AP Calculus BC Course: AP Calculus BC Course Overview: This course is taught over a school year (2 semesters). During the first semester on a 90 minute mod, students cover everything in the Calculus AP topic AB/BC outline as it appears in the course planner below. All AB material is covered in the first semester. The second semester is reserved for the completion of BC material and AP review. Graphing Calculator skills are reviewed in the beginning of the course and emphasized in every section thereafter. In all sections of the course students are assessed with regular quizzes as well as previous AP free response items. These AP items are particularly important as they encourage students to produce solutions both verbally and in written sentences. Basic Teaching Strategies: Students are taught to understand the concepts graphically, numerically and analytically. They are epected to relate the three concepts throughout the course. Students are taught in large group settings and small group settings. In both environments, students are encouraged to verbally present solutions and guide each other. Students are invited to attend review sessions in March, April and May up to the AP test day. These sessions focus on practice tests including both multiple choice and free response items. Primary Technology: A TI-89 is strongly recommended for use in this course. Many students still use TI-83+ and TI-84+ calculators. An overhead panel and TI-SmartView are used for both types of calculator (the TI-89 SmartView should be available early net year). It is important that students are able to use a calculator to guide in accurate graphical, numeric and analytic responses Secondary Technology: Rick Parris, WinPlot, Peanut Software, Eeter, NH (latest version, Feb. 2007) This software is used for graphical presentations in class and for creating graphs for assessments and worksheets. Primary Tetbook: Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus of a Single Variable. Eighth Edition. Houghton Mifflin Company, New York, 2006.
Review Books and Other Resources: All AP Calc Free Response Items (1969 to present) -These encourage students to produce solutions both verbally and in written sentences. -To best solve AP free responses, students need to consider a combination of graphical, numeric and analytical responses. Foerster, Paul. Calculus Eplorations. Key Curriculum Press, Emeryville, Ca. 1998. -The student uses the graphing calculator to help solve problems, eperiment, interpret results and support conclusions. -These eercises require the student to communicate mathematics both verbally and in written sentences. Solow, Anita. Learning by Discovery: A Lab Manual for Calculus. The Mathematical Association of America, Washington, DC. 1999. -Students are given a model to make a mathematical lab report. These lab reports have students write brief eplanations and etensive paragraphs to draw conclusions. -Students use graphical, numeric and analytic responses to discover many Calculus concepts. Additional resources are also researched and incorporated into the course over the school year. Course Planner: Fall (2 Preparation for Calculus (1 week) Graphs and Models o A library of basic functions is reviewed. Students are taught to have a minds eye view of all common functions. n n 1 (,,,,ln( ), e,sin( ),cos ( ), tan( ), etc. ) Linear Models and Rates of Change o y m b and y y1 m 1 are emphasized for their importance in Calculus Functions Models to Data o This section is vital to review the use of function notation. II. Limits and Their Properties A Preview of Calculus o Slope as rate of change. o Area under a curve as distance traveled. Finding Limits Graphically and Numerically Evaluating Limits Analytically
sin( ) o lim 1 0 o Four indeterminate forms: 0,,0, are dealt with. 0 Continuity and One-Sided Limits. o Check for continuity at a point o The Intermediate Value Theorem is presented. Infinite Limits o Using limits and one-sided limits to identify asymptotic behavior (2 Differentiation In all sections students are relating f to the slope of the function. Finding a derivative is a step in answering the Tangent Line Problem ( 3 weeks) The Derivative and the Tangent Line Problem f ( ) f( ) o The difference quotient: f '( ) lim 0 o Alternatives for calculating f (c). f ( ) f( c) f '( c) lim c c o Use a tangent line to approimate function values. (local linearity) Basic Differentiation Rules and Rates of Change o Identify where a derivative eists and does not eist Product and Quotient Rules and Higher-Order Derivatives The Chain Rule Implicit Differentiation Related Rates. o This is a mini unit in itself. 3 days is devoted to this section alone. IV. Applications of Differentiation Etrema on an Interval Rolle s Theorem and the Mean Value Theorem Increasing and Decreasing Functions and the First Derivative Test. o Justify your answer many AP free response items are shown to students with this direction to show the importance of the First Derivative Test. Concavity and the Second Derivative Test o Points of Inflection o Testing for rel. etrema Limits and Infinity o This is the final piece in curve sketching. A Summary of Curve Sketching Optimization Problems o finding the best, fastest, cheapest, largest, or smallest is an important use of differential calculus.
