AP Calculus AB and BC



AP Calculus AB and BC

Book Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy. Calculus: Graphical, Numerical, Algebraic: AP Edition. Boston: Pearson Prentice Hall.

While every chapter has multiple teacher-made worksheets on the material, students are expected to complete the online homework within two days after the end of the lectures on each section. The homework program that is used is from Pearson.

Calculator The majority of my students use the TI 89. About 5% of my students use the latest TI 84. I teach to both of those calculators. I have a set of 89’s in my classroom.

Handouts on first day – Packet of Free Response Questions from 2003 till present, Pre-Calculus Packet, Complete Course Outline (Compiled by Vivek Verma and Dominic LaBella)

There is a Video Bank available for almost every section in the book on my web site. These videos include ones created by other math teachers as well as ones I have created.

1. Pre-Calculus Review - The first weeks of class are dedicated to the remediation of all the pre-requisite material that the students will use in calculus.

2. LIMITS AND CONTINUITY

Rates of Change and Limits

• Including basic substitution, basic properties (adding, subtracting, multiplying by a constant, multiplying, dividing) tables, reduction of fractions (by algebra or trigonometric reductions), dealing with indeterminate form and infinite limits, symbolic representation of limits

Limits Involving Infinity

• Including end behavior models, dealing with both x approaching both positive and negative infinity, sandwich theorem.

Continuity

• Including one-sided limits, piecewise functions, IVT, functions that are always continuous, types of discontinuities

Rates of Change and Tangent Lines

• Including both the definition of derivative and the alternate form of the definition (without the actual word derivative) and the normal line. Students will graph (by hand and with calculator) to see/show that the instantaneous rate of change is the slope of the tangent line

Materials

PPS presentations for every lecture.

The following worksheets: Sandwich Theorem / Limits involving infinity; Algebra and the Limit, and The Everything That You Could Possibly Be Asked About Limits In Chapter 2 Worksheet.

Evaluations

Quick Quiz on one-sided limits, Quick quiz on memorized theorems (Sandwich and IVT) with uses of each. Jeopardy Quiz (A graph is drawn and students are asked, “In terms of the three conditions of continuity, what was the student asked to draw?”). Homework on each section (). Chapter Test on all of limits.

3. DERIVATIVES

Derivative of a Function

• Including definition of derivative; alternate definition of derivative; symbols used to represent derivatives, average rate of change vs. instantaneous rate of change

Differentiability

• Including the connection between derivatives and continuity, where derivatives don’t exist, graphing derivatives and finding derivatives on the home screen of the calculator, symmetric difference quotient. Brief covering of horizontal asymptotes connected to max’s / min’s and the increasing function means a positive slope of the tangent line.

Rules for Differentiation

• The shortcuts, notation for higher order derivatives

Velocity and Other Rates of Change

• Including how long particles are in the air, when does a particle hit the ground, position of the particle when velocity equals a value.

Derivatives of Trigonometric Functions

Chain Rule

Implicit Differentiation

• Including derivatives of inverses of functions

Derivatives of Inverse Trigonometric Functions

Derivatives of Exponential and Logarithmic Functions

• Including logarithmic differentiation, derivatives of logs / lns, exponentials.

* In this chapter, students will show that they understand derivatives by computing them and also by graphing f’(x) and f”(x) when given a drawing of f(x). Students are expected to use their calculators to graph derivatives and to use their home screens to find derivatives (including implicit d).

Materials

PPS presentations on each section. Worksheet “The Long Way” (finding derivatives using the definition, alternate definition, and the symmetric difference quotient), “The Pain of Algebra” (Reducing derivatives after using the product / quotient rules). The answers to the problems must match the answers that the students get when they use their calculators. Discovery worksheet for the derivatives of inverses of functions, worksheet on difficult implicit d questions, Worksheet at the end of the chapter that has difficult chain rule combined with exponential / logarithmic problems.

Evaluations

Small quizzes on memorized derivatives. Cumulative quiz on memorized theorems with one application of each theorem. Students must get a perfect on this quiz for it to count. Test on entire chapter.

4. APPLICATIONS OF DERIVATIVES

Extreme Values of Functions

• Including finding cp’s / max’s / min’s. Increasing / Decreasing. Setting up sign charts / using mathematical notation to describe inc/dec.

Mean Value Theorem

• Including the lecture of the myth of the EZ Pass tickets, graphically and algebraically find c value that meets standards. This section also includes Rolles Theorem.

Connecting y’ and y” with the graph of y

• Including 2nd derivative sign charts, concavity, graphing 1st and 2nd derivatives, graphing original equation from derivative information. This section also includes speed / finding whether speed is increasing or decreasing using 1st and 2nd derivatives. Finding max / min velocity, acceleration, speed.

Modeling and Optimization

• Including Popcorn box example (Students in groups are given different sizes of paper and they create their own boxes. They want the largest box possible because they get to fill it with popcorn). They have to graphically represent their model.

