Syllabus for BC Calculus Course Overview My students enter BC Calculus form an Honors Precalculus course which is extremely rigorous and we have 90 minutes per day for 180 days, so our calculus course does much more than the standard BC Calculus course. Students will receive a strong foundation in the calculus with conceptual understanding and the ability to perform all necessary skills. Students will be able to work with technology and they will be able to determine when this use is appropriate. Course Planner Our primary textbook is Calculus: Graphical, Numerical, and Algebraic, Third Edition by Ross L Finney, Franklin D. Demana, Bert K. Watts, and Daniel Kennedy. The following is an approximation of the time that we spend on each topic. The schedule is adjusted to meet student needs and abilities. There will be a variety of assessments including tests, quizzes, and problem sets. We will allow about 2 weeks for review for the AP exam, at the end of the year, but there will be constant review using problem sets as we progress through the course. Teaching Strategies Much of the course is taught in an interactive method. The lessons are planned, but we go where the students lead. With this approach, I learn things about their reasoning that I would not learn otherwise. Students are encouraged to work together to explore ideas and solve problems. Students will be expected to write explanations, to show diagrams and graphs and to connect numerical and analytic approaches. There are formal groupings which I choose from time to time and informal study groups which the students determine. When students are assigned a problem or project in a formal group, they must discuss the assignment among themselves and come to a conclusion which represents the group as a whole. Once they are in agreement, they write a summary report which must explain their reasoning and support their conclusions. Students explore ideas using technology and some basic manipulatives. As an example of low tech exploration, I will explain how I introduce volumes. I have learned that volumes with known cross sections generalize easily, so I start there. Students are assigned to groups and given functions which define a base. The cross sections are described and students build these cross sections using clay (Actually I use plasticene.) The dimensions are written on the cross sections and then used to write a Riemann Sum and eventually an integral. This generalizes readily into the disk method, so I do not have to teach the disk method. However, I do use the clay to develop the shell method as they do not discover that by themselves. The calculator is used daily in class to experiment and interpret results. Students know that a series converges if and only if its sequence of partial sums converges. Therefore, I have written a very simple program which gives an accumulated sum for the terms of a series (the sequence of partial sums) Students are able to add these terms one at a time using this program or simply let the program run and see what happens. Before they use this program they will have added terms of a geometric series and found a general tem for the sequence of partial sums. One of the things that they will do with this program is to try different p-series and make conjectures as to which ones converge. This works quite well and they will be able to confirm their conjectures later when they
learn the integral test. We also use the calculator answers to test the reasonableness of answers. Another example of using the calculator to make conjectures is done when they put a function in the calculator and for the next function they put in the difference f ( x h) f ( x) quotient and try this for small values of h. A favorite exploration is to h make conjectures about the derivatives of f ( x) a. They will determine what happens for different values of a and decide that a value of e is the best option. x Limits (8 days) Syllabus We begin with a basic question, Can we guarantee that this function can be made as close to a given number as possible? From this question students will eventually develop a definition of a limit of a function. They have done limits of sequences in precalculus, so they are able to develop a definition. Limits by definition Show by epsilon-delta definition whether a given stated limit is correct Proofs of the limit theorems Show by using limit theorems whether a given number is the limit of a function Limits at infinity Limits of infinity Asymptotes Definition of continuity in epsilon-delta form and the theorems of continuity Extreme value theorem Intermediate value theorem Find examples of functions which are never continuous and functions which are continuous at only one point. Evaluate limits by changing the form algebraically (later we learn other methods) sinh Special limits such as lim and their proofs h0 h Derivative (11 days) The derivative is motivated as a rate of change using such topics as speed and velocity. Derivatives by definition Proofs of the derivative theorems (sums, products,...) Chain rule Tangent lines and their accuracy of approximation Implicit differentiation Higher derivatives Derivatives using the theorems Derivatives of trigonometric functions Increasing and decreasing functions Critical points, relative and absolute maximum and minimum Concavity and inflection points Application of the derivative (9 days)
