AP® Calculus AB; Course Syllabus Course Overview Students you enter this class with a variety of backgrounds and levels of confidence. With your help, I will provide the level of work that you need. The goals in this class are to learn about calculus of one variable (differentiation, integration and how to use the two together to solve applied problems), but more importantly to learn to think critically, or exercise your brain, in addition to successfully passing the AP®

Exam. You will learn how to use your calculator to do any integration or differentiation, however, knowing how to push the button does not achieve the goals above. Therefore, all exams will be composed of a non-calculator portion and a calculator portion, similar to the expectations on the AP®

Exam. Your calculator will be used to support your answers and to investigate calculus concepts. I recommend that you work together on the homework. If you explain a problem to another student, you will understand it better. Furthermore, this will give you practice in clearly justifying your solutions because I will often ask you to explain your answers and justify your solutions in your homework and on assessments. It is also useful to have a friend look at your solutions to make sure that you have done the problems correctly and adequately explained your solution. The class is not competitive: helping a classmate learn the material will not adversely impact your grade. Be sure to include all of your work when writing up the homework problems. No credit will be given for simply writing down the answers. Answers are posted online and are in the back of your book. I am interested in the why of the problem, not simply the answers. Partial credit will also be given on exams, but you must give enough information so that I can see what you are doing and why you made a mistake. All exams are graded with a scoring rubric, similar to the scoring of the open response portion of the AP® Exam. All topics covered will be studied from the four rule (graphical, numerical, analytical, and verbal) approach. That means you must have complete conceptual understanding of the material. For example, just knowing how to find a derivative from a given function will not be enough if you are asked to find the derivative at a point and all you are given is a set of data. Examples of the four rule approach will be addressed during class discussions and will be a part of your regular home assignments. A sample homework assignment demonstrating student conceptual understanding of derivatives and students’ ability to effectively communicate mathematics is attached. (Sample handout #1) Calculators and Technology A TI-NspireCX CAS handheld is the preference; alternatively a TI-89 or TI-84 Plus graphing calculator. If you do not have access to one of these graphing calculators, let me know right away. You can check out a calculator from me to use for the year. The assessments and homework are designed around your access to a graphing calculator. A TI-NspireCX CAS handheld will be used for presentations and you will be instructed in its use during the year. I will provide instruction on alternative graphing calculators on an as needed basis. In past years, every student has opted to purchase or check out a TI-NspireCX CAS handheld due to its symbolic nature and its ability to check answers to homework. Solutions to homework problems, practice sheets, and powerpoint presentations, are posted online for your review. Spend time previewing presentations of upcoming concepts prior to class to help you be

engaged in the discussion of new concepts. This generates a richer dialogue between students and teacher, and students and students throughout the year. A sample activity using the calculator to discover the chain rule is attached. (Sample handout #2) Student Evaluation Unit tests will be the primary tool for determining levels of conceptual understanding and skill. You will not be graded on the average of tests scores, rather you will have a cumulative checklist of concepts and skills for each test. If you keep up on your daily assignments, you will have an additional opportunity through retakes to demonstrate understanding and mastery. It is expected that you will work hard to improve in areas which are initially difficult. Retakes: Since my goal is for you to master understanding, there will be opportunities for you to retake test questions. The purpose of this is to encourage you to continue studying the concepts until you have mastered concepts at an acceptable level. Occasionally I will give a retake for a quiz. The entire quiz must be retaken. Retakes replace original scores at a rate of 90%. You will only have retake opportunities if you have completed your homework by its due date. No homework, NO RETAKES. Quizzes will be approximately once a week, announced well in advance. You will be given a specific time limit for all quizzes. This will help me to determine what you do and do not understand. Quiz questions are generally taken from the book and handouts. There are occasional retakes on the quizzes. Quizzes are primarily skill based, while tests are primarily problem solving and conceptually based (this is a generalization). Homework: There is no way to learn without doing, so you will be assigned daily written work. Only certain assignments will be turned in for credit, but I will be glad to look at your practice work at any time. I will collect your non-graded homework on tests days to learn whether you have completed the assigned work in order to qualify for the retakes. Homework problems can be tough. I want you to struggle with ideas and to be clear about what you do and do not understand. It is not my goal to have everyone complete the work as quickly and perfectly as possible. I do want to know what is confusing you and preventing you from arriving at an understanding or a solution to a given problem. The class will benefit when incomplete solutions are explored and a variety of approaches are shared. These will be shared with the class to launch discussions. You will do much of your work in small groups. In order to encourage collaboration and to increase the quality of feedback on your work, occasionally I will collect your assignments as a group, and one paper will be chosen at random and completely corrected. The other papers will simply be checked off. The group will need to discuss the corrections on the one paper and each student will need to reevaluate his/her own work according to those corrections. A sample collected homework assignment is attached. (Sample handout #3)

