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Calculus AB2

VIEWS: 63 PAGES: 10

									Calculus AB                                                   Instructor: M Challender

Text Book: Calculus, concepts and applications. Paul A. Foerster, 1998 Key Curriculum
Press.

We will cover almost all the material in the first ten chapters of the text in the sequence
outlined in the attached syllabus. Broadly speaking, the topics will be functions, graphs
and limits. The study of continuity as a property of functions, the concept of the
derivative, application of the derivative and computation of basic derivatives is studied.
Basic integrals and their application to differential equations and the fundamental
theorem of calculus will be taught as well as numerical methods of integration.

We will concentrate on learning the material in preparation to successfully challenge the
College Board Calculus AB exam. Homework will be one part of your grade; quizzes
will make up the second part, and the final is the third. You must pass all the components
of the course to earn a letter grade of C or better. There is no remediation of the material
in this course.

The grading scale for collected homework assignments is on a 4-point scale. Work turned
in late will earn no more than half credit. All assigned work must be completed in order
to earn a passing grade. It is expected that the homework be completed correctly.
Collaboration on homework is encouraged but remember that understanding how
someone else does a problem does not necessarily mean that you will be able to
reproduce the solution. The purpose of my reading the homework assignments is to better
prepare you to write complete solutions to problems.

Quizzes over the homework material will be graded using a rubric style system in
preparation for the Calculus AB exam. Some of the quiz questions will be open ended
and require many steps similar to the homework problems. There will also be multiple-
choice questions requiring you to recall previously learned concepts. Your understanding
of the homework material will be determined by the quiz questions that are very similar
to the homework problems. You are expected to pass the quizzes on a regular basis.

The final exam will be very similar to the College Board Calculus AB exam. You will
know in advance what the questions will cover. We will prepare for this exam over the
course of several class meetings.

One way to master math is to do many problems. You should plan to spend about two or
more hours of work out side of class for every hour in class. The time spent by each
individual will vary. Problem assignments for each class day are listed in the syllabus.
Certain parts of assignments will be collected. I suggest you do all the problems. Have
assignments prepared for collection the first class meeting of the week following the
week after they are assigned. If the material is presenting difficulties, ask questions.
People are encouraged to work together. You can learn from each other, however group
learning may only make you feel you understand. I will be available at 6:45 A.M. during
the week for help and by appointment. This time is set aside to answer your questions;
you should come prepared to get help.
Apart from homework that will be collected as outlined above you will be doing
problems in class sometimes with collaborators, please bring your book. There will be
quizzes of about 10-30 minutes duration almost daily to help us assess how you are
coming with the current material. Be prepared for the quizzes they will be over the most
recently covered material. I will inform you of the material that will be covered on the
next quiz, your input is valued. If you feel you need more practice in a certain area we
will practice these concepts. On quizzes and exams collaborations cease, of course, and
people do their own work.

The material covered in this course is the Calculus material outlined in the College Board
Calculus AB syllabus. I am confident that we can reach a reasonable level of mastery. It
will take commitment from you. Since this is an advanced placement class, you are
expected to work at the college level. If you do not understand a concept it is your
responsibility to see to it that you do what it takes to learn the material. There is no
remediation of assessments. Continued failure of assessments will result in a grade lower
than you may be used to.

An additional grade point is offered for this course. Students earning a B or better in the
course should be able to score a 4 or 5 on the AB exam. Students earning a C or lower are
encouraged to challenge the AB exam, but will not earn the additional grade point if their
score is below a 3 on the AB exam. I will not be surprised if a student earning below a B
passes the AB exam with a respectable score. Students earning a C are meeting the
minimum level of expectation for the course.

The purpose of this course is to produce highly prepared students for a college
environment. Students successfully completing this course with a B or better will have
demonstrated mastery the high school mathematics curriculum. All students will have an
understanding of the effort that is required of them to meet the expectations of a very
challenging curriculum. The knowledge about themselves and methods they use to
acquire knowledge will help them meet the challenges ahead.

The following is an outline of the problems you are to complete to meet the minimum
expectations for the course. You are encouraged to work ahead, come to class prepared to
ask questions regarding your concerns about particular concepts.

Chapter 1

1.1 The concept of instantaneous rate. The idea of a limit, average rate, instantaneous rate
of change and the derivative are introduced. Problem 1.1.1 requires the use of a graphing
calculator in radian mode. Problems 1.1.1 and 1.1.2.

