Advanced Placement Calculus (BC)
text: Anton, Howard, Irl Bivens, and Stephen Davis.
Calculus 7th edition (with early transcendentals).
New York, NY: John Wiley and Sons, Inc. 2002
resource text: Berkey, Dennis, Paul Blanchard.
Calculus 3rd edition.
Orlando, FL: Harcourt Brace Jovanovich, Publishers. 1984
Prerequisite Material as listed in the ‘06-’07 Program of studies: “ intended for students with exceptional backgrounds in mathematics” … including analytic geometry, advanced function analysis, trigonometry, probability and statistics, and a working knowledge of the TI graphing calculator.
Calculators: A graphing calculator is required in this course. The calculator will be used on most tests, including portions of the final exam.Many tests will have a separate calculator and non-calculator section.
The TI 83, TI83+, TI 84 are recommended.
The TI 89 may be used on some tests and on the A.P. exam. However, due to the capabilities of this machine, it will be prohibited on more tests than the TI 83/84.
(Note: A TI-84 or TI-83 PLUS calculator will be used for all classroom demonstrations.)
The calculators will be used to assist in analyzing data graphically and numerically. However, the graphing calculator is not the answer to all problems! Students will be asked to analyze and interpret the results of what they see happening on tables, graphs, statplots, and the in the Statistics editor. Moreover, they will be assigned specific problems that establish the limitations of the graphing calculators. Continual attempts are made at establishing the relationships among a functions’ graphical, numerical, analytical, and verbal representations.
The use of the graphing calculator to graph functions within an arbitrary viewing window, to find roots and points of intersection, to determine numeric derivatives, and to approximate definite integrals will be stressed.
Preparation for the A.P. Exam: Students will be assigned A.P. example problems which have been released at the end of each unit. An outline of some of the problems chosen is included at the end of this document. Students are encouraged to work together on and discuss these problems; however copying is discouraged. Students will be asked to present some of these problems in class. Others will be graded using the rubric presented by The College Board. Emphasis is placed upon justification which supports their answers.
We, also, will take a few “sample” exams, in early May.
Assessment: Tests and quizzes are given routinely as denoted in the following syllabus. In addition to traditional assessment, student presentations, homework assignments (including sample AP problems), and graphing calculator activities are routinely graded. Some of the graded assignments include:
1) An investigation of limits numerically involving “non-routine” functions
2) An inductive discovery used to determine the means of finding an inverse function’s derivative by
analyzing a function’s points and slopes numerically and graphically and comparing them to those of its inverse.
3) An analytical and graphical link of Newton’s method to Tangent Line Approximation.
4) A thorough sequence of background derivations needed to derive the Surface Area formula.
Please note that the order of presentation established in the Anton textbook is not always followed:
Course Outline/ Assignment Sheets
CHAPTER 1 Function Review (time frame: 5 days)
The italicized items below you are to read and review on your own.
Appendix C - equations of lines,
Appendix E Trigonometry Review
Sec 1.1 Read pp. 8 - 14 Analysis of Graphical information
Sec 1.3 Read pp. 27 - 38 How to graph a function on your graphing calculator
Sec.1.5 Read pp. 61 - 62 Equations of lines and direct proportionality.
Sec. 1.6 Read pp. 63 - 75 Families of functions. and inverse proportionality.
Note: Problems assigned are representative of testable questions.
YOU MAY ALWAYS DO MORE PROBLEMS THAN THOSE ASSIGNED, AND ASK ABOUT THEM DURING CLASS OR SOME OTHER TIME.