Linearization and Newton's Method

• (While Newton’s method isn’t on the AP test, it is an interesting kick-off to show value of the tangent line) Including tangent lines and differential equations. This section also has beginning antidifferentiation with the concept of the family of antiderivatives

Related Rates

• Including problems using proportions and all manner of shapes changing as time changes.

Materials

PPS presentations on each section. Worksheet after 3rd section with difficult example that can only be done using the calculator. Popcorn box materials (paper / scissors / tape / popcorn / a broom…). Worksheet on Newton’s method. Worksheet with 40 additional real world related rates problems, worksheet called “Volpe goofs” with AP-style questions and answers that are not acceptable. Students have to determine why the work shown / answers given are wrong. Students will use the packet of former AP questions starting in this chapter to see AP-style related rates multi-part problems.

Evaluations

Test on entire chapter 4. Cumulative quiz on memorized theorems with one application of each theorem. Students must get a perfect on this quiz for it to count.

Starting with this chapter, students will have “quests” two days after the test on material from two chapters ago (so this quest will be on limits)

5. THE DEFINITE INTEGRAL

Estimating with Finite Sums

• Including RRAM, LRAM, MRAM, using charts, equations, and graphs, Students are required to find a calculator program that will find RAMs. Students will learn over-estimating / underestimating using 2nd derivative. Students will inscribe and circumscribe rectangles.

Definite Integrals

• Including the connection between an infinite number of rectangles and the integral.

Definite Integrals and Antiderivatives

• Including all basic antidifferenitaion rules. Average Value / Mean Value Theorem for Integrals, and finding the x value associated with the MVT both algebraically and graphically. Finding distance traveled as the integral of velocity. Finding average velocity using MVT

Fundamental Theorem of Calculus Pts. 1 and 2

• Including both the derivative of the integral (and the integral of the derivative) and the Integral Evaluation Theorem. Finding tangent lines when given the function as an integal

Trapezoidal Rule

Materials

PPS presentations on each section. Worksheet to kick off the chapter on rectangles and trapezoids. (I have them drawn. The kids have to find the areas under the curves using the shapes that I have drawn. Students must create a table to model the graphs / geometric shapes I have made), worksheet on FTC with variable in both limits of integration, worksheet on some basic integrals. Multiple problems from Free Response Packet.

Evaluations

Cumulative quiz on memorized theorems with one application of each theorem. Students will demonstrate that they understand when a function meets the necessary requirements for a theorem to hold. Students must get a perfect on this quiz for it to count. Test on all of 5; Quest on Chapter 3

The December break usually happens mid-Chapter 5 for AB – There is a pack of multiple choice AP problems that are due two days after we return from break.

The Pennsylvania required Student Learner Objectives (SLOs) for AB will take place after Chapter 5.

Part 1 –Students are expected to get a 90% or above on the Chapter 3 Quest.

Part 2 – Students will be given a worksheet with a table of x’s and their associated y’s. Students will answer a variety of questions on the derivative

6. DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING

Slope Fields and Euler's Method

• Including creating slopefields by hand and on their 89s. Students are required to get a calculator program (that is easier to use than the one built into the calculator). This is a flipped lesson. Students must watch the video that I have created online before they come to class. Students must be able to sketch a graph given initial conditions and “connecting” the slopes on the slopefield.

Antidifferentiation by Substitution

Antidifferentiation by Parts

• This is a flipped lesson. Students must watch two videos before coming to class.

Exponential Growth and Decay

Logistic Growth

• Including analysis of differential equations / solving differential equations with initial conditions and domain restrictions. Using logistic growth models.

Material that we cover that is not (officially) in book

Antidifferentiation of trig functions to different powers

Antidifferentiation by trig substitution

Materials

PPS presentations on each section. Packet of material for the “Not in book” material above.

In our school, one of the rites of passage is “The Integral Packet.” Both AB and BC students learn all of the methods above even though not all are required for AB. Students get 70-80 integrals (depending on the luck of the draw for their year – I have multiple versions of this assignment). Every integral requires at least two of the types of integration that they learned for each problem. We spend ten days in AB and eight in BC working on these packets in class. Students are required to use their calculators to show that their answers are correct (either on the home screen or by using the grapher). Technology stronger than their calculator is frowned upon. Students must get a 52/104 or higher on the test to have their packet accepted. Students with grades lower than that must start the whole process over again without using class time.

Evaluations

Test on all of 6. Students must get 75% on the integral packet to have the assignment count as 100/100.

Quest on Chapter 4

The December break usually happens mid-Chapter 5 for BC – There is a pack of multiple choice AP problems that are due two days after we return from break.

7. APPLICATIONS OF DEFINITE INTEGRALS

Integral as Net Change

• Including total distance traveled, displacement. Using the initial condition to find the particle’s position.

Areas in the Plane

• Area between curves using geometry and integration.

Volumes (Disks, Washers, Shells and Cross Sections)

• Lengths of Curves

Materials

PPS presentations on each section that include numerous computer-generated presentations of revolutions of planes around lines. Worksheet on sketching the initial planar regions that are the start of numerous 3 dimensional figures / finding the volumes of these figures by one of the methods described in the chapter. Worksheet on finding lengths of curves first by “guessing” by analyzing the graph, then by using analytical methods.