1.2 Rate of change is studied by using equations, graphs and tables. The meaning of
derivative and the definition of limit are learned by approximating rate of change from
graphs and tables. Problem 1.2.25 illustrates how to use the TI83 to solve problems
related to rates of change in f(x). Problems 1.2.1 to 1.2.27.
1.3 The idea of integral as area under a curve. The meaning of definite integral is shown.
Problems 1.3.1 to 1.3.13.

1.4 Estimating the value of a definite integral. The area under a curve is found using the
trapezoid rule. Problem 1.4.7 requires that the student write a program to estimate a
definite integral using the trapezoid rule. Problems 1.4.1 to 1.4.5 odd and 1.4.9 to 1.4.17
odd.

1.5 The formal definition of a limit. Indeterminate forms, step discontinuity, removable
discontinuity are introduced. Problem 1.5.19 uses the Ti 83 to experiment with the limit
of sin(x)/x as x0. Problems 1.5.1 to 1.5.19 odd.

1.6 Calculus journal. A selection of problems serves as examples to key concepts.
Problems 1.2.14, 1.3.6, 1.3.10, 1.4.2, 1.5.12, 1.5.14, 1.5.16.

1.7 Chapter review. Problems R1 to R6 all.

Chapter 2

2.1 Numerical approach to the definition of a limit. The formal definition of a limit is
continued using an intuitive approach and introducing delta and epsilon values. Problems
2.1.1 to 2.1.3.

2.2 Graphical and algebraic approach to the definition of limit. The values of delta ()
and epsilon () and the limit L are calculated as well as estimated from tables and graphs.
Problem 2.2.7 to 2.2.11 will need the table feature of the TI83 to confirm delta values for
a given epsilon value. Problems 2.2.1 to 2.2.13 odd.

2.3 The limit theorems. The properties of limits are used in proofs, and explained why
they are true. Indeterminate forms and undefined limits are discussed. Problems 2.3.1 to
2.3.25 odd.

2.4 Continuity. The definition of continuity, step discontinuity, one-sided limits, cusp,
and asymptotes is studied. Problems 2.4.1 to 2.4.69 odd.

2.5 Limits involving infinity. Definition of limit as x goes to infinity and as limit goes to
infinity. Problem 2.5.11 experiments with the limit as applied to an integral using the
trapezoid program to conjecture a limit. Problems 2.5.1 to 2.5.13 odd, 2.5.12, and 2.5.14.

2.6 Intermediate value theorem and its consequences. The notion of continuous functions
is reinforced and the idea of extreme values is exercised. Problems 2.6.1 to 2.6.14 all.

2.7 Chapter review Problems 2.7.R0 to 2.7.R6 all.
Chapter 3

3.1 Graphical interpretation of the derivative. The connection to the slope of the tangent
line in connection to the instantaneous rate of change is explored. Problems 3.1.1 to 3.1.9
all.

3.2 Difference quotient and one definition of derivative. The definition of the derivative
as x goes to c is used. The idea of local linearity is explored using the TI83 zoom feature.
Problems 3.2.1 to 3.2.20 all

3.3 Derivative functions. Symmetric difference quotient, forward and backward
difference quotients, and the idea that x is a tolerance. Problems 3.3.5 and 3.3.6 require
students to make a conjecture about the derivative of sin(x) based on the graph of the
numerical derivatives graph. Problems 3.3.1 to 3.3.9 odd, 3.2.12 to 3.3.13 all.

3.4 Derivative of the power function and another definition of the derivative. The use of
h0 is used, derivative of a power function, derivative of a sum, a constant times a
function and a constant function are covered. Problems 3.4.1. to 3.4.29 odd.

3.5 Displacement, velocity and acceleration. The connection between acceleration,
velocity and displacement are established. The idea of speed, and relative versus absolute
values is covered. Problems 3.5.1 to 3.5.12 all.

3.6 Introduction composite functions using sine and cosine. The concept of inside
function and outside function are established. Problems 3.6.1 to 3.6.8 all.

3.7 Derivative of a composite function; the chain rule. The chain rule as it applies to
trigonometric and polynomial functions is learned. In problems 3.7.3 to 3.7.22 students
use the numerical derivative feature of the TI83 to support their conclusions. Problems
3.7.1 to 3.7.26 all.

3.8 Proof of the derivatives of sine and cosine. The squeeze theorem is shown, sinusoidal
function applications are explored using the TI83 calculator. Problems 3.8.1 to 3.8.13
odd.

3.9 Anti-derivatives and indefinite integrals. The constant of integration is learned and
the use of initial conditions to find a particular equation is shown. Problems 3.9.1 to
3.9.25 all.

3.10 Chapter review and journal entry. Problems 3.10.R0 to 3.10.R9 all.

Chapter 4

4.1 Introduction to the chain rule, product rule, and quotient rule. Problems 4.1.1 to 4.1.6.
4.2 Derivative of the product of two functions. The formula for the derivative of a
product is proven. Problems 4.2.1 to 4.2.21 odd require that the student check their
answers by comparing the graphs of the calculated derivative and the numerical
derivative. Problems 4.2.1 to 4.2.35 odd.

4.3 Derivative of a quotient of two functions. The formula for the derivative of a quotient
is shown. Problems 4.3.1 to 4.3.25 use the TI83 is used to confirm solutions. Problems
4.3.1 to 4.3.31 odd.

4.4 Derivative of the trigonometric functions other than sine and cosine. The derivative of
the six trigonometric functions is shown. Problems 4.4.1 to 4.4.43 odd.

4.5 Derivative of the inverse trigonometric functions. The arctangent, arccosine, etc. are
reviewed. Implicit differentiation is introduced. Problem 4.5.29 is key to discovering the
derivative of the inverse of a function. Problems 4.5.1 to 4.5.24 all, 4.5.25, 4.5.26, 4.5.29.

4.6 Differentiability and continuity. The idea of differentiability is covered, including
differentiability at a point and on an interval. Differentiability implies continuity is
studied. Problems 4.6.1 to 4.6.38 all.

4.7 Derivative of parametric equations is not covered.

4.8 Graphs and derivatives of implicit relations. Differentiation of implicit relations is
covered in detail. Problems 4.8.1 to 4.8.27 all.

4.9 Chapter review and journal write. Problems 4.9.R0 to 4.9.R8 all.



Chapter 5

5.1 A definite integral problem. This section shows other ways to find the area under a
curve. Problem 5.1.4 requires the students to use the TI83 to estimate a definite integral
using a midpoint sum. Problems 5.1.1 to 5.1.7 all.

5.2 Review of antiderivatives. The meaning of antideivative, indefinite integral and the
need for a constant of integration is covered. Problems 5.2.1 to 5.2.17 odd.

5.3 Linear approximations and the differentials. Linearization of a function is used find
approximations for functions. Problems 5.3.1 to 5.3.39 odd.

5.4 Formal definition of antiderivative and indefinite integral. The integral sign (  ) is
introduced, and the integral sign becomes an operation. Problems 5.4.1 to 47 odd.

5.5 Riemann sums and the definition of definite integral. Definition of Riemann sums are
introduced along with an extensive vocabulary list. Partitions, lower, midpoint, and upper
sums are covered. The idea of integrability is shown, and the integral as a limit of a
Riemann sum is covered. Problem 5.5.10 requires the student to write a program that will
be used to estimate integrals using the upper, lower, and midpoint Riemann sums.
Problems 5.5.1 to 5.5.12 all.

5.6 The mean value theorem and Rolle’s theorem. The mean value theorem is proven
algebraically, Rolle’s theorem and the mean value theorem are used in a variety of
questions. Problem 5.6.33 uses the TI83 it illustrate the proof of Rolle’s mean value
theorem. Problems 5.6.1 to 5.6.18 all, 5.6.19 to 5.6.33 odd.

5.7 Some special Riemann sums. The idea that a Riemann sum can be used to find and
exact value of a definite integral is discovered. Problems 5.7.1 to 5.1.12 all.

5.8 The fundamental theorem of Calculus. The fundamental theorem is proven using an
intuitive approach, by applying the Riemann sum and the squeeze theorem. Problems
5.8.1. to 5.8.9 all.

5.9 Definite integral properties and practices. Evaluating the definite integral using the
fundamental theorem is covered. The properties of integral are established, such as
positive and negative integrands, reversal of the limits on integration, sum of integrands,
etc. Problems 5.9.1 to 5.9.37 odd.

5.10 A way to apply the definite integral. A variety of questions exercise writing
Riemann sums and evaluating definite integrals using the fundamental theorem. Problems
5.10.1 to 5.10.7 all.

5.11 Numerical integration by Simpson’s rule and a graphing calculator. Simpson’s rule
is derived and applied to questions involving tables of values. Students write a program
for the TI83 calculator to evaluate integrals by Simpson’s rule. Problems 5.11.1 to
5.11.13 odd.

5.12 Chapter review and journal write. Problems 5.12.R0 to 5.12.R12 all.

Chapter 6

6.1 Integral of the reciprocal function, a population growth problem. The foundations for
question involving direct proportion, and separating variables is introduced. Problems
6.1.1 to 6.1.7 all.

6.2 Antiderivative of the reciprocal function. The  1/x dx = ln x + C is introduced in a
discovery lesson aided by the TI 83 calculators graphing capabilities. Problems 6.2.1 to
6.2.12 all.

6.3 Natural logarithms, and another form of the fundamental theorem. The derivative of
an integral is investigated. The definition on the natural log is illustrated. Problems 6.3.1
to 6.3.57 odd, 6.3.58, and 6.3.60.
6.4 ln x really is a logarithmic function. The uniqueness theorem for derivatives is used to
establish the properties of logarithms. Problems 6.4.1 to 6.4 13 all.

6.5 Derivatives of exponential functions and logarithmic functions. The technique of
logarithmic differentiation is established. Problems 6.5.1 to 6.5 37 odd, 6.5.38.

6.6 The number e and the derivative of base b logarithm functions. The definition of e is
covered, along with the change of base property, and the interchange of limits within
continuous functions. Problems 6.6.1 to 6.6.21, and 6.6.20.

6.7 The natural exponential function and the inverse of ln. The derivative and the integral
on f(x) = ex is shown. Problem 6.7.63 uses the TI 83 to discover L’Hospital’s rule.
Problems 6.7.1 to 6.7.63 odd.

6.8 Limits of indeterminate forms: I’Hospital’s rule. The application of l’Hospital’s rule
is covered and a variety of algebraic manipulations are exercised. The indeterminate
forms are covered. Problems 6.8.1 to 6.8.29 odd, 6.8.31 to 6.8.35 all.

6.9 Derivative and integral practice for transcendental functions. The focus is to master
integrals ad derivatives of exponential functions, and the integrals of tangent, cotangent,
secant and cosecant. Problems 6.9.1 to 6.9.90 all.

6.10 Chapter review and journal write. Problems 6.10.R1 to 6.10.R9 all.

Chapter 7

7.1 Exponential growth and decay. The concept of a differential equation is introduced.
The direct proportion property of exponential functions is explored. All the problems in
this chapter require the TI83 to interpret the students’ results. Problems 7.1.1 to 7.1.6 all.

7.2 Direct proportion property of exponential functions. General solutions to differential
equations is covered, separation of variables and initial conditions are used to find the
particular solution to a differential equation. Problems 7.2.1 to 7.2.9 all.

7.3 Other differential equations for real-world applications. Differential equations are
further explored using rates of change that are directly proportional to functions. More
complicated problem scenarios are used. Problems 7.3.1, 7.3.4, 7.3.5, 7.3.7 plus a
selection of problems at a later date.

7.4 Graphical solution of differential equation using slope fields. Given a slope field the
graph of the approximate particular solution is determined. The conjecture is confirmed
algebraically when possible. Slope fields, and logistic equations are introduced. Problems
7.4.1 to 7.4.7 odd, 7.4.10.
7.5 Numerical solutions of differential equations by Euler’s method. This topic is not a
part of the AB curriculum and will be covered at the end of the course.

7.6 Predator prey population problems. This is not a part of the AB curriculum and will
be covered at the end of the course.

7.7Chapter review and journal write. Problems 7.7.1 to 7.7.4 all.

Chapter 8

8.1 Cubic functions and their derivatives. The connection between functions and their
derivatives is introduced. Concavity, points of inflection and the second derivative are
discussed in a discovery lesson. Problems 8.1.1 to 8.1.5 all.

8.2 Critical point and point of inflection. Critical points as they apply to local and global
extremes are discussed. Points of inflection and cusps are also learned. The idea of
infinite versus undefined is presented. The notation for the first and second derivatives is
presented. Students will become familiar with the different notations and their uses.
Problems 8.2.1 to 8.2.41 odd.

8.3 Maximum and minimum in plane and solid figures. The idea of extremes is applied to
plane and solid figures. Problems 8.3.1 to 8.3.31 odd, additional problems are suggested
if student needs further practice.

