South Miami Senior High



Advanced Placement Calculus BC Syllabus

Ms.Lemes

Room 288

305-532-4515 ext.2288

jlemes@

website:

Textbook: CALCULUS; Sullivan, Miranda; Bedford, Freeman & Worth, 2017.

Supplementary Material:

1. If applicable – textbook, CD, software, study guide, etc.

2. CALCULUS; Larson, Hostetler, Edwards; Houghton Mifflin Company,2002.

3.CALCULUS, GRAPHICAL, NUMERICAL, ALGEBRAIC; Finney, Demana, Waits, Kennedy; Prentice Hall, 2003. (On website)

4. Labs and Handouts obtained from speakers at summer institutes held by College Board.

5. Review Materials for the AP Test: Released copies of previous exams obtained from AP Central; BARRON’S How to Prepare for the AP Calculus, Shirley O. Hockett and David Bock, 7th. Edition; Multiple Choice & Free Response Questions in Preparation for the AP Calculus (AB) Examination, David Lederman, D& S Marketing Systems, 7th. Edition.

Prerequisites: Before studying calculus students should be familiar with linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric and piecewise defined functions. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions.

Course Overview: This course consists of a full high school academic year of work that is comparable to calculus courses in colleges and universities. It is expected that students who take this course will seek college credit, college placement, or both. This course explores the “mathematics of change”. We will examine, analyze, and describe the precise way in which changes in one variable relate to changes in another. The goal of this class is to have students

• Work with and understand the connections among functions represented graphically, numerically, analytically, and/or verbally (C3)

• Understand the meaning of the derivative in terms of a rate of change and local linear approximation (C3)

• Use graphs of 1st. and 2nd. Derivatives to approximate the graph of f(x) and likewise use the graph of f(x) to approximate the graphs of the corresponding derivatives(C3)

• Use graphs of 1st. and 2nd. Derivatives to approximate the graph of f(x) and likewise use the graph of f(x) to approximate the graphs of the corresponding derivatives(C3)

• Understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of a rate of change (C3)

• Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus (C3)

• Communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems (C4)

• Model a written description of a physical situation with a function, a differential equation, or an integral (C3)

• Use graphing calculators to help solve problems, interpret results, and verify conclusions (C5)

• Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measure and be able to explain solutions orally with adequate mathematical terminology (C4)

• Emphasize that students explain and justify their verbal and written responses and therefore demonstrate an understanding of the subject matter (C4)

• Students are encouraged to voice their concerns and problems with peers or previous calculus students familiar with the topics, either in study groups or on a one to one basis (C4)

• Develop an understanding of convergence and divergence of series based on partial sums and terms of a series as areas of rectangles; relationship to the Integral Test (C3)

• Develop an understanding of the Calculus of Polar, Parametric and Vector Valued Functions by using an algebraic, graphic and numerical approach (C3)

• Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment (C2)

Graphing Calculator Policy: Each student is required to have a graphing calculator (TI-84) for home use and in class. Graphing calculators are used daily to explore, discover, and reinforce the concepts of calculus. Calculator programs are uploaded and used throughout the year. Calculator use is restricted on some examinations and on certain AP essays. With the use of the calculator, students are encouraged to sit in groups and explore and discover as much as possible by seeing how functions behave without a traditional didactic approach. When sitting in groups they are encouraged to act as facilitators to help each other in the learning process. More specifically, calculators are used to explain the topic of local linearity, limits, connections between f(x), f’(x) and f”(x) when exploring relative maximums and minimums and concavity, investigating sequences and partial sums numerically and graphically.

• Using the trace and zoom features of the graphing calculator it is easy to see that most functions become linear when viewed close in; in addition, students write linear equations that approximate the zoomed in graph and then zoom out to see that it closely resembles the tangent to the curve at a specific point (C3 C5)

• With the use of tables where delta x is very small, students are able to find the limiting value of the function as x approaches a specific value such as zero; and likewise by letting delta x be very large they are able to explore if the function has a limiting value as x approaches infinity (C5)

• By entering f(x), f’(x) and f”(x) as y1, y2 and y3 respectively, the student is able to visualize on the table where relative maximums and minimums occur by identifying where f’(x) changes signs and likewise finding changes in concavity where f”(x) changes signs; also by graphing it is possible to detect that the maximum or minimums in f(x) occur where f’(x) has zeros and changes in concavity in f(x) take place where f”(x) has zeros ( C3 C5)

• Generate several terms of a sequence with the sequence command; by the fifteenth term if the sequence converges it will approach a value of zero; then use a table to represent the sequence and see the value approach zero (C5)

• Finding partial sums of a series on the calculator using the summation symbol is helpful in understanding the behavior of the infinite series (C5)

Assignments: There will be an assignment every time the class meets. An assignment may be problems from the textbook or AP resources, a worksheet, or a project, or a laboratory. There will also be assignments online from Khan Academy, to better prepare you for the AP Test. Homework is not accepted more than one class late. Late homework will be assigned a highest possible grade of 80%. With an admit, the student may turn in missed assignments up to one week late (if it’s been more than a week, it will not be accepted, so don’t ask!). Class projects are usually assigned for applications such as optimization, related rate or volume problems of the type encountered in the free response section of the AP test. Labs are usually group activities done during class time.

