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AP CALCULUS AB 2007-2008

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AP Calculus AB

 

AP CALCULUS AB SYLLABUS

 

Course Design and Philosophy

 

The objective of this course is help students focus on conceptual understanding and thinking skills in calculus. In addition, students receive a strong foundation, which enables them to succeed in future mathematics courses. Students will learn calculus by doing it – algebraically, graphically, numerically and verbally and in turn discover the areas their talent lies and appreciate the beauty of calculus. As they are introduced to new concepts and techniques they will have the opportunity to participate in cooperative groups, class discussion, peer learning, peer tutoring and to incorporate technology. The course is challenging to most students and demands hard work and high expectations of one-self. This prepares them to take on future challenges in a competitive society.

C3The course provides students with the opportunity to work with functions represented in a variety of ways—graphically, numerically, analytically, and verbally—and emphasizes the connections among these representations.

 

 


 

Teaching Strategies

The first few weeks I review pre-calculus concepts and familiarize students with the TI83+ graphing calculator.  Students work in groups / pairs when doing an activity or working on a problem. Ideas are investigated analytically, graphically and numerically. It is important for students to understand that graphs and tables are not sufficient to prove an idea, verification always requires an analytic argument. They often work alone initially and then turn to their partners to collaborate. [C3] [C5].

C5—The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions.

 

 

 


 

 Communication is one of the major goals of the course. [C4]   

Students are expected to communicate verbally and on paper using proper vocabulary and terms.   Students discuss new concepts and share their thoughts. They are encouraged to verbalize their thoughts and interact among themselves. Quite often they are asked to come up to the board and work out the problem (some of them love it); explaining their work to the class and clarifying any question(s) their classmates had while working on the problem. [C4]

Science activities using the Texas Instruments CBL/CBR are incorporated to reinforce and understand the concepts better. They are encouraged strongly to work in study groups and check out each other’s work. [C4]  [C5] 

 

 

Calculator Ideas 

The graphing calculator is a good tool to help students visualize various concepts and ideas. Helps solving problems, interpret results and support conclusions. [C5]

It is used to perform some functions such as:

Graphing a function in a specified window

Finding the slope and equation to the tangent line

Finding the derivative numerically

Investigating continuity- discontinuous function, continuity on an interval    

Exploring trigonometric functions and their derivatives

Approximating the value of a definite integral using numerical methods

 

 

 

Activities

 

  • Students will draw a graph representing average length of their hair as a function of time during the past months [C3]
  • Explore function graphs on the graphing calculator, transformations and    compositions [C5]
  • Line of best fit, finding the function that best fits the data – using the regression equation [C3] [C5]
  • Use the TI Graphing Calculator, a CBL System and a motion Detector to measure distance and velocity and produce graphs of motion
  • Analyze and interpret graphs
  • Explore data collection for a velocity vs. time graph                                 
  • Calculate average velocity  using the CBL. A ball is tossed and the height-versus- time  graph is generated and examined . Students compute the average velocity over  a time interval. Then they determine the velocity of the ball at exactly .05 seconds after the ball was tossed and explain how their answer was obtained . After this  they zoom in on the graph near t=.05 until the graph looks like a line. Students find the slope of the line and compare it to their estimation of the instantaneous velocity at t=.06  [C3][C5] 
  • Students come to the board and plot the derivative of the sine curve using the local maximum, local minimum and inflection points to arrive at the cosine curve and explain the relationship . At this point there are all sorts of discussions. [C4]
  • Hands on Activities such as Differentiating Bingo helps reinforce differential rules.
  • Introduce the calculator program RSUM, to obtain left hand and     right hand Riemann sums. [C5]
  • Hands on activities to illustrate the concept of finding the volume of a solid with known cross-sectional areas – using Play-Doh they will sculpt various shapes and slice it to get the cross-sectional area
  • Volume of a cylindrical shell: I use an activity from the Instructors guide to accompany Calculus – concepts and calculators.  A stack of typing paper is rolled into the form of a cylindrical   shell. We then find the volume of the paper used to make the shell. 