Course Anticipated Timeline

First Semester AP Calculus AB

Section Topics Timeline

Handouts Precalculus Review 2 weeks

Lines; Slope as rate of change, parallel and perpendicular lines, equations of lines, secant lines and tangent lines, applications

Functions and graphs; functions, domain & range, families of functions, piecewise functions, composition of functions, average and instantaneous velocity, parametric functions and motion, applications

Exponential and Logarithmic functions; Exponential growth and decay, inverse functions, logarithmic functions, properties of logarithms, applications

Trigonometric functions; Graphs of basic trigonometric functions, domain and range, transformations, inverse trigonometric functions, applications

Unit 1 Limits and Continuity 3 weeks

2.1 Rates of Change and Limits; Properties of Limits, One and Two-sided limits, Sandwich Theorem

2.2 Limits Involving Infinity; Asymptotic behavior, end behavior, visualizing limits

2.3 Continuity; continuous functions, continuity at a point, discontinuous functions, types of discontinuities

2.4 Rates of Change and Tangent Lines; Averages rates of change and instantaneous rates of change

Handouts for unit 1

Exploration to introduce limits; Limits Involving Infinity; The Hiker; Rates of change- Ford vs Chevy; Introduction to Continuity; Limits, Continuity and Secant Lines; Difference Quotient revisited

Unit 2 The Derivative 5 weeks

3.1 Definition of the Derivative

3.2 Differentiability; Local linearity, numeric derivatives using the calculator, differentiability and continuity

3.3 Rules of Differentiation; Power rule, product rule, quotient rule, modeling motion, graphical interpretation of derivatives

3.4 Velocity and Other Rates of Change; Applications to velocity and acceleration, business, science, intro to tangent line approximation

3.5 Derivatives of trigonometric functions

3.6 Chain Rule; Parametric derivatives

3.7 Implicit Differentiation; Differential method, y’ method, introduction to related rates

3.8 Derivatives of Inverse Trigonometric Functions

3.9 Derivatives of Exponential and Logarithmic Functions

Handouts for unit 2

Product Proof puzzle; Doing a Lot with a Little; Match the function with its Derivative; Recognizing Derivatives; Sorting them Out; Explorations Modeling motion; Sparse Data; Velocity worksheet; Concepts around average rates of change vs instantaneous rates of change; Introduction to Tangent Line Approximation; Spread of an Infection, Interpretations; Discovery of the Chain Rule; Derivative Interpretations; Parametric Derivatives; Unbroken Chain; Sliding Ladder Problem; Implicit Differentiation of tangent lines and ellipses

Unit 3 Applications of the Derivative 5 weeks

4.1 Extreme Values of Functions; Local (relative) extrema, global (absolute) extrema

4.2 Mean Value Theorem; Rolle’s theorem, increasing and decreasing functions, f, f’, f” graphical connections

4.3 Concavity and the Second Derivative; Critical values, first derivative test for extrema, concavity and points of inflection, second derivative test for extrema

4.4 Modeling and Optimization; Applications from college resources

8.1 L’Hopital’s Rule

4.5 Linearization and Newton’s method; Over and under-estimates

4.6 Related Rates

Handouts for unit 3

Increasing and Decreasing functions on the TI-89; Concavity and the Second Derivative on the TI-89; Supplemental related rates problems; Supplemental optimization problems; Writing activity; Supplemental problem sets

Unit 4 The Definite Integral 3 weeks

5.1 Estimating with Finite Sums; Riemann sums right and left endpoints

5.2 Definite Integrals; Riemann sum as a definite integral, connection between integrals and areas under curves

5.3 Definite Integrals and Antiderivatives; Introduction to Fundamental Theorem of Calculus, the Average Value Theorem

5.4 Fundamental Theorem of Calculus

Handouts for unit 4

Estimating with Finite sums; TI-89 exploration of Riemann sums; The Naked Mile; Newspaper article (finding articles in the paper demonstrating rates of change and accumulation); The Area Function; Riemann Sum as a Definite Integral; Area Under the Curve; Exploring Definite Integrals and Compare Those Integrals; The Fundamental Theorem of Calculus-Intro; What is Fundamental about the Theorem; The Fundamental Theorem concepts; Accumulation and Functions defined as integrals; Valentine worksheet; Homer and Truckasaurus application

