AP Calculus BC Mr. Hein
Course Syllabus
Course Objectives: This class will use the topics learned in Math Analysis 1-2. The three major topics of this course are derivatives, integrals, and series with an emphasis on developing a conceptual understanding of these topics and the ability to apply them to problem situations. We will be studying all topics graphically, numerically, analytically, and verbally and students will learn to make connections between these representations. We will use the ti-83 or ti-84 graphing calculator throughout the course. Each student is required to have his or her own graphing calculator. Student will learn how to do many things on their graphing calculator including: graphing a function in an arbitrary window, finding zeros and intersection points, finding derivatives at a point, graphing slopefields and approximating definite integrals using Riemann sums. One goal of this class is for all students to be prepared to succeed on the Advanced Placement Calculus BC exam.
Teaching Strategies
I promote excitement in my class for each topic that is coming up. I think it is really important for students to see that I enjoy the mathematics we are learning. I believe strongly that students need to see where things come from, and so we derive almost every formula we come across. I also expect students to be able to write proficiently about the mathematics they are learning. When there is an answer, I require an explanation about what that answer means, and/or where it came from. You can see this in assignments, test, and projects. Calculators are used throughout the course and students are allowed to use them during most tests and quizzes. Students are encouraged to use their calculator to support conclusions made whenever possible. For example, when students compute the derivative analytically we use the calculator to look at the two graphs and determine if our ‘derivative’ makes sense. We also use the calculator to check the derivative at a few points to make sure it is giving us the same value. In addition we will also use the calculator to experiment with possibilities. This is not done always, but typically when we are beginning the topic and as needed afterwards. In this same way students are taught to look at a problem in multiple ways at the same time. Topics are taught in such a way that students will learn how to see connections between graphs, functions, tables and verbal expressions. Students are asked to explain their answers verbally when giving a solution. Students will spend time working together to support each other’s learning inside and outside of class. Students are encouraged to form study groups. This will increase as the year goes on. I expect students to work together as they review for the AP test. Communication is a goal of this course. We meet at the food court at a grocery store to review for the AP test and the students get into groups at the tables.
Resources:
Finney, Ross L., Franklin Demana, Bert Waits, and Daniel Kennedy.
Calculus: Graphical, Numerical, Algebraic. Reading, Mass.: Addison-
Wesley, 2007.
Released Free-Response questions from previous years as found on the AP central website.
David Lederman, Multiple Choice and Free-Response Questions in Preparation for the AP Calculus (AB) Examination, 8th edition, 2004
Course Outline
Chapter 1: Prerequisites for Calculus (7 days)
· Elementary functions and their graphs
o Piecewise
o Linear, power, exponential/logarithmic,
o trigonometric/inverse trigonometric
o Parametric equations
· Getting familiar with the graphing calculator
o Graphing
o Windows, and zoom
o Solving equations
Chapter 2: Limits and Continuity (9 days)
· Limits:
o Limit at a point, limit at infinity, infinite limits
o Properties of limits
· Continuity
· Rate of change
· Tangent line to a curve
· Slope of a curve at a point
Chapter 3: Derivatives (14 days)
· Definition of f’
· Derivative at a point
· Relating the graphs of f and f’
· How f’ (a) fails to exist?
· Intermediate value theorem
· Rules for differentiation:
o Sum, product, quotient
o Chain Rule
· Velocity as rate of change
· Derivatives of trigonometric, inverse trigonometric, exponential and logarithmic functions.
· Implicit differentiation
Chapter 4: Applications of Derivatives (15 days)
· Critical Values
· Mean Value Theorem
· Using the derivative to find:
o Critical points and extreme values
o When the function is increasing or decreasing
o Inflection points
o When the function is concave up or concave down
· Modeling and optimization problems
· Linear approximation
· Newton’s Method
· Differentials
· Related rates
Chapter 5: The Definite Integral (12 days)
· Rectangular approximation methods
o RRAM, LRAM, MRAM
· Riemann sums and sigma notation
· Finding an antiderivative
· Average value of a function
· Using integrals and graphs to find area, volume, average value of a function
· Fundamental Theorem of Calculus
· Trapezoidal and Simpson’s Rule
· Error Analysis
Chapter 6: Differential Equations and Mathematical Modeling (14 days)
· Slope fields
· Antiderivatives and the indefinite integral
· Techniques of integration:
· Substitution, integration by parts, trigonometric substitution, partial fractions
· Separable differential equations
· Euler’s Method
· Exponential growth and decay
· Logistic growth
Chapter 7: Applications of Definite Integrals (12 days)
· Integral as net change
· Displacement vs. distance traveled
· Motion on a line
· Areas between curves
o Differentiate with respect to x and y.
· Using the integral to find volumes
o Cross sections
o Revolving around the x axis and y axis
o Disc method
o Washer method
o Shell method
o Length of a curve
Chapter 8: L’Hospital’s Rule, Improper Integrals, Partial Fractions (11 days)
· Indeterminate forms
· L’Hospital’s Rule
· Relative rates of growth
· Improper integrals
· Infinite discontinuity
· Partial fractions
Chapter 9: Infinite Series (14 days)
· Geometric series
· Power series:
· Term-by-term differentiation and integration to find power series of new functions
· Taylor’s series
· Maclaurin series
· Lagrange form of the remainder
· Error using calculator
· Tests for convergence/divergence:
o nth term test
o Direct Comparison
o Ratio Test
o Integral Test
o Limit Comparison Test
o Alternating Series Test (Leibniz’s Theorem)
· Radius and Interval of convergence
Chapter 10: Parametric, Vector, and Polar Functions (11 days)
· Parametric functions:
o Derivative at a point
o Length of a curve, surface area of a solid of revolution
· Vectors:
o Angle between vectors
o Scalar product
· Using vectors to describe motion in the plane
· Polar coordinates and pole graphs:
o Slope, horizontal and vertical tangent lines
o Area, length of a curve
AP Calculus Driving Distance Project name______
Due Date: December 1st
Project check off list.
q You can have groups of up to 4 but no more. Group selection is up to you.
q You need to drive some predetermined route that will take you more than five minutes.
q You must set your odometer to 0 or record your starting mileage.
q You must record your ending mileage when you are done.
q You will record your speed in MPH regularly. (I would choose every 10 seconds or something) But it’s your choice.
q You need to determine how far you traveled just using the speeds during each interval.
q This needs to be done using left and right rectangular approximation. You should have two different estimates. (show work.)
q You need to have a speed/time graph
q You need to have a drawing of the rectangles for each of the estimates.
q Answer the following questions in complete sentences as part of a minimum one paragraph summary to your project:
o How far did you travel according to your odometer?
o How far did you travel according to your estimates?
o Which estimate is the most accurate in this case?
o How could you have made your estimates for the distance traveled more accurate?
o How do you find the distance traveled when only given speeds?
q Present your project to the class displayed in some way.
o Creative freedom is given for displaying and presenting your project.
q Please take your time to put your project together well.
o DO NOT bring in a piece of notebook paper with a couple roughly sketched graphs and chicken scratch.
o Make sure that everything is done from above.
o Make sure that you indicate each different part with adequate labels.
o Answer the above questions in short form with work before you include them in your summary.
o Please write a summary that is clear and well thought out.
q Be prepared to answer for how you contributed to your group and how others in your group participated as well.
Others in my group: