BTWHSPVA MATH DEPT
Instuctors: Sue Cutrer and Niloo jalilvand
Advanced Placement Calculus AB Syllabus
This is an Advanced Placement Course and every student is expected to take the AP Exam. Students are also preparing for their next mathematics course.
We want students to understand the calculus in the following four ways:
Numerically, Algebraically, Graphically, and Verbally.
Chapter 1: Prerequisites for Calculus
Chapter 2: Limits and Continuity
Chapter 3: Derivatives
Chapter 4: Applications of Derivatives
Chapter 5: The Definite Integral
Chapter 6: Differential Equations and Mathematical Modeling
Chapter 7: Applications of Definite Integrals
Chapter 8: L'Hôpital's Rule, Improper Integrals, and Partial Fractions
Chapter 9: Infinite Series Chapter 10: Parametric, Vector, and Polar Functions
Local Linearity – we explore various functions and discover that by zooming in, the function may appear to be a straight line. We can find the slope of a line with two points and we chose points that are very close together. Students work in pairs to find the slope of the segment connecting the two points. Each pair is responsible for finding the slope at several points on a given function. We explore the various functions as each pair reports on their findings. We refer back to this activity often in our later studies. We use this activity to review concepts needed for calculus.
Our students use the TI-89 and some also use the TI-83/84. Calculators are used often to explore and discover. We emphasize the four capabilities that students are allowed to use on the exam – plot the graph of a function within an arbitrary viewing window, find the zeros of functions, numerically calculate the derivative of a function, and numerically calculate the value of a definite integral. We use other features of the graphing calculator in our explorations.
Assessments are given both with and without the graphing calculator. Limits – an intuitive understanding, using algebra, and estimating limits from graphs and tables of values. We use the graphing calculator to find limits on the graph screen and from the table.
(TEST 1) Continuity – definition of continuity, one-sided limits, Intermediate Value Theorem. Infinite Limits – Vertical Asymptotes, finding vertical asymptotes and determining infinite limits.
(TEST 2) The Derivative and the Tangent Line Problem – students are reminded about the opening activity, local linearity. We define the derivative of a function and find some derivatives using this method. We want students to recognize the definition of the derivative (limit of the difference quotient) and the alternate form. This exploration allows us to discover some of the rules for differentiation that we will study. We compare this work to the slopes we found when we used local linearity. Differentiability and Continuity – vertical tangent lines and points that have no tangent line, instantaneous rate of change, graphical, numerical, and analytical Differentiation.
(TEST 3) Rules for Basic Differentiation: The Constant Rule, The Power Rule, The Constant Multiple Rule, The Sum and Difference Rules, Sine and Cosine, and Rates of Change. (SIX WEEKS TEST , AP style questions, both multiple choice and free response) Rules for Differentiation, continued: The Product Rule, The Quotient Rule, Derivatives of Trig Functions, Higher-Order Derivatives. The Chain Rule Derivatives of natural log functions and exponential functions.
(TEST 1) Implicit Differentiation Related Rates
(TEST 2) Extrema, Relative Extrema, Critical Numbers Rolle’s Theorem and Mean Value Theorem. The First Derivative Test, increasing and decreasing functions
(SIX WEEKS TEST) Concavity, Points of Inflection, The Second Derivative Test. This lesson concludes with a matching game. Students match graphs of functions with their derivatives. Also descriptions of the function and derivative are matched with the graphs.
Students verbalize what they are looking for and how they know they have a match. This is a very rich conversation filled with talk such as “The first derivative is positive, so the function must be increasing over this interval.” Derivative as a rate of change – position, velocity, acceleration, and the question of speed. Connecting in tables and graphs. Limit as x approaches infinity, horizontal asymptotes, comparing relative rates of growth to find limits. Curve Sketching without the graphing calculator. Many of our students will attend universities that do not allow the graphing calculator in the next course they take. We work on increasing our power to sketch the functions without the calculator. We use the first and second derivative to make accurate sketches. We emphasize the rational functions and improve our ability to make a sketch so that we identify intervals of increasing/decreasing and concave up/concave down quickly.
(TEST 1) Optimization Linear Approximations
(TEST 2) Antiderivatives and Indefinite Integration – general solutions and particular solutions. (SIX WEEKS TEST) The Semester Exam is given at the end of the Third Six Weeks. It is a shorter version of an AP EXAM. Students have two hours for this exam. Question format includes multiple choice and free response with and without the graphing calculator. Slope fields – students explore several differential equations and draw the slope fields by hand. They learn characteristics of various differential equations. Students identify the slope field by the differential equation that created it and by the function that it could represent. We combine the graphical approach with the algebraic approach, when possible, and compare our results. Area and Definite Integrals – Riemann Sums, left, right, and mid-point The Mean Value Theorem for Integrals and the Average Value of a Function
(TEST 1) The Fundamental Theorem of Calculus, I and II. Integration by Substitution – Pattern Recognition, Change of Variables for Definite Integrals, Integration of Odd and Even Functions. (TEST 2) Trapezoid Approach for finding Area Total Distance Traveled
(TEST 3) Inverse Functions – Derivative of an Inverse Function. Integration of natural log and exponential functions
(SIX WEEKS TEST) Differential Equations: Growth and Decay Models
(TEST 1) Inverse Trig Functions - Derivatives of inverse Trig Functions Integrals Involving Inverse Trig Functions.
(TEST 2) Area between curves. Volumes of solids – Disc, Washer, and Known Cross Sections – The first day of this section is known as Candy Day. We anticipate it for days. As we revolve the various functions around the x-axis, we see known candy shapes. When we see a shape, everyone in the class gets a piece of candy – kisses, peanut butter cups, eggs. This helps us visualize the 3 dimensions and the circular cross sections we need for problem solving. We also use play-doh as a base on poster board and we place shapes of the known-cross sections in the play-doh so that they stand up perpendicular to the appropriate axis. We also use Calculus in Motion to help students find the volumes and PowerPoint demonstrations.
(SIX WEEKS TEST) Review for the AP Exam – Our major assessments have been AP style. We continue to work on Multiple Choice Questions and Free Response Questions. We used released AP Tests. Sequences and Series (After the exam)