Final exam Final exam review Second Semester AP Calculus AB

Section Topics Timeline

Unit 4 continued Definite Integrals and Antiderivatives 2 week

5.4 Fundamental Theorem of Calculus

5.5 Trapezoid Rule

Handouts Applications of integrals

Mixed problem sets consisting of everything learned thus far, released old AP exam questions

Unit 5 Differential Equations and Mathematical Modeling 4 weeks

6.1 Antiderivatives and Slope Fields; Reading slope fields, explorations of slope fields with graphing calculators

6.2 Integration by Substitution

6.3 Integration by Parts

6.4 Exponential Growth and Decay; Separable differential equations, Newton’s Law of Cooling

6.5 Population Growth; Logistic differential equations, revisiting slope fields

6.6 Numerical methods; Eulers method, differential equations on the TI-89, visualizing Eulers approximations

Handouts for unit 5

Group exploration-reading slope fields; Explorations – Using the TI-89 with differential equations; The Logarithmic Function; Clearing the Hill; Slope Fields revisited; Huey, Dewey, and Louie, Pie Napping; Group review of differential equations

Unit 6 Applications of Definite Integrals 3 weeks

7.1 Integral as Net Change; Summing rates of change, displacement vs total distance traveled and the relationship to the integral, revisit linear motion

7.2 Areas in the Plane

7.3 Volumes; Solids of revolution--washer method, shell method, volumes of solids with known cross sections

Handouts for unit 6

Area review problem; Misc AP Review problems; Collecting Washers; Astroid Fragments; Misc volume application problems

Review This schedule allows for some flexibility depending on student needs. 3 weeks

A variety of review methods are implemented during this time including one extensive story problem (covers 5 topics) and a group game review which takes 4 class periods, and has an extensive multi –step problem from each unit.

Unit 7 After the AP Exam 4 weeks

6.3 Integration by Parts; Revisited

7.4 Lengths of Curves

7.5 Applications from Science and Statistics; Center of Mass, work and work pumping liquids, normal probabilities

8.2 Comparing Rates of Growth

8.3 Improper Integrals; Infinite limits of integration, tests for convergence and divergence, integrands with infinite discontinuities, applications

8.4 Partial Fraction Decomposition; Trigonometric substitutions, integral tables

Handouts Mixture problems Teacher Resources Primary Textbook: Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy. Calculus: Graphical, Numerical, Algebraic: AP Edition. Boston: Pearson Prentice Hall. Student Activities: A variety of sample activities are attached which; i) demonstrate student exposure to working with functions represented in a variety of ways and emphasizing the connections among those representations (Handout sample #4, #7, and #8). ii) demonstrate students communicating mathematics and explaining solutions to problems both verbally and in written sentences (Handout sample #1, 3 and #5). Students work on these problems in groups of 4 discussing their solutions as they move through each question. iii) demonstrate students using graphing calculators to help solve problems, experiment, interpret results, and support conclusions (Handout sample #2 and #6). Additional resources: Additional Calculus textbooks from a variety of authors are also used as teacher supplements. Several worksheet questions, quiz and test questions, and in-class activities are taken from the University of Washington and University of Michigan mathematics websites of released exams and course activities, in addition to using past AP® Exam questions. My experience has shown this to be an effective tool for student success in college mathematics and success on the AP® Exam. The evidence to support this practice are my students’ exam scores and former students success in college mathematics courses.

AP Calculus Sample #1 Derivative Interpretations Purpose: To clarify student understanding of derivatives and their units. 1. The temperature, H, in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by

H = f t( ) , where t is in minutes since the coffee was put on the counter.

a. Is f ' t( ) positive or negative? Give a reason for your answer.

b. What are the units of ( )'f t ?

c. Using complete sentences explain what ( )' 20f means in practical terms. 2. The temperature, T, in degrees Fahrenheit, of a cold yam placed in a hot oven is given by

T = f t( ) , where t is the time in minutes since the yam was put in the oven.

a. What is the sign of f ' t( ) ? Why?

b. What are the units of

f ' 20( ) ?

c. What is the practical meaning of the statement

f ' 20( ) = 2 ?

3. The cost, C (in dollars) to produce g gallons of ice cream can be expressed as C = f g( ) . Using units,

explain the meaning of the following statements in terms of ice cream.

a. f 200( ) = 350 b.

f ' 200( ) = 1.4

more problems not included in sample

The Chain Rule Sample #2 AP Calculus Purpose: To use the TI-89 and a variety of examples to discover the procedure for computing the derivative of a composite function. 1. To discover the Chain Rule, first practice taking derivatives of a few functions using the TI-89. Since each function will soon be an inner and outer function in the derivative of a composite, it will be helpful to keep a catalog of these derivatives in front of you.

