

·  Plane curves and parametric equations and calculus

·  Parametric equations and vectors: motion along a curve, position, velocity, acceleration, speed and distance traveled

·  Analysis of curves given in parametric and vector form

·  Polar coordinates and polar graphs

·  Area of region bounded by polar curves

·  Numerical solution of differential equations using Euler’s Method

·  Limits of infinite sums as definite integrals

Review and Practice for AP Tests until Test date