8.4 Area of a plane region. The area of regions bounded by two curves is studied and the
application of the definite integral and its properties is reviewed. Students use the
numerical integration feature of the TI83 to estimate the area of a region bounded by
curves. Problems 8.4.1 to 8.4.24 all.

8.5 Volume of a solid by slicing. The methods for finding the volumes of solids by cross
sectional area and the disk and washer methods of revolution are covered. Students will
use the TI83 calculator’s numerical integration feature to determine the validity of their
results. Problems 8.5.1 to 8.5.27 odd.

8.6 Volume of a solid of revolution by cylindrical shells. The method for finding the
volumes of solids using shells is covered. Problems 8.6.1 to 8.6.18 all.

8.10 Chapter review and journal write. Problems 8.10.R0 to 8.10.R6 all.


Chapter 9 (This chapter is not part of the AB curriculum and will be done at the end of
the course.)

9.1 Introduction to integration of a product of two functions. This is an introduction to
integration by parts. A numerical solution to an integral is used to discover the technique
of integration by parts. Problems 9.1.1 to 9.1.7 all.
9.2 Integration by parts-A way to integrate products. The technique of integration by
parts is covered. Problems 9.2.1 to 9.2.10 all.

9.3 Repeated integration by parts. A method of integration by parts is introduced.
Different reoccurring patterns are explored, making the original integral repeat, use of
trigonometric properties, regrouping factors between steps. Problems 9.3.1 to 9.3.49 odd.

9.4 Reduction formulas and computer software. An iterative processes is used the powers
of trigonometric functions. Problems 9.4.1 to 9.4.23 odd.

9.5 Integration of special powers of trigonometric functions. Trigonometric identities and
“u” substitution techniques are exercised in this section. Problems 9.5.1 to 9.5.33 odd.

9.6 Integration by trigonometric substitution. Techniques that utilize trigonometric
substitution are used, reinforcing the techniques learned in section 4.5. Problems 9.6.1 to
9.6.29 odd.

9.7 Integration of rational functions by partial fractions. A few simple methods of
breaking rational functions into partial fractions are covered. Problems 9.7.1 to 9.7.19
odd.

9.8 Integrals of inverse trigonometric functions. Integration techniques for the inverse
trigonometric functions are covered. Problems 9.8.1 to 9.8.6 all.

9.9 Calculus of hyperbolic and inverse hyperbolic functions. This topic is not part of the
AB curriculum.

9.10 Improper integrals. The ideas of convergence and divergence are used to determine
if a solution to an improper integral exists. Problems 9.10.1 to9.10.25 odd.

9.11 Miscellaneous integrals and derivatives. A selection of problems that will serve to
review techniques of integration and derivatives in a random order. Problems 9.11.1 to
9.11.102 every other odd.

9.13 Chapter review and journal entry. Problems 9.13.R0 to 9.13.R11 all.




Chapter 10

10.1 Introduction to distance and displacement for motion along a straight line. Given a
velocity function distance traveled and displacement are determined. Through out the
chapter the TI83 calculator will be used to graph and analysis problems. Students will use
the numerical integration and derivative features of the TI83 in a routine manner while
problem solving. Problems 10.1.1 to 10.1.5 all.

10.2 Distance, displacement, and acceleration for linear motion. Given velocity or
acceleration the distance traveled and the displacement are determined. Problems 10.2.1
to 10.2.15 odd.

10.3 Average value problems in motion and else where. The concept of average velocity
of a function and average velocity are discussed. Problems 10.3.1 to 10.3.15 odd.

10.4 Related rates. The relationship between different variables and the rates at which
they are changing with respect to each other is covered. Problems cover changing volume
of a variety of solids as well as linear rates. Problems 10.4.1 to 10.4.21 odd.

10.5 Minimal path problems. Exercises involve setting up functions that can be used to
find minimum values. Problems 10.5.1 to 10.5.13 odd.

10.6 Maximum and minimum problems in motion and elsewhere. A variety of problems
involving related rates and maximum volume and minimize surface area are solved.
Global minimum and maximum are also included. Problems 10.6.1 to 10.6.11 odd.

10.8 Chapter review and journal write. Problems 10.8.R0 to 10.8.R5 all.

To review for the Calculus AB exam, students will work all open-ended questions from
the 2001 to 2007 exams. The review of basic concepts will be done using the Barron’s
AB Calculus review.

								
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