Function Inner Outer ddx

(inner) ddx

(outer)

1+ x2

sin 2x( )

x −1( )3

3x + 2( )4

tan x2( )

sin2 x

2. Use the TI-89 to compute the following derivatives.

Function Derivative

1+ x2

sin 2x( )

x −1( )3

3x + 2( )4

tan x2( )

sin2 x

3. Based on these examples, can you see a pattern? Write out your guess by filling in the right side of the following equation.

ddx

f g x( )( )( ) = ______________________________________________

4. Try these out (Use your calculator to check your results):

ddx

tan2 3x( ) = ddx

16 − 4x2 =

AP Calculus Sample #3 Newton’s Law of Cooling

Huey, Dewey and Louie: Pie-napping Daisy Duck has baked a pie for Donald and has left it on the window sill to cool. The temperature of the pie when it was removed from the oven was 200°F and the temperature outside is 60°F. Donald’s mischievous nephews have their eyes on the pie and plan to steal it as soon as it has cooled off enough form them to handle. Daisy plans to serve the pie to Donald in forty minutes. The nephews can’t steal the pie until it cools down below 80°F. The question is, “who will eat the pie?” a) Sketch a graph of the temperature of the pie as a function of time. Be sure to clearly indicate any interesting features of the graph. Suppose you are told that twenty minutes after the pie was placed on the window sill the temperature of the pie was 120°F and the pie was cooling at an instantaneous rate of 2.5°F/min. (b) Illustrate this information on your graph. (c) If the pie continues to cool at this rate, predict the temperature of the pie at the end of forty minutes. (d) Is this prediction an over- or an under-estimate of the actual temperature of the pie at forty minutes? Explain your response in complete sentences.

Additional questions not included in sample

AP Calculus In class discussion Continuity and Limits Sample #4 Purpose: To introduce the definition of continuity and differentiate between continuity at a point vs continuity of a function. 1. Use the graph of the function f at the right to answer the following questions. Assume this is the entire graph of f. (a) What is the domain of f? (b) Find ( )lim

3f x

x→

(c) Find ( )lim

0f x

x→

(d) What is ( )3f ? DEFINITION: A function g is called continuous at a if two things are true: i) a is a point in the domain of g and ii) ( ) ( )lim f x f a

x a=

→

A function is just called continuous if it is continuous at every point in its domain. 2. Answer the following question for the function f whose graph is above. Explain your answer. (a) Is f continuous at 3?

(b) If f continuous at 0?

(c) Is f continuous at 1?

(d) Is f continuous? Not every function is continuous, but many are. It’s often important to know ahead of time that a function we want to work with is continuous, because that makes limit calculations easy. The following kinds of functions are always continuous:

polynomial functions trigonometric functions exponential functions

There are many other kinds of continuous functions; some will be explored later in the course. You can also combine continuous functions to make new continuous functions. For example, you can add, multiply or compose continuous functions and be guaranteed to get a continuous function. Division, on the other hand, sometimes creates complications, as we’ll see below.

3. Find 2lim 3 2 42x x

x− +

→. Explain your solution.

4. Can you find 2 1lim11

xxx−−→

by plugging in 1x = ? Explain.

Sometimes we can find a limit of a function where it isn’t continuous or defined by first simplifying the function, so that it resembles a continuous function. 5. Simply the function in problem number 4, and find the limit of the resulting expression as 1x→ . Why can you say that this is the same as the limit you were asked to find in problem number 4? Use your calculator and zoom in around x = 1. What do you see? Use the table function on your calculator with Δx increments of 0.01. What do you notice?

6. Let ( ) 2 3125 6.25c x x x= −

(a) Simplify the expression ( ) ( )( ) ( )5 55 5

c h ch

+ −+ −

as much as possible; there should not be an h in the

denominator of any fraction in the simplified expression.

(b) Is ( ) ( )( ) ( )5 55 5

c h ch

+ −+ −

continuous at 0h = ? Why?

Is the simplified expression in part (a) continuous at 0h = ? Why? Additional questions not included in sample

Sliding Ladder Problem Sample #5 AP Calculus (adapted from UW math) 1. A ladder 25 feet long is leaning against the wall of a building. Initially, the foot of the ladder is 7 feet from the wall. The foot of the ladder begins to slide at a rate of 2 ft/sec, causing the top of the ladder to slide down the wall. The location of the foot of the ladder at time t seconds is given by the parametric equations ( )7 2 ,0t+ . (a) The location of the top of the ladder will be given by parametric equations ( )( )0, y t . Find the formula for ( )y t . What is the domain of t values? (b) The graph of the function ( )y t is given below. Compute the average velocity of the top of the ladder on these time intervals: [0,2], [2,4], [6,8], [8.9]. Explain in words what the average velocity is telling you in terms of the picture.