LabsAssigned:

1. Calculator lab to develop an understanding of the capabilities of the calculator and the syntax

2. Limit lab that involves finding the same limit in three different ways: graphical, numeric (by use of the calculator) and algebraically

3. Lab on graphing the first and second derivatives of a function from the graph of f(x) without the use of the calculator or vice versa

4. Lab on graphing the definite integral by thinking of the integral as an accumulator of area under the curve

5. Lab on problems relating to the Fundamental Theorem of Calculus and limits that exemplify the definition of the derivative

6. Lab on analyzing the function’s relative maximums and minimums, points of inflection and concavity with the use of the calculator

Projects Assigned:

1. Optimization problems obtained from different sources that relate to more real life situations and involve minimizing cost and maximizing profit

2. Related Rate problems obtained from different sources requiring greater critical thinking skills and a more in depth analysis

3. Volume project where students create a 3-D model representing a self created solid of revolution or other type of solid developed from an accumulation of some geometric figure such as an isosceles triangle; the problem is then solved geometrically and graphically

4. Concepts Worksheet on the Taylor Series which connects the approximated calculator value obtained for a function such as ln x at a particular x with the value obtained by using the Taylor Polynomial; understanding how error decreases with as degree of series increases

Assessments: Chapter quizzes are assigned to take home as a test review (but it is not the only source of test review!) . A test is given at the end of each chapter. IMPORTANT: if a student is absent on the day of the test, the admit must be brought in the following class in order to make up the test, which will be a more difficult version (if you don’t want to take the more difficult version, take the test on time or before your anticipated absence!). No admit = zero grade for your test. Daily homework is checked. Free response and multiple choice questions are randomly used from released exams to test certain topics more comprehensively throughout the year. The questions are graded similarly to how they are graded on the AP exam. Approximately a month before the exam students work on AP review tests, such as the BARRON’S, Acorn booklet, D&S book or other released exams and are graded in the AP style to give students an indication of how they will perform on the actual AP test.

Grading Policy: Grades are determined by:

Tests/Quizzes 70% (Test weight: 3, Quiz weight:1)

Notebook 10%

Classwork/Homework 20%

100%

*Grading Category/Weight for Projects and Labs will be decided at the time they are assigned.

Grading Scales:

A = 100% - 90% A = 4.00 – 3.50

B = 89% - 80% B = 3.49 – 2.50

C = 79% - 70 % C = 2.49 – 1.50

D = 69% - 60 % D = 1.49 – 1.00

F = 59 % - 0 % F = 0.99 – 0.00

Bellworks:

At the beginning of every class there will be Bellwork on the board. It is your responsibility to: come in each class, sit down, take out your homework, and start working on the Bellwork as I take attendance. In your notebook, the Bellworks section must be separate from the notes section!

Hall Passes:

No passes will be given during an exam. Otherwise, grab the pass and step out without disturbing the class. No passes will be given the first or last 10 minutes of class.

Course Materials:

* Notebook or binder * Pen, pencil *Calculator (TI-84, or TI-nspire)

Topics and Timeline:

I. Prerequisites of calculus and review of graphing calculator (2 hours)

A. Lines and Functions

B. Functions (inverse, greatest integer, piecewise, odd, even)

C. Trigonometric Functions

D. Topic Review – Function transformations (shifts, dilations, f[x] vs. [f(x)], etc.)

III Limits and Continuity (5 hours)

A. Rates of change and limits, including one-sided limits, removable discontinuities, jump discontinuities (1 hour)

B. Limits involving infinity, including sandwich theorem proof of

lim as x → 0 of sin x / x , horizontal asymptotes, end behavior models (1 hour)

C. Continuity and Differentiable -> continuous theorem (1 hour)

D. Rates of change and tangent lines, average velocity (1 hour)

E. Topic Review – emphasizing graphical and analytical computation of limits (1 hour)

IV Derivatives (7 hours)

Derivative of a Function – definition and derivation of polynomial, rational, and radical functions

A. Differentiability and local linearity, introduction of corners and cusps, vertical tangents

B. Derivation and application of rules for differentiation

C. Instantaneous velocity, acceleration, and other rates of change with an introduction to analyzing graphs of the derivative

D. Derivatives of trigonometric functions

E. Chain Rule for composite functions

F. Implicit Differentiation

Sections on differentiation of exponential, logarithmic, and inverse trigonometric functions addressed later.