You will find “** **” around the problems I consider most important
Sec.1.2 Functional Notation Read pp. 16 - 25 pp. 25 - 27 Nos. 1, 3, 4c, 5,
**13, 14, 15, 16, 17, 19, 20, 21**
Appendix App. A10 Nos. 27, 29, 37, 41
Sign Lines
Appendix B p. A11-A15pp. A15 Nos. 7, 21, 23, 33
Absolute Value Functions
** Throughout the course, there will be an emphasis placed upon how to deal with “piecewise functions”
Sec.1.4 SymmetryRead pp. 38 - 48 pp. 48 - 49 Nos. 31 - 37 odd, 45,
And Composite Functions 69, 75, 76, 78
Sec.1.8 ParametricsRead pp. 88 - 95 pp. 95 - 96 Nos. 1 - 9 odd, 19,
21, 36, 37
Quiz on the above sections
CHAPTER 2 Limits and Continuity (time frame: 7 days)
Sec.2.1 Limits (graphically and numerically – with calculator as an aid)
Including a discussion of asymptotes and end behavior
Read pp. 108 - 118 pp. 118-121 Nos. 1-19 odd, 25, 29
(evens for extra practice)
Sec.2.2 Computation of Limits (analytically)
- including a discussion of one sided limits, and asymptotes
Read pp. 122 - 129 pp. 129-130 Nos. 1, 3, 4, 5 - 29 EOO, ***31, 33-40***
Sec.2.3 Computing End Behavior (analytically)
- including horizontal asymptotes
Read pp. 131 – 136 pp. 136-137 Nos. 1, 3, 4, 5-25 EOO, ***29, 30, 31***
41, 43, 45
Sec. 2.5 Continuity
- definition in terms of limits
Read pp. 147 - 153 pp. 156-158 Nos. 1 - 4, 5, 9, 11, 13-23 odd, 24-30
Sec. 2.6 Continuity and Limits of Trig. Functions
Read pp. 159 - 162 pp. 153-164 Nos. 1, 5, 9, 11, 13-33 EOO, 37-43 odd
TEST Sections2.1 - 2.6 (skip sec. 2..4)Limits and Continuity
Assign slope worksheet
CHAPTER 3 The Derivative (time frame: 10 days)
Sec. 3.1 Slope of the secant line and Slope of the tangent line
(Average and Instantaneous Rates of Change)
note: be familiar with both notations
Read pp. 169 - 175pp. 175-177 Nos. 1, 2, 3, 5, 6, 7-13 odd, 15-17
Sec.3.2 Definition of the Derivative
-the limit of the difference quotient
-the relationship between differentiability and continuity
Read pp. 177 -188pp. 188-190 Nos. 9, 11, 13, 14,15, 16,18
Application/Reading Probspp. 188-190 Nos. 1-5, 7, 8, 21, 23, 25, 26, 27, 31,33, 35 (to be done after Section 3.3)
-including vertical and horizontal tangent lines
-determining the equation of tangent lines
-including graphs of f(x) and f’(x)
QUIZ Sec. 3.1 and 3.2 Derivatives of a polynomial, a rational function, a radical by using the definition of the derivative
(work must be shown!!)
Sec3.3 Differentiation Techniques
Read pp. 191-197Power Rule pp. 198-199 Nos. 1-14 all, 33-36
Product Rule Nos. 15-20, 71, 73, 74
Quotient Rule Nos. 21-26
Combined Rules Nos. 27, 28
Higher Order Nos. 45-49 odd
Application/Reading Problems
pp. 199-200 Nos. 37 - 43 odd, 51, 55, 57-61 all
Differentiabilityp. 190 Nos. 41-44
p. 200 Nos. 75-78 and worksheet
-including worksheet on the “Mean Value” theorem for derivatives
and applications of this theorem (“EZ pass system”)
Sec.3.4 Differentiating Trig. Functions
Read pp. 200-202pp. 202-203 Nos. 1-23 odd (at least)
Application/Reading Problems pp. 199-200 Nos. 31, 32, 33
Sec. 3.5 The Chain Rule (for composite functions)
Read pp. 204-208pp. 209-210 Nos. 7-39 EOO, 51 (at least)
Application/Reading Problems p. 209 Nos. 1, 3, 5,43-47 odd, 69, 71, 73
QUIZ Sec. 3.3, 3.4, 3.5Power, Product, Quotient, Chain Rules, Algebra to simplify. Derivatives of Trigonometric Functions
NO APPLICATION/READING PROBLEMS
APPLICATION PROBLEMS (time frame: 9 days)
Sec. 3.6 Implicit Differentiation
Read pp. 211-217pp. 217-218 Rational Exponents Nos. 1-7 odd
Implicit Diff Nos. 9 - 25 odd, 43
- including higher order derivatives
Sec.3.8 Local Linearity (Differentials)
Read pp. 227-232pp. 232-233 Nos. 13, 15, 22, 23, 24-26 all, 29, 45a,
46a, 47a, 48a, 50a
TEST Sec. 3.6 & 3.8Implicit Diff., Rational Exp., Linear Approx
Sec 3.7 Related Rates
Read pp. 219-223 pp. 223-226 Nos. 1, 3, 5, 7, 8, 9, 10, 11-19 odd, 21-
25, 27-37, 39
- Students are encouraged to research and to share a non-traditional problem.