Optimization Related rates Mean Value Theorem (Average rate of change and instantaneous rate of change) Applications to business and economics Newton s method Antiderivatives (guess and check) Curve sketching (Graph the derivative of a function when its graph is given. Differentials and linear approximations Show that any differentiable function is continuous Integrals (14 days) Integration is motivated by finding displacement using a velocity function. I make sure that some of the velocity functions have positive and negative values as I want the students to see a limit of a sum of products and not just an area under a curve. They should also see how units help them to determine which products are needed. With trial and error we write a formal definition of an integral. Write an integral as a limit of a sum Integration by definition Write a limit of a sum as an integral and show that different resulting integrals are equivalent Discover the properties of an integral from the definition Use properties of the integral to graph f ( x) t dt Write a justification of the property that any function defined as an integral is continuous. Use the definition of to set up an integral which can be used for an application such as area or volume. Fundamental theorem of calculus x 1 Application of the integral (12 days) Every application that we do begins with the definition of the integral. We do a large number of applications to let students see that all they need is to have a situation explained and apply the definition of an integral. Area Volume Arc length Surface area Average value of a function Torque Moment of Inertia Mass and center of mass centroids Fluid Pressure Business applications Work Improper integrals
Inverse functions (9 days) From the definition of an inverse of a function we develop the derivatives of exponential and logarithmic functions and inverse trig functions. Derivatives of exponential functions Derivatives of logarithmic functions Derivatives of the inverse trig functions (derivations and their use) Antiderivatives of functions that are in the form of a derivative of inverse trig functions Hyperbolic functions L Hopital s Theorem (proof and applications) Techniques of integration (8 days) Algebraic substitution Integration by parts Partial fractions Trigonometric integration Trigonometric substitution Differential equations (14 days) We develop the meaning of a differential equation. Students must understand what the solution of a differential equation is. We write differential equations to fit a situation described. Variable separable differential equations. First order linear differential equations Higher order differential equations with constant coefficients (homogeneous and nonhomogeneous) Homogeneous differential equations Partial derivatives and differentials Exact differential equations Slope fields Euler s method (technique and accuracy) End of First Semester Summary, organization and review of AP tests of AB Calculus (5 days) Infinite Sequences and Series (22 days) The series and its sequence of partial sums are connected. We write sequences of partial sums from series and series from sequences of partial sums. Most of the theorems we use are proved. Direct comparison test Limit comparison test Ratio test Root test Alternating series test and its accuracy of approximation
Absolute and conditional convergence Cauchy Condensation theorem Yasser S. Abu-Moustafa Theorem Integral test and its accuracy of approximation Taylor polynomial Taylor Series Lagrange error analysis Alternating series error analysis Alternate methods of producing power series Binomial expansion, long division, Geometric series,... Series from known series by differentiation, integration, substitution, addition, using identities,... Series solution of differential equations (Power series and approximating by Taylor polynomials) Parametric equations and polar coordinates (10 days) This is familiar to the students from their precalculus course, so there is only a cursory review and then the calculus is applied. Area in polar coordinates Arc length in polar coordinates Slopes and angles of intersection of curves given in polar coordinates Arc length of functions defined parametrically Area of regions defined with parametric functions of vector equations Volume of revolution for regions defined parametrically Surface area of revolutions of regions defined parametrically Vector Functions (5 days) Students know all of the vector properties from their precalculus, so only the calculus needs to be added. Derivatives and integrals of vector functions Velocity and acceleration Arc length Partial derivatives (14 days) Graphs of functions defined in space Intersections of the space curves and planes to find rate of change in the intersection Definition of partial derivatives Partial derivatives Tangent lines Tangent planes and linear approximations Quadratic surface approximations Saddle points Relative and absolute maximum and minimum points Surface area Multiple Integrals (10 days)
Cylindrical coordinates Spherical coordinates Double integrals Triple integrals Volumes using double and triple integrals Mass using double and triple integrals Center of mass and center of volume using multiple integrals Change of variable in multiple integrals Vector Calculus (overview) (10 days) We do not get this covered every year. Vector fields Line integrals Green s Theorem Stokes Theorem Organization and review for AP Exam (The time available) I try to provide two weeks of review to remind the students of the differences between some topics and let them work through a released exam. Samples of Evidence Students are regularly required to write up problem sets showing analytic work, tabular data, graphical representations and explanations. I have learned that students have trouble with tabular data, so I regularly give them problems presented with tabular data and also problems presented graphically. Calculators are used to experiment (See the section on teaching strategies.) The calculators are used to find numerical integration, Riemann Sums, create slope fields, draw solutions curves, illustrate that an inflection point is a point where the curve changes concavity (that the tangent crosses there).. Primary Textbook: Finney, Ross. Calculus: Graphical, Numerical, and Algebraic. 3 rd ed. Pearson Education, Inc., 2010.