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A.P. Calculus BC

Syllabus

Course Overview:

In my class, Calculus BC is the continuation of the exploration of students’ exposure to further study in higher level mathematics. It is my goal that the students succeed in their future mathematics courses because of a strong foundation and procedures, rather than the memorization of rules.

The following outline is a list of topics covered in this course. I have listed the approximate number of days it takes to usually cover the material in each section. Also, the order of the topics is not necessarily sequential. Past material is regularly reviewed to help build upon and enrich the students understanding of the concepts.

Course Outline:

|TOPICS/CONCEPTS |SECTIONS/CHAPTERS |DAYS |

| | | |

|1. Functions, Graphs, and Limits | |10 Days |

|A. Analysis of graphs |Throughout |Daily |

|B. Limits of functions (including one-sided limits) | | |

|Intuitive understanding of the limiting process |Chapters 1 and 2 | |

|Algebraic calculation of limits |Sections 2.1, 2.2, 2.3, & 2.5 | |

|Graphical or tabular estimation of limits |Chapter 1 and Section 2.2 | |

|C. Asymptotic and unbounded behavior | | |

|Asymptotes in terms of graphical behavior |Section 2.5 | |

|Limits involving infinity |Section 2.5 | |

|Comparison of relative magnitudes of functions and |Chapters 3 and 6 | |

|their rates of change. | | |

|D. Continuity as a property of functions | | |

|Intuitive understanding of continuity |Section 2.4 | |

|Continuity in terms of limits |Section 2.4 | |

|Intermediate Value Theorem and Extreme Value Theorem |Section 2.4 | |

|Parametric, polar, and vector functions. | | |

|Analysis of planar curves completed in parametric form, polar form, and vector form. | | |

| |Sections 4.7, 8.7, & 10.7 | |

|II. Derivatives | | |

|Concept of derivative | | |

|Derivatives presented graphically, numerically, and analytically. | | |

|Interpretation as instantaneous rate of change | |25 Days |

|Limit of the difference quotient |Sections 3.1, 3.2, 3.3, & 3.4 | |

|Relationship between differentiability and continuity |Sections 1.2, 3.5, and throughout | |

|Derivative at a point |Sections 3.2 & 3.4 | |

|Slope of a curve at a point |Section 4.6 | |

|Tangent line to a curve and local linearity | | |

|Instantaneous rate of change as limit of average rate of change |Sections 3.1 & 8.2 | |

|Approximate rate of change graphically and tabularly. |Section 3.2 | |

|Derivative of a function |Sections 1.2, 3.2, & 3.4 | |

|Characteristics of [pic] and [pic] | | |

|Relationship between increasing and decreasing of [pic] and the sign of [pic] |Sections 1.2 & 3.3 | |

|Mean Value Theorem (with geometric consequences) | | |

|Equations involving derivatives (interchangeable verbal and numerical) |Sections 3.3 & 8.2 | |

|Second derivatives | | |

|Characteristics of [pic], [pic], and [pic] |Sections 3.3 & 8.2 | |

|Relationship between concavity of [pic] and the sign of [pic] | | |

|Points of inflection |Section 5.5 | |

|Applications of derivatives |Sections 7.2 & 7.3 | |

|Analysis of curves (notion of monotonicity and concavity) | | |

|Analysis of planar curves given in parametric form, polar form, and vector form, | | |

|including velocity and acceleration. | | |

|Optimization (absolute (global) and relative (local) extrema) |Section 8.2 | |

|Related-rates problems | | |

|Implicit differentiation to find derivative of inverse function |Section 8.2 | |

|Relationship between and with velocity, speed, and acceleration | | |

|Geometric interpretation of differential equations (slope fields and solution curves for|Section 8.2 | |

|differential equations) | | |

|Numerical solution of differential equations using Euler’s method |Section 8.2 | |

|L’Hospital’s rule, including determining limits and convergence of improper integrals | | |

|and series. | | |

|Computation of derivatives |Sections 4.7, 8.7, & 10.7 | |

|Knowledge of basic derivative functions | | |

|Basic rules for derivatives (sums, products, quotients) | | |

|Chain rule and implicit differentiation |Sections 8.2, 8.3, & 10.5 | |

|Derivatives of parametric, polar, and vector functions | | |

| |Section 4.9 | |

|III. Integrals |Section 3.9 & 4.5 | |

|Properties of definite integrals | | |

|Computation of Riemann sums (left, right, midpoint) |Sections 1.2, 3.5, & 4.9 | |

|Definite integral as a rate of change of a quantity | | |

|Basic properties of definite integrals (additivity and linearity) | | |

|Applications of integrals |Sections 7.4 & 7.6 | |

|Area under a curve (including regions bounded by polar curves) | | |

|Volume of a solid with known cross sections |Section 7.5 | |

|Average value of a function | | |

|Distance traveled by a particle along a line |Section 6.5, 9.10, & 12.7 | |

|Length of the curve, including a curve given in parametric form | | |

|Fundamental Theorem of Calculus |Sections 3.4, 3.8, 3.9, 4.4, & 4.5 | |

|Evaluation of definite integrals using the Fundamental Theorem of Calculus |Sections 4.1, 4.2, & 4.3 | |

|Represent a particular antiderivative using the Fundamental Theorem of Calculus |Sections 3.7 & 4.8 | |

|Analytic and graphical analysis of functions defined by using the Fundamental Theorem of|Section 4.7, 8.7, & 10.7 | |

|Calculus | | |

|Techniques of antidifferentiation | | |

|Antiderivatives from derivatives of basic functions |Section 5.4 | |

|Antiderivatives by substitution of variables, parts, and simple partial fractions |Sections 5.4 & 5.6 |19 Days |