TEST Section 3.7
CHAPTER 4 (time frame: 9 days)
INVERSE , EXPONENTIAL , and LOGARITHMIC FUNCTION
Review Sheet on logs and exponents
Sec. 4.1 Inverse Functions
Read pp. 242-250pp. 250-252 Nos. 1a, c, 3, 5a, c, 9, 11a, b, 13, 19, 23,
35, 37, 45, 47
- including two ways of finding the derivative of an inverse function
Sec. 4.2 Exponential and Logarithmic Functions
Read pp. 252-260Properties of…
p. 260 Nos. 1 - 15 odd at least
p. 261 Nos. 16 - 33, 42, 48
Sec. 4.3 Differentiation of Exponential and Logarithmic Functions
Read pp. 262 - 267p. 267 Nos. 1 - 33 odd at least, 47, 49, 51
(Logarithmic Differentiation)pp. 267-268 Nos. 35 - 47 odd
Sec. 4.4 Differentiating Inverse Trig. Functions
Read pp. 268-273 p. 273-274 Nos. 5, 13, 23-31 odd, 35, 37
TEST Sec. 4.1 - 4.4 Properties & Derivatives of Inverse, Exponential and Logarithmic Functions
Sec. 4.5 L’hopital’s Rule
- including an analysis of relative growth rates of functions
Read pp. 276-285p. 283 Nos. 1 - 35 odd, 41, 43, 45, 51
TEST Sec. 4.5 L'Hopital’s Rule
Chapter 5 (time frame: 15 days)
FUNCTION ANALYSIS
Sec. 5.1 Increase/Decrease, Concavity
Read pp. 290-297 pp. 297-300 Nos. 1, 2, 3, 6, 7, 9, 15, 23, 56,
**33, 63**
Sec. 5.2 Relative Extrema
Read pp. 300-305 pp. 305 Nos. 1, 3, 5, 7, 9 **13, 15, 17, 19 ** 21, 35,55
Sec. 5.5 Absolute Extrema
- including the “Extreme Value” theorem
Read pp. 332-339 pp. 339 Nos. 1, 5, 10, 15, 24 ** 31, 33 **
Sec. 5.3 Curve Sketching
Read pp. 308-319 pp. 320 polynomial function, rational function, piecewise function, function involving an absolute value, function involving a rational exponent, exponential/logarithmic function
TAKE-HOME ASSIGNMENT
In Class TEST Sec. 5.1-5.3
OPTIMIZATION PROBLEMS
Sec. 5.6 Applied Extrema Problems
Read pp. 341-350 pp. 350-353 Nos. 1, 4, 5, 6, 8, 10, 11, 15, 17, 19, 20, 21, 22, 29, 35, 39, 41, 42, 49, 50, 54, 55, 56, 62
p. 340 No. 44, 50
- students are encouraged to research and to share a non-traditional problem.