Evaluations

Test on all of Chapter 7. Quest on 5.

8. SEQUENCES, L'HÔPITAL'S RULE, AND IMPROPER INTEGRALS (BC only material in blue)

Sequences

• Defining convergence / divergence in terms of sequences.

L'Hôpital's Rule

• Including how to handle indeterminate forms

Relative Rates of Growth

Improper Integrals

Materials

PPS presentations on each section. Worksheet on examining graphs and equations to determine where an integral would be considered improper. Worksheet on graphically and algebraically (with limits) determining which function grows faster.

Evaluations

Quiz for AB on L'Hôpital's Rule and growth rates. Test for BC on all of Chapter 8.

9. INFINITE SERIES

Power Series

• Including introductory worksheet with a calculator exploration on why we need series. Geometric Series. Telegraphing Series. Finding sums of series. Determining convergence / divergence in terms of series. What “centered” means in terms of series.

Taylor Series

• Including construction of series and the common Maclaurin Series / the manipulation of these Maclaurin Series. Beginning integration with series.

Taylor's Theorem

• Including finding error using LaGrange and number of terms needed to control error using the calculator

Radius and Beginning Interval of Convergence

• Including ratio and nth root tests.

Testing Convergence at Endpoints

• This section contains the brunt of the chapter. Alternating series. Error bound with Alternating Series. Harmonic Series. P-Series. Integral test. Absolute vs. Conditional convergence. Limit comparison tests. Direct comparison test.

Materials

PPS presentations on each section. Multiple worksheets on the different concepts, Multiple problems from Free Response Packet.

Evaluations

Quiz on memorized Maclaurin Series (with domains). Group project at the end with multiple series questions. Exam on all of Chapter 9.

The Pennsylvania required SLOs for BC will take place after the Maclaurin Series is taught.

Part 1 – (Aforementioned) Memorized Quiz on Maclaurin Series (Students required to get 42/42 on this assessment)

Part 2 – Take home assignment on Maclaurin Series with mulitple problems using Maclaurin (Example: Re-write sin(x2) using the appropriate Maclaurin Series and integrate)

10. PARAMETRIC, VECTOR, AND POLAR FUNCTIONS

This is a student driven chapter.

Students are placed in three groups. They must create two videos – One to explain the pre-calculus behind their assigned section and one to describe the calculus involved with their assigned material. Each student must present in the video. Each group must provide videos created by other sources (PatrickJMT, Kahn, et al) and worksheets. Each group must provide a former free response problem with answers available. Students must earn a 90% or above on their own material before their section can be posted for the others in class to view.

I wait till after presentations and use remediation as needed.

Parametric Functions

• Including finding the first and second derivatives, finding tangent lines, finding speed, length of curve / total distance traveled.

Vectors in the Plane

• Including finding the velocity and acceleration vectors, speed, distance traveled / position of the particle given velocity.

Polar Functions

• Including finding the slope (dy/dy), areas enclosed by graphs, area between two polar curves

Materials

Access to computer carts. Access to microphone for kids-recorded videos.

Evaluations

Quizzes for each section (for students to prove that they know their own material before they go on to teach the others in class). Student-produced homework / worksheets. Test on all of Chapter 10.

AP Review

Once the new material ends, we begin the review. The calendar has different exercises each day.

Students do approximately 500 multiple choice questions before the AP test. They are given a list on the first day of the review and each student knows which question s/he is to be an expert. They then fill in “Signature sheets.” For example – I cannot figure out how to do problem number 6 on Day 5 of the material. I will get credit for showing the teacher that I tried to do the problem, but during class, I must find the student in charge of problem number 6. That person has to explain it to me in a way that I can understand and sign off on my paper that I proved that I understand. While I am getting signatures for the ones that I do not know, people are also coming to me to get help on the questions that are considered to be mine.

We did / do about half of the free response questions that have been given since 2003 during class. In the first few days of the review, we look at some of the posted responses that other students have given. There are two after-school review sessions, and one Sunday morning session where we do some of the alternate exams. These are driven by the students who determine where they are weakest.

The Friday before the exam, students can stay after school to do a compressed version (2 hours – each section is dramatically shortened) of the AP test using one of the super-secret tests that the CollegeBoard has online. I score these and send the predicted AP score to the students on Saturday.

Days that we get a little off-topic… but we are still concentrating on calculus.

There are a few days that we get off topic in class.

1. We will watch a movie in the two days before Thanksgiving break. Stand and Deliver.

2. In the two days before the December holiday break, we sometimes play a game that I have created called Calculopoly. Do not let the word “game” fool you. It is constant calculus while moving ahead on the board.

3. To kick off the first calculus chapter, we have a brief day to have a history lesson on what was going on in the world during the advent of calculus. While I do throw in lessons on Rolles and Newton during the regular lectures, this lecture is about 30 minutes of a full 42 minute class.

Snow Days!

Snow Days are chock full of calculus. Students must check their email in the morning to find out what they need to do for class.

This 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.

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

This course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

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