Unit 1A: Review of Integration, Differentiation and the ...



Name:September 6, 2016AP Calculus BC Syllabus and OverviewMr. T. GuntherMission StatementOur course is designed to prepare our students with the knowledge of univariate calculus they will require for careers in the sciences, engineering, medicine or business. Our students upon completion will have a broad base of tools to analyze change, compute volumes, and examine infinite sums. In addition, our students will be able to better communicate mathematical concepts pictorially, graphically, verbally, and in writing.Course OverviewStudents taking our AP Calculus BC course are expected to complete a full-year of AP Calculus AB. During Advanced Math and the Calculus AB course, students should have mastered the basics of graphing and analyzing the behavior of functions, computing derivatives and utilizing the concept of a derivative as a rate of change, applying derivatives to compute local and global maxima, computing Riemann sums and approximating the area underneath a curve, and computing antiderivatives and utilizing the concept of accumulated change.Important topics applied during our Calculus BC course include applying the Fundamental Theorem of Calculus and identifying and applying accumulation functions, using the Intermediate, Mean, and Extreme Value Theorems, approximating, creating, or solving functions from rates of change through differential equations, computing work, force, and volume of objects through integration, computing limits and comparing the relative rates of growth of functions, determining the convergence of infinite sum and writing arbitrary functions as an infinite sum (Taylor series), expressing curves in polar coordinates, andusing position, velocity, and acceleration vectors to describe the 2-D motion of particles.Students will have access to TI-84, TI-89, or equivalent calculators, the use of which is an integral (pun intended!) part of the course. Students who do not have their own graphing calculators will be given a TI-84 to sign out and use for the school year. Students during the course will have the opportunity to use the calculators to visualize curves created using Cartesian, rectangular, and polar coordinates. Calculators are also used regularly to numerically compute approximations for derivatives and integrals for both analytical functions and functions represented using tables. In addition, Microsoft Excel is used for “lab experiments,” e.g., to apply Euler’s Method for estimating the results of various differential equations for population, including exponential and logistic growth.AssessmentAssessment comes in a variety of forms. Assigned problems are checked for completeness and graded 10 points each once each week – generally after double-periods. Tests at the end of each unit are graded out of 300 points. Quarterly multiple choice quizzes will also be given to assess student learning. Lab “experiments” are graded from 25-100 points and involve giving students a “comfort factor” with concepts before being presented more formally. The labs also require students to communicate calculus concepts, thus facilitating greater understanding. Papers will be assigned roughly once each marking period, and are used to encourage students to both use technology and writing skills to express mathematical concepts algebraically, graphically, and “linguistically.” Papers will be graded out of 200 points. In addition, a research project in an area of mathematics decided by the student will be assigned, with both a paper and oral presentation based on the research to be completed. A midterm and final will each be worth 300 points — the midterm will be a multiple choice test of problems from a selected AP Exam, while the Final will be an essay on each student’s mathematical secondary education.Course OutlineUnit 1A: Review of Integration, Differentiation and the “Big 3” Value Theorems (Sept.)Review of basic concepts – derivative as rate of change, integral as area under the curve.Review of computation and analytical methods for derivatives and antiderivatives.Review of graphical analysis of curves – derivatives as slopes of tangent line, relationship between second derivative and concavity.Review of the “Big 3” Value theorems – Intermediate, Extreme, and Mean Value Theorems Essays: Extreme Value Theorem ApplicationAfter several example problems, students will first write an essay in groups of two applying the EVT to the choice of where to place a secondary line of pipe going to a secondary tank in order to minimize the pressure of the system according to Poiseuille’s law. Students will then apply the EVT to a unique problem of their own choosing individually.Unit 1B: Accumulation Functions and Fundamental Theorem of Calculus (Late Sept.-Early October)Creating accumulation functions.Graphing accumulation functions and finding extreme values.Relationship between derivative of accumulation function and original function – Fundamental Theorem of CalculusExperiment: Graphing Accumulation FunctionsStudents first graph accumulation functions by estimating areas underneath graphed functions using pencil and paper. Results are compared to graphs generated by a graphing calculator using numerical integration. Extreme values of the accumulation function are compared to roots of the original function until students are convinced of the Fundamental Theorem of Calculus – the derivative of the accumulation function is just the original function. Students are then required to explain this relationship orally and communicate the observation in writing.Unit 2: Differential Equations (Oct.)What is a Differential Equation?Creating and Interpreting Slope Fields for Differential EquationsSolving Simple Differential Equations by Anti-differentiationFinding Antiderivatives through SubstitutionFinding Antiderivatives through Integration by Parts (IBP)Finding Antiderivatives through Repeated IBP – Tabular Integration by PartsSolving Separable Differential EquationsApplications of Differential Equations – radioactive decay, population modeling (including exponential and logistic growth), Newton’s Law of CoolingNumerical Approximations for Differential Equations – Euler’s MethodExperiment: Applying Euler’s Method – Excel LabStudents use an Excel spreadsheet to repeatedly perform the equations necessary to apply Euler’s Method to give a tabular estimate of a Differential Equation. Examples are then related to population models described by differential equations, including logistic growth.Unit 3: Spatial Applications of Definite Integrals (Nov.)Interpretations of Definite Integral – net change, marginal cost, population and population density“Tabular” Integration – estimating net change from tabular data through Riemann sumsComputing Areas between CurvesUnit 3: Spatial Applications of Definite Integrals (Nov.) — cont.Interpreting Areas between Curves as an AccumulationVolumes with Geometric Cross-SectionsApplying Cavalieri’s TheoremComputing Volumes of Solids of Rotation using Disks/WashersComputing Volumes of Solids of Rotation using Shells (“Dog Food Can Label method”)Computing Lengths of CurvesExperiment: Estimating Volume of a Light Bulb by Riemann SumsStudents trace light bulbs or similar solids of rotation of various shapes. The traced light bulb is then “sliced” to represent roughly disk-shaped cross-sections whose volume can be computed as the volume of the cylinder. After this Riemann Sum is computed, the light bulb is submerged in a graduated cylinder to check the estimate of its volume. Students are then required to explain in writing the process by which they have estimated the volume and to explain what errors in estimation can occur.Essay: “The Great Escape”Students will investigate the filling of a conical tank with water. Using methods from solids of rotation, students will find a functional form for the amount of water in the tank, and for the rate at which the height of the water is increasing, along with deciding an important variable for the individual filling the tank.Unit 4: “Tricks” with Differentiation and Integration (Dec.)Using L’Hopital’s Rule and differentiation to find paring the Growth Rate of Functions using O( ) and o( ) notationComputing Antiderivatives of Rational Functions using the Method of Partial FractionsComputing Antiderivatives of Radical Expressions using Trigonometric SubstitutionComputing Antiderivatives using Inverse Trigonometric FunctionsComputing Antiderivatives of Powers of Trigonometric FunctionsMidterm Preparation and Exam (Jan.)Unit 5: Infinite Series & Polynomial Representations of F’ns (Jan.-Feb.)Geometric Series, Proof of Convergence and Properties of Geometric SeriesPower Series Constructed from Geometric Series by Differentiation and IntegrationApproximating Functions with Polynomials Using Derivatives – Taylor PolynomialsApproximating Functions with Infinite Degree Polynomials – Taylor SeriesUsing the Ratio Convergence Test – Series that are “like” Geometric SeriesUsing Other Convergence Tests for SeriesConvergence of Power Series – Radius of Convergence and Convergence of Endpoints.Experiment: Geometric Series Calculator LabStudents use their calculators in line entry mode to create a program that will compute the partial sums of a geometric or other series in order to investigate its convergence.Unit 6: Parametric, Vector, and Polar Functions (Feb.-Mar.)Graphing, Finding Slopes of Tangent Lines, and Path Lengths for Objects with Paths Described ParametricallyModeling Projectile MotionGraphing, etc. for Objects with Paths Described Using Polar CoordinatesVectors and Vector Valued FunctionsUnit 6: Parametric, Vector, and Polar Functions (Feb.-Mar.) — cont.Experiment: “Blammo the Human Cannonball” (from Anton,)Students use parametric equations to study ballistics and determine the position of a net and cannon angle necessary to propel “Blammo the Human Cannonball” safely into a netting without hitting the roof of a circus big top. Students must describe in writing and defend orally the assumptions they have made, the equations that they choose to use in order to calculate Blammo’s trajectory, and, critically, the parameters for the firing angle and placement of nets to ensure the safety of Blammo’s flight.Experiment: “Finding the Length of a Cycloid”Students follow a bug that has been run over by a tire to derive the parametric equations for a cycloid. Students then use the arc length formula along with graphing calculators to find the distance that our poor insect friend has traveled.Unit 7: Applications of Def. Integrals from Science and Statistics (Mar.-Apr.)Improper Integrals – Convergence and ComputationUsing the Normal Probability Distribution FunctionComputing Work when Force or Distance is VariableComputing Fluid Force and Pressure on SurfacesExperiment: Die Rolls, Normal Probability and the Central Limit TheoremStudents roll a set of results using various dice (d4, d6, d8, d10, and d12) and various “statistics” are computed such as “sum of all dice,” “range,” “product of high and low.” TI-83 Calculators are used to compute the mean and standard deviation. The normal probability density is input into the TI-83 and the estimated percentage of each sample that should fall (if the sample were normally distributed) between two chosen values is computed. These are compared with the experimental results to find which result (usually “sum of all dice”) gives the most “normal” data.The month of April will be dedicated to to taking practice multiple choice and free response problems from prior AP Exams, andpreparing for the University of Scranton Integration Bee. After spending some time working previous exam questions, we will have a mock BC exam using previous year problems in late April. The final exam is a special essay, and will be gven in June during the regularly scheduled final period.AP Calculus BC Exam for 2017 is Tuesday, May 9th – the day after Bio/Physics .Reference SourcesMain Textbook:Finney, Ross, Franklin D. Demana, Bert K. Waits, and Daniel Kennedy. Calculus: Graphical, Numerical, and Algebraic. Upper Saddle River, NJ: Pearson/Prentice-Hall 2003.Other ResourcesAnton, Howard, Calculus: Brief Edition, A New Horizon. Sixth Edition. New York: John Wiley & Sons, Inc.White, Michael. Advanced Placement Summer Institute. Wilkes University, 2004. ................
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