Test on Applied Extrema
Sections 2.5 and 5.8 “Named Theorems”
INTERMEDIATE VALUE THEOREM (page 154)
INTERMEDIATE VALUE THEOREM Corollary(page 154)
ROLLE'S THEOREM (page 360)
MEAN VALUE THEOREM (page 360)
Sec. 2.5 Read pp. 153 - 156 p. 158 Nos. 42, 43
Sec. 5.8 Read pp. 359 - 362p. 363-364 Nos. 1 - 17 odd, 18, 20, 24, 26, 29
Quiz on Named Theorems
Chapter 6 INTEGRATION (time frame: 16 days)
Sec. 6.2 Antiderivatives
Read pp. 377-384pp. 385-386 Nos. 7-20 all
pp. 385-386 Nos. 21-33 odd, 53, 54, 55, 56 (Trig)
p. 386 Nos. 39-44, 46-49 odd, 57 p. 456 Nos. 31-34
(Initial Value Problems)
Sec. 6.3 “U-substitution”
Read pp. 387-392-pp. 392-393 Nos. 1 - 6, 7-52 odd at least
p. 393 Nos. 61-64, 67, 68 (Initial value Problems)
Test Section 6.2 & 6.3 Antidifferentiation
Sec. 6.1 Intro. To Summation
Read pp. 372-377p. 377 Nos. Using n = 4 do problems #5 (left)
#6 (right)
#7 (midpoint)
Sec. 6.4 Continue Summation using Formulae
Read pp. 394-404p. 404 Nos 17, 23 Using n= 100 do #38,41
(both left and right)
Sec. 6.5 Riemann Sums and Integrability
Read pp. 406-410p. 405#38,39,42 p.414 #9,10
p. 414 # 11-15 (using geometry)
p. 414 #16-22 (properties of definite integrals)
Worksheet (Expressing sums as a definite integral) -- over the next week!
Sec. 6.6 The Fundamental Theorem of Calculus
Read pp. 415 - 429
Evidence worksheet of connection between definite integral and Sum
Derive part I
p. 424 Nos. 1-25 odd, 33, 34, FTC Part I
day 2: Applications (velocity/ total distance)
(Accumulation -- stock market -- Net signed area!! )
do p. 426 #59,60 (MVT for integrals)
p. 425 Nos. 29a, 30a, 31a,b
p. 458 Nos. 7, 8 p. 460 # 35
What does the definite integral yield? the indefinite integral?
Sec. 6.8 Definite integrals using u-substitution
Read pp. 440 - 443p. 444 Nos. 9, 10, 19-22, 45, 46
p. 444 Nos. 1-18 optional
Practice!!
Sec. 6.6 Fundamental Theorem Part II
- including functions “defined by” integration
Read pp. 422-424p. 426 Nos. 47-58
p. 455 No. 13, 14
p. 445 Nos. 59, 60, 61
derive Part ii and discuss composite functions!
- Students are encouraged to research and share a non-traditional problem.