[0,2] = [2,4] = [6,8] = [8.9] = (c) The foot of the ladder is moving at a constant rate; how about the top of the ladder? Explain.

2. A ladder 25 feet long is leaning against the wall of a building. Initially, the foot of the ladder is 7 feet from the wall. The foot of the ladder begins to slide at a rate of 2 ft/sec, causing the top of the ladder to slide down the wall. The location of the foot of the ladder at time t seconds is given by the parametric equations ( )7 2 ,0t+ . The location of the top of the ladder will be given by parametric equations

( )( )0, y t and ( ) ( )2625 7 2y t t= − + . The graph of ( )y t is pictured below.

(a) Sketch the graph of the derivative of ( )y t below. What is the domain of ( )y t′ ? (b) Compute the formula for the derivative function ( )y t′ . (c) In practical terms and using complete sentences, what does the derivative tell us about the ladder?

Logistic Models Exploration Sample #6 AP Calculus

Population….logistically speaking.. In a logistic model of population growth, the rate of growth of the population is proportional both to the population itself and to the difference between the carrying capacity and the population. 1. State this relationship as a differential equation, using C as the carrying capacity and P for the population.

2. Let

dPdt

= ′P = 0.0002P 1800 − P( ) represent the growth of a population P of rabbits on an isolated

island after t months.

a) How many rabbits can the island sustain? Explain.

b) Use the given differential equation to determine what will happen to P if the initial population is i) 1800. Why? ii) 2200. Why? iii) 300. Why? 3. a) Use technology to sketch the slope field for this differential equation. Record and explain your

choice of window for the given differential equation.

b) Does this slope field confirm your answers from (2)? Explain.

Get your teacher’s signature before proceeding. _______

c) Draw the three solution curves on the slope field (provided by your teacher) corresponding to initial populations of 300, 1200, and 2200. Check your solution curves with your calculator. Correct in red the ones you got wrong. Paste your slope field below. d) What is the population (approximately) when it appears to be growing most rapidly? Label this point F on the paper slope field for the initial population of 300 and explain your selection.

4) a) Sketch the graph of ′P = kP C − P( ) with P as the independent variable and with P’ as the dependent variable. What is the value of P when P’ is at a maximum? Why? Additional questions follow not included in the sample

Area under the Curve review Sample #7 AP Calculus (adapted from UM math) Per:___ Ms. S is already tired of winter. She is dreaming of her beach house and days rafting on the local river near the beach. Not all days on the river are beautiful, though. Last summer a storm dumped about a year’s worth of rainfall on the area in a couple of days. A man-made lake held back by a dam near the beach rose as the swollen rivers rushed toward the lake. The graph below gives the rate R, in thousands of cubic meters per hour, that water was entering the lake during that day as a function of t, in hours since midnight. The volume of the lake at midnight was 400,000 cubic meters. The maximum volume that can be held by the dam is 460,000 cubic meters. Due to an oversight, the floodgates of the dam were kept closed until 6:00 am when they were opened to full capacity. The gates allowed water to leave the lake at a constant rate of 2000 cubic meters per hour.

a. Approximate the volume of the lake when the floodgates were opened. Show your reasoning. b. When did the lake reach its highest volume? Explain. c. Approximately what was the highest volume of the lake on that day? Explain. d. At what rate was the volume of the lake changing at 6:00 pm?

AP Calculus Sample #8 Assume that f(x) and g(x) are differentiable functions about which we know very little. In fact, assume that all we know about these functions is the following table of data:

x f(x) f’(x) g(x) g’(x) -2 3 1 -5 8 -1 -9 7 4 1 0 5 9 9 -3 1 3 -3 2 6 2 -5 3 8 ?

This isn’t a lot of information. For example we can’t compute f’(x) with any degree of accuracy. But we are still able to some things, using the rules of differentiation.

1. Let )()( xfxxh = . What is )1('h ?

2. Let )()(4)( xgxfxj −= . What is j’(0)?

3. Let)()()(xgxxfxk = . What )2(' −k ?

4. Let )()( 3 xgxxl = . If )2('l = -48, what is )2('g ?

5. Let 3)(1)(xf

xm = . What is )1(' −m ?

6. Let )()()( 2 xgxfxxn = . What is ?)0('n