H. L’Hopital’s Rule and why it works, to use with taking limits

I. Relative rates of growth, to help in finding limits as x → infinity

J. Topic Review – emphasis on analyzing derivative graphs and sketching

f ‘(x) from the graph of f(x)

V. Applications of Derivatives (11 hours)

A. Finding global and local extrema, analyzing critical points (1 hour)

B. Rolle’s Theorem and the Mean Value Theorem (1 hour)

C. Connecting f ’ and f ” with the graph of f(x), first derivative test, concavity, increasing and decreasing functions (2 hours)

D. Modeling and optimization (2 hours)

E. Linearization (1 hour)

F. Related rates (2 hours)

G. Topic Review – emphasis on graph analysis, applications problems are covered thoroughly through weekly assignments (1 hour)

VI. The Definite Integral (5 hours)

A. Estimating with finite sums: LRAM, RRAM, MRAM, and random

points using graphing calculator and hand computations (1 hour)

B. Definite integrals (1 hour)

C. Definite integral rules, antiderivatives, and average value of a

function (1 hour)

D. Fundamental Theorem of Calculus and area under the curve

(1 hour)

E. Trapezoidal Rule (1 hour)

VII. Differential Equations and Mathematical Modeling (4 hours)

A. Antiderivatives and initial conditions (text) and Slope Fields

(handout from AP Central) (1 hours)

B. Integration by substitution (2 hours)

C. Topic Review – antiderivatives only (1 hours)

VIII. Applications of Definite Integrals (10 hours)

A. Integral as net change (2 hours)

B. Areas between two curves (2 hours)

C. Volume: disk, washer, cross-sections in that order (6 hours)

Exponential and Logarithmic Functions (7 hours)

A. Derivatives of exponential and logarithmic functions, including logarithmic differentiation (2 hours)

B. Handouts on applications of exponential growth and decay (2 hours)

C. Selected applications problems (1 hour)

D. Population growth: logistic growth, carrying capacity, more slope fields. (1 hour)

E. Differential Equations from past AP exams (1 hour)

Techniques of Integration (19 hours)

A. Integration by parts (1 hour)

B. Tabular integration by parts (1 hour)

C. Integrating powers of trigonometric functions (2 hours)

D. Integration by trigonometric substitution (3 hours)

E. Integration by partial fractions including Heaviside method (4 hours)

F. Integrals of inverse trigonometric functions (2 hours)

G. Improper integrals (2 hours)

H. Mixed integration techniques-identify the method (2 hours)

I.. Topic Test (2 hours)

Review and administer midterm exam (7 hours)

Power Series (27 hours)

A. Introduction to power series exploration (1 hour)

B. Geometric sequences and series (2 hours)

C. Mathematical models of geometric series and sequences (3 hours)

D. Power series for an exponential function (1 hour)

E. Power series for other elementary functions (3 hours)

F. Taylor and Maclaurin series and operations on these series in closed and open forms (5 hours)

G. Interval of convergence for a series-Ratio method and root test (3 hours)

H. Nth term test for divergence, direct comparison test including p-series (2 hours)

I. Integral test, limit comparison test, absolute and conditional convergences including telescoping series (3 hours)

J. Remainder estimation: alternating series, Lagrange form, geometric series (2 hours)

K. Topic Review (2 hours)

Polar coordinates (6 hours)

A. Review of polar coordinates handout (2 hours)

B. Lengths and areas by polar coordinates (3 hours)

C. Topic Review (1 hour)

Parametric and Vector Functions (6 hours)

A. Parametric Functions (2 hours)

B. Vector Functions and Speed (3 hours)

C. Topic Review (2 hours)

Review for AP Exam (15 hours)

AP Exam and aftermath (15 hours)

A. Projects assigned

Student presentations of AP essays after the 48 hour wait

Review and administer final exam (5 hours)

Class Rules + Expectations:

• Always be respectful, prompt, and prepared!

• No phones in class!

• Ask lots of questions! (even if you think they are dumb)

• ALWAYS check the class website for HW and announcements!

I, ____________________________, Period:_____, have read the entire syllabus and agree to comply with Ms.Lemes’ classroom rules and expectations.

Student’s Signature: ___________________________________ Date:_______________

Parent’s Signature: ____________________________________ Date:_______________

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