Differentials o Local linear approimation is emphasized in this section. MIDTERM EXAMINATION: This test is very AP like (but needs to fit into 90 min). It contains about 30 M.C. and 3 Free Response Items. V. Integration ( 3 weeks) Antiderivatives and Indefinite Integration o Students see Indefinite Integrals as an opposite operation to Derivatives. Area o Concept of accumulation of area to find: distance travel, volumes of water spilled over a dam etc. o All different types of Sums are defined all at once: Left, Right, Midpoint, Inscribed, Lower, Upper. Also Trapezoid Rule Riemann Sums and Definite Integrals The Fundamental Theorem of Calculus 1 b o Average Value = ( ) b a f d, also presented a Integration by Substitution VI. Logarithmic, Eponential, and Other Transcendental Functions The Natural Logarithmic Function: Differentiation. o Logarithmic Differentiation is solid review of basic Log properties. The Natural Logarithmic Function: Integration Eponential Functions: Differentiation and Integration Bases Other than e and Applications VII. Applications of Integration Area of a Region Between Two Curves Volume: The Disk Method Volume: The Shell Method- if time allows Volume: Solid with known cross section VIII. Differential Equations and Calculus Methods Newton s Method Slope Fields and Euler s Method Differential Equations: Growth and Decay Separation of Variables and the Logistic Equations First-Order Linear Differential Equations
FINAL EXAMINATION: (for the first semester) This test is very AP like (but needs to fit into 90 min). It contains about 30 M.C. and 3 Free Response Items. Spring Note: Students have approimately 14 weeks at this time to prepare for the BC portion of the AP Calculus eam. IX. A Trig review and Calculus Inverse Trig Functions: Many Trig Inverse topics are reviewed in this section. o Rt Triangle Trig, Range of Inv. Functions, Basic Calculations and Properties Inverse Trigonometric Functions: Differentiation o All Si Trig Inv. Functions (Chain Rule is emphasized in this section) Inverse Trigonometric Functions: Integration du du du o,, 2 2 2 2 a 2 2 a u u u a X. IntegrationTechniques, L Hôpital s Rule, and Improper Integrals Basic Integration Rules o A comprehension review Integration by Parts Trigonometric Integrals Trigonometric Substitution Partial Fractions Indeterminate Forms and L Hôpital s Rule Improper Integrals XI. Conics, Parametric Equations, and Polar Coordinates I switch and do this section first. (chapter 10 and chapter 9) Conics and Calculus o A review of the standard form of the different conic sections Plane Curves and Parametric Equations o Graphing with parametric equations o Eliminating the Parameter Parametric Equations and Calculus 2 dy d y o Finding and (also what these values mean with parametric 2 d d equations) o Identifying: velocity vectors and magnitude (speed) o Identifying: acceleration vectors and acceleration
o Arc Length of a Parametric Equation Polar Coordinates and Polar Graphs o Plotting Points o Sketching Graphs. o Converting between Rectangular and Polar o Identifying Basic Polar Graphs: Lines, Spirals, Circles, Rose Curves, Limacons and Cardiods. (the period of these graphs is stressed where appropriate) Area and Arc Length in Polar Coordinates o The upper and lower bounds for each of these formulas is stressed. MIDTERM EXAMINATION: This test is very AP like (but needs to fit into 90 min). It contains about 28 M.C. and 2 Free Response Items. X. Infinite Series (4 weeks) Sequences Series and Convergence o Finite and infinite series summation formulas The Integral Test and p-series o Also the harmonic series Comparisons of Series Alternating Series o Alternating series test o Identifying error bound when possible The Ratio and Root Tests Taylor Approimations Power Series o Radius and Interval of convergence Representation of Functions by Power Series Taylor and Maclaurin Series o Maclaurin series functions for: e,sin,cos and 1 - Identify interval 1 and radius of convergence for each. o Show basic differential and integral formations for these series o Lagrange error bound for Taylor polynomials Throughout the course, assessment is based on a spiral review of Pre-calculus and Calculus. Students are given a weekly quiz