            Use the above shell method for rotations around the y-axis and     other vertical lines

  • As an introduction to slope fields I use  the activity from the Instructors guide to accompany Calculus – concepts and calculators. For the differential equation y’= y- x students will calculate the slope for the different coordinates given to them in the range -4<= x<= 4 and -3<= y<=3. For instance when x=3 and y=5, y’= y – x =2.  And when x= -2 and y= -3, y’= y- x= -1. The student with coordinate (3,5)  will go up to the  board and draw a short line segment with a slope of 2 at the point (3,5). The student with coordinate (-2,-3)  will draw a small line segment with slope -2 at the point(-2,-3). Continuing in this manner the class would complete the slope field . This activity gives rise to all sort of discussions. [C4] 
  • Since the task of sketching a slope field by hand can be tedious and time consuming , students will be introduced to the calculator program  slopefield . [C5]

                         

 

Student Evaluation

 

Students will be graded on the Quizzes and Tests, which account for 70% of the grade. The remaining 30% goes towards Assignments, Homework, Class work and Presentations. The tests and quizzes incorporate multiple choice and open - ended questions. The final exam and mock AP exams follow the format of the AP Exams. 

 

 

 

 

 

 

 

 

 

                                                                                                                        

 

AP Calculus AB Course Outline

Unit 1: Pre-calculus Review: Functions and Graphs (4 weeks)

Basic Functions and Transformations

Linear Functions and Models

Exponential Functions

Inverse Functions

Logarithms

Polynomial and Rational Functions

Composition of Functions

Trigonometric Functions

Applications

 

Unit 2: Limits and Continuity (3 weeks)

 Rates of change

 Average and Instantaneous Velocity

 Limits at a point

 Properties of limits

 Two-sided

 One-sided

Limits involving infinity

 Asymptotic behavior

 End behavior

 Properties of limits

Continuity

 Continuous functions

 Discontinuous functions

  Removable discontinuity

  Jump discontinuity

  Infinite discontinuity

 

Unit 3: The Derivative (5 weeks)

Definition of the derivative

Average and Instantaneous Velocity

Derivative of a function at a point

The Derivative Function

Calculating Derivative Numerically

Derivatives of algebraic functions

     The product and quotient rule

     The chain rule

Derivatives of trigonometric functions

Implicit derivatives

Derivatives of inverse trigonometric functions         

Derivatives of logarithmic and exponential functions

Related Rates of change

Approximation

 

Unit 4: The Definite Integral (3 weeks)

  Calculating Distance Traveled

  Calculation of Areas

 Riemann sums

 Trapezoidal rule

 Definite integrals

  The Fundamental Theorem of Calculus I

 

Unit 5: Applications of the Derivative (3 weeks)

            Increasing and decreasing functions,

Critical values – Relative Max/ Min

Concavity and points of inflection

            Applications of the second derivative

Mean value theorem

Rolle’s theorem

     Optimization problems

 

Unit 6: Integrals, Finding Antiderivatives (5 weeks)

Functions defined by Integrals                        

The Fundamental Theorem 2

Areas of plane regions

Antiderivatives

Integration using the chain rule

Method of substitution

Integration by parts

Trigonometric functions and inverses

Numerical Integration

 

 

Unit 7: Using the Definite Integral (2 weeks)

Net and total distance traveled

Volume by slicing

Volume by shell method

Average value of a function

More applications of the Definite Integral

 

 

Unit 8: Differential Equations (2 weeks)

 

 

 

Introduction

Slope Fields

Euler’s Method

Separation of Variables

 

The time schedule allows for 2-3 weeks of flexibility and review for the Exam.

 

Major Text

Best George, Stephen Carter, Douglas Crabtree. Calculus, Concepts and Calculators.

Andover, MA: Venture Publishing, 2006

 

Supplemental Text

Larson Ron, Robert P. Hostetler and Bruce H. Edwards. Calculus I

Boston: Houghton Mifflin,2006

 

Additional Resources

Instructors guide: Calculus, Concepts and Calculators.