Sec. 6.9 Logarithmic and Exponential Functions
Read pp. 446 - 455p. 456 Nos. 15-26 odd at least
p. 460 Nos. 42, 44, 46, 47
Test Sec. 6.5 – 6.9
Applications of the Definite Integral (time frame: 16 days)
Sec.6.6 (More on Average Value)
Read pp. 421-422p. 426 Nos. 59, 60 (MVT for integrals revisited)
pp. 444-445 Nos. 29-32, 60 p. 439 Nos. 55, 57
p. 461 No. 50
Area & signed Areap. 425 Nos. 35, 39-43, 45
pp. 444- 445 No. 23-28, 62, 63
Sec. 7.1 Area:
Read 468-472p. 473 Nos. 1-4, 7, 11, 13, 15, 23, 25, 27, 33, 37, 39
Test on avg value, accumulation, and area
Sec. 7.2 Volumes using disks
Read pp. 474-479p. 480 Nos. 1-4, 5, 9, 13, 16 (about the x axis)
Nos. 17,23,25 (about the y axis)
***29-34 all***(about other axes)
Nos 41-44 (with calculator)
Volumes by slicingp. 480 Nos. 35-40
Sec. 7.3 Volumes using Shells (not an A.P. topic)
Read pp. 482-485p. 485 Nos. 1-4, 5, 7, 9, 11, 13 (x and y axes),
***20, 21***(other axes)
Sec. 8.7 Approximation Techniques and error bounds
- use of graphing calculator to assist in approximation
Read pp. 565-576p. 576 Nos. 1, 3, 5 (using midpoint and
trapezoidal rules)
Nos. 7,9,11 (determining error bounds)
Sec. 7.4 Arc Length
Read pp. 487-490p. 490 Nos. 1, 3, 5
Sec. 7.5 Surface Areap. 494 #3-17 odd, 21,23
Read pp. 491-494
TestSec. 7.2 - 7.5 and 8.7
Sec. 5.4 Rectilinear Motion (Differentiation)
Read pp. 322-329position, velocity, speed, acceleration
Use of parametric mode to model such motion
When is a particle “speeding up”/ “slowing down”
pp. 329- 332 Nos. 1, 3, 7, 11, 13, 19, 21, 23, 35, 37
Sec. 6.7 Rectilinear Motion (Integration)
Read pp. 427-434distance and displacement
pp. 436-439 Nos. 1-17 odd, 27, 65-68
p. 445 No 61 p. 461 No 49
note: switching text to Berkey and Blanchard 3rd ed. for vector- valued functions ( Chapters 15-17) (time frame: 9 days)
** This is because we use the Anton books in our following course to discuss this topic and we don’t want too many of the same problems.
Parametrics
Sec. 15.4 The Calculus of Parametric Functions
Read pp. 745-753pp. 755-757 Nos. 1, 5, 9, 11, 13, 15, 17, 31-37odd,
45,46,53
Sec. 15.5 Arclength and Surface Area involving Parametrics
Read 758-761p. 763 Nos. 3, 7, 11, 17, 19, 27
Sec. 16.1 General Vector info.
Read pp. 767 - 780 Definition of a vector: (as opposed to a scalar)
Vector notation(s) <a,b> or ai + bj
(Physics – Review?)Formula for magnitude of a vector ||v||
Analyzing a vector in terms of its x, y (,and z) components
“Normalizing” a vector (making it one unit long)
Resultant vectors
p. 781Nos. 1, 5, 9, 11, 15, 17, 21, 36, 39, 43
Sec. 16.2 Intro to “vector-valued” functions – Lines in space
Read pp. 782 - 788 x = x0+ at y =y0 + bt where (x0 ,y0 ) is a point the line passes through and the line parallel to the non-zero vector <a,b> .
Note: t is a multiplier that will determine the points that are being traced.
An equivalent form of such a line would be the vector form of the equation:
r(t) = <x0,y0> + <a,b> t
p. 788Nos. 1, 3, 5, 7, 9, 17, 19, 21, 23, 25, 29, 33
Sec. 17.1 More on Vector Valued Functions – Curves in space
Read pp. 814 - 821 p. 821Nos. 9, 11, 13, 15, 17, 29, 33, 37,42, 46
Domain, notation, continuity
Sec. 17.2 The Calculus of Vector Valued Functions
Read pp. 823 - 829 p. 829Nos. 1, 3, 7, 9, 13, 21, 23, 31, 35, 37, 41, 43
Determine where a vector valued function is differentiable
Determine the derivative of a vector valued function
Determine the integral of a vector valued function
(indefinite and definite)
Sec. 17.3 Tangent Vector and Arc Length along a curve
Read pp. 831 - 839 p. 839Nos. 1, 3, 5, 7, 9
Determine the vector tangent to a given curve at a given point
Determine the unit vector tangent to a given curve at a given point
Sec. 17.4 Position, Velocity, and Acceleration Vectors
Read pp. 840 - 847 p. 847Nos. 1- 17 odd, 22
Anton p. 909 Nos. 25, 27
Determine the velocity, acceleration vectors given the position vector.
Determine the position vector given the acceleration, velocity vectors
Determine the distance and displacement traveled by a particle traced
by a vector valued function.
Test Berkey Chapters 15-17 (only the above sections)
Chapter 12 and 13 (Extra Practice problems)
Vectors and Vector Valued Functions
** This group of problems could be used as an alternative within the Anton book
Anton 7th Ed
Sec. 12.2 Read pp. 795-801 p. 804 Nos. 1, 5, 7, 9, 11, 13, 17
Vectors
Sec. 12.5 Read pp. 825-828p. 829 Nos. 3, 5, 9, 11, 15
Parametric Equations of Lines
Sec. 13.1 Read pp. 860-863p. 864 Nos. 1, 3, 6, 8, 9
Introduction to Vector Valued Functions
Sec. 13.2 Read pp. 866-870, 872-873p. 874 Nos. 1-7 odd, 11, 13, 23-29 odd, 35, 39, 41-49 odd
The Calculus of Vector valued Functions
Sec. 13.6 Read pp. 899-901 p. 908 # 1-17 odd 25,27
Position, Velocity, and Acceleration Vectors
CHAPTER 8 Other Integration Techniques(time frame: 7 days)
Sec. 8.2 Integration by Parts
Read pp. 526 - 533p. 533 Nos. 1, 5, 10, 13, 15, 17, 25, 29,
AP Exam Review ***47-51***
Sec. 8.3 More complex Trigonometric Integrals
Read pp. 534 - 540 p. 541 Nos. 4, 5, 17, 24, 25, 26, 27, 30,41
Sec. 8.5 Partial Fractions
Read pp. 548 - 555 p. 554 Nos. 9, 11, 13, 19, 22, 25, 27, 28, 33-36
Test 8.2 ,8.3, 8.5 Parts, Partial Fractions, Trig integrals
CHAPTER 9 Integration, Slope Fields, and Euler's Method
(time frame: 7 days)
Sec. 9.1 Separable Differential Equations
- including initial value problems
Read pp. 600-602p. 605 Nos. 7, 8, 15-23 odd, 27, 29, 31, 33, 34
Sec. 6.2Slope Fields
Read pp. 384-385p. 386 Nos. 35-38
Sec. 9.2 Slope Fields and Euler’s Method
Read pp. 608-613p. 613 Nos. 1, 3, 4, 9
p. 613 Nos. 13, 14, 20
Sec. 9.3 Modeling w/ Differential Equations
Read pp. 615-622p. 622Exponential Growth Nos. 1 - 14
Logistic Equation Nos. 21-26
Newton’s Law of Cooling Nos. 29-31
Quiz – Differential Equations and Slope Fields
CHAPTER 10 Sequences and Series (time frame: 23 days)
Background needed for Series
Sec. 8.8 Improper Integrals
Read pp. 579 - 585 p. 585Nos. 1, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 27, 29
p. 586Nos. 46, 47, 48, 49, 50, 51 (comparison
test)
Sec. 10.2 Sequences (a function whose domain is the set of whole numbers)
Read pp. 647 - 656
Pay particular attention to the definition of sequence, the notation used, and the convergence or divergence of sequences.
even/odd terms must approach limit for sequence to converge
"squeezing theorem"
pp. 656-657 # 1-30
Sec. 10.3 Monotone Sequences
mention the definition of strictly monotone, and least upper bound,
greatest lower bound.
(Pay particular attention to theorem 10.3.3 and 10.3.4)
eventually increasing/decreasing (first few terms don't necessarily have
to follow the pattern of the "tail")
Read pp. 658-662
no written assignment
Sec. 10.4 Infinite Numeric Series
Read pp. 664-670
Carefully define : series, partial sum, sequence of partial sums,
convergence and divergence of a series, interval of convergence.
Develop the idea: begin with writing repeating decimals as a fraction, square division
Writing a series...
in open form: a1 + a2 + a3 + ...+ ak +...
in closed form:
Sum of a geometric series
List the "partial sums" of such a series
Telescoping series
Harmonic series
pp. 670-671 #1-21 odd,28
24,25 (interval of convergence)
Sec. 10.1 Taylor and MacLauren Polynomials (Approximating a function using a polynomial)
Read pp. 638 - 646
Revisit Geometric Series mapping successive approximations against graph
of f(x) = 1/(1-x)
"Building" polynomial representations of a function
local linearization
local "quadratization"
include the local nth degree polynomial
Use of calculator to verify that each successive partial sum (Taylor polynomial) yields a better approximation of the function (for a particular interval of convergence)
f(x) = a Use of ratio test to determine "interval of convergence"
1 - x
then relate to p. 646 #12, 11
Analysis of successive Taylor polynomials for the function f(x) = sinx
Derive the general case Taylor/ Maclauren Approximations
Other "key" functions:
F(x) =
How to get other MacLauren polynomials from these "key" polynomials
p. 646 - 647 #7-16
p. 646 - 647 #17-23odd
(following instructions for #27-30)
Discuss the Upper Bound on the Error (Remainder) a Taylor Polynomial approximation will yield:
(Remainder Estimation Thm. - Lagrange Error Bound)
Which Pn is needed to be accurate to a certain decimal place (for a given function)?
pp. 646 - 647 #31,32,38
Sec. 10.10 Deriving One Series from Another
Read pp. 712-715
More use of substitution, differentiation, and integration to get one series from another.
pp. 718-719 #5,9,11,19,27,33
Sec. 10.8 MacLauren and Taylor Series; Power Series
Read pp. 693 - 695
MacLauren/ Taylor Polynomials with an infinite number of terms are called MacLauren/ Taylor Series
Note: the difference between a series as described in section 10.4
and power series in this section
Question: For which values of x will the series converge? (called the
convergence set or the interval of convergence)
note: It is important to pick a Taylor polynomial for which x0
is close to x to assure the series converges.
p.700 #1 -24
Sec. 10.9 Taylor Series (Remainders)
The sources of "error":
1)error which may occur because you are not within the interval of
convergence
2)error which may occur by stopping with a "partial sum" (the error is
the sum of the "tail")
3) rounding error
Objective #1: Using a partial sum to represent the sum of an infinite series,
determine an upper bound on the error at x
1) Use of the Remainder Estimation Thm (LaGrange remainder)
notation: (polynomial of degree n + LeGrange remainder "after nth degree term")
or Pn(x) - B < f(x) < Pn(x) + B
2) Use of n+1st term of an alternating series (will be considered later)
Is there a relationship between these two "error bound" techniques?
Objective #2: Given a particular tolerance , find the degree polynomial needed
to be within this range
Objective #3: To determine the maximum value for fn+1(c)a<c<x
note: necessary when (n+1)st derivative is varying.
pp. 709 #9-15 odd
Test on Series (nonconvergence – except for geometric series) -- part #1
Sec. 10.8 Convergence
Read pp. 695 - 700
Begin with Example #3a (relate to 24,25 from 10-4)
Prove the Ratio Test
Then, do the rest of Example #3 (d is where alt. harmonic is introduced) ,
#4 (p-series review.)
refer to convergence test sheet. (begin to memorize)
Note: Now is the time to memorize the "key" series (see compiled version or "know cold" sheet) -- if you haven't already done so.
pp. 700-701 #25-49 odd
Other ways to test numeric series convergence and the convergence of “endpoints of an interval”
Sec. 10.5 Convergence Tests - Divergence Test, Integral Test, p-series Test
- linking the idea of Riemann Sums to Series and the use of the integral test
Read pp. 672-677
pp. 677-678 1-23 odd
Sec. 10.6 Convergence Tests -- Comparison Test, Limit Comparison Test,
Ratio Test for Absolute convergence, Root Test
Read pp. 679-683
pp. 684 #1-37 odd
Sec. 10.7 Convergence -- Alternating Series, Absolute Convergence, Conditional Convergence