|Improper integrals |Sections 5.3 & 5.7 | |

|Applications of antidifferentiation | | |

|Using specific conditions to find specific antiderivatives including motion along a line| | |

|Solving separable differential equations and using them in modeling | | |

|Solving logistic differential equations and using them in modeling |Sections 5.8, 5.9, 8.5, 8.7 10.1, 10.2,| |

|Numerical approximations to definite integrals |& 10.3 | |

|Use Riemann sums and trapezoidal sums to approximate definite integrals of functions | | |

|algebraically, graphically, and tabularly. | | |

| | | |

|IV. Polynomial Approximations and Series | | |

|Concept of series. | | |

|Series defined as partial sums | | |

|Convergence defined in terms of the limit of the sequence of partial sums. |Sections 5.6 & 5.7 | |

|Series of constants | | |

|Motivating examples (decimal expansion included) |Sections 5.6 & 5.7 | |

|Geometric series with applications | | |

|The harmonic series | | |

|Alternating series (with error bound) | | |

|Terms of series as areas of rectangles as connected to improper integrals (including the| | |

|integral test and use in testing the convergence of the p-series. |Section 5.3 | |

|The ratio test (convergence/divergence) |Section 5.3 | |

|Comparing series to test for convergence/divergence | | |

|Taylor Series |Section 9.10 | |

|Taylor polynomial approximations (graphical demonstration of convergence) | | |

|Maclaurin series and the general Taylor series centered at [pic] |Sections 5.3, 5.8, 10.1, & 10.2 | |

|Maclaurin series for the functions: [pic], [pic], [pic], and [pic]. | | |

|Formal manipulation of Taylor series and shortcuts to computation including: |Sections 7.2 & 7.3 | |

|substitution, differentiation, antidifferentiation. | | |

|The formation of new series from previously known series |Section 7.6 | |

|Functions defined by power series | | |

|Radius and interval convergence of the power series | | |

|Lagrange error bound for Taylor polynomials | | |

| |Sections 1.3, 1.4, & 5.4 | |

| | | |

| | | |

| |Chapter 12 | |

| | | |

| | |16 Days |

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| | | |

| |Chapter 12 | |

| | | |

| |Section 12.2 | |

| |Section 12.7 | |

| |Section 12.7 | |

| |Section 12.7 | |

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| | | |

| |Section 12..6 | |

| |Section 12.7 | |

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| |Section 12.5 | |

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| |Section 12.5 | |

| | | |

| |Section 12.5 | |

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| | | |

| | | |

| |Section 12.5 | |

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| |Sections 12.3, 12.4 | |

| |Section 12.6 | |

| |Section 12.8 | |

Note: Our school is on the alternating day schedule, so every other day we have Calculus class. This schedule is designed to leave 1-2 weeks for exam review and exploration of a few topics outside of the Calculus BC curriculum. After the AP Exam, other calculus related topics are investigated and discussed.

Pedagogical:

As topics are discussed throughout this course, students work through a continual review of material from courses preceding calculus, rather than all at once at the beginning of the year. Utilizing the best practices in teaching, students are expected to work within different groupings and collaborative discussions. Students are also expected to investigate multiple representations of the above topics. For example, when exploring derivatives of functions, students are expected to

✓ sketch tangent lines and compare the graphs of [pic] and [pic]

✓ use information presented tabularly to approximate derivatives using the slope of the secant line

✓ use the derivative to find the slope of the tangent line, and

✓ communicate their results and the relationships that they found.

Students are expected to do the following throughout the year:

✓ Show all of their work

✓ Use complete sentences when explaining solutions either verbally or in writing

✓ Give the setup that is used on the calculator as well as the calculator result (example: when finding the area between curves, students must show the definite integral as well as the answer)

✓ Students are expected to show the mathematical steps that lead to their final solution on non-calculator work

✓ Justifications must be in complete sentences

Students are expected to complete a Calculus Journal throughout the year. The prompts for the journal are given in the text, as well as, some additional prompts provided by myself based on the types of learning I am seeing in class. Students are asked to justify all entries in complete sentences and in depth discussions.

Assessments:

Students are expected to demonstrate their knowledge on a variety of assessments throughout the year. During each class period I utilize formative assessments to determine where my students are on the concepts and procedures being discussed in class that day. I also utilize the summative quizzes and cumulative exams at the end of each unit. Each unit’s exam includes material from previous and is organized into a non-calculator and a calculator portion. It varies from exam to exam which part of the test is longer, but each exam includes portions of both types of problems. My students quickly figure out that even though it is on the calculator portion of the exam, it does not mean that the questions will be easier to do with a calculator!

Calculator Usage:

Students are required to use a graphing calculator and TI-83, TI-84, and TI-89 calculators are made available for the students to use. Students are required to perform the following tasks on graphing calculators:

✓ Graph a function within an arbitrary viewing window

✓ Find roots of functions

✓ Numerically calculate the derivative of a function

✓ Numerically calculate the value of the definite integral

Graphing calculators are used regularly to help investigate problems, experiment with situations, interpret results, solve problems, and support conclusions.

Textbook:

Each student has a copy of the primary textbook for this course.

Foerster, Paul A. Calculus: Concepts and Applications. Emeryville, CA: Key Curriculum Press. 2005. 2nd Edition.

Reference Textbook:

Finney, Ross L., Franklin Demana, Bert Waits, and Daniel Kennedy. Calculus: Graphical, Numerical, Algebraic.

Reading, Mass.: Addison-Wesley, 2007.

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