EHS AP C Materials Sp, 04

Calculus AP C materials (Draft 12-29-03)

Calculus web sites for surfing.
Calculus @UTK Courses at the Univ. of Tenn. that use the Stewart text book (Calculus: Concepts and Contexts) They provide some useful supplementary materials for Math 109,110,210.
Visual Calculus (Univ. of Tenn.) calculus (first year) tutorials, etc.
Online Mathematics Tutorials at Harvey Mudd College! ...many excellent tutorials...Calc I-III, DE's, Linear Algebra.
Live-Math interactive on Calculus and Differential Equations files using Live-Math plug-in.
Calculus@Internet Many references to resources.
Winplot from Rick Parris.
Notes for Winplot authored by Al Lehnen (Madison Area Technical College in Madison, Wisconsin)
Applied Calculus
WWW Interactive Mathematics Server- on-line and popup calculators for calculus!
SOS Mathematics - Calculus materials
Math 1002 calculus at University Of Aberdeen: Calculus Notes from Ian Craw.
Professor Biles' (HSU) Introductions to WinPlot, X(PLORE), and MSExcel

CALENDAR SCHEDULE by week.

Introduction & Review overview of DE's Ch. IV

More review.
Differential equations and Direction Fields IV.D

Euler's Method IV.E

More euler's method and Direction Fields.

estimate from (1+1/n)ⁿ.
Begin 7.2 and Models for (Population) Growth and Decay:
y' = k y; y(0)=1. k = 1.
The exponential function.VI.A

Applications to graphing.

More on the relation between the DE y'=y with y(0)=1 and e^x.

Quick summary I of exponential function.
The natural logarithm function:I.F.2

y = ln (x) and ln(2)
ln|x| and integration of 1/x.

Models for learning.
y' = k / x; y(1)=0. k =1
VI.B

Taylor Theory I.IXA

Taylor theory II IXA..
IXB
Applications: Definite integrals and DE's.

Taylor theory III. IXB MacLaurin Polynomials

Taylor theory III. IXB
Taylor Theory for remainder proven!

IX.C
Taylor Theory derivatives, integrals, and ln(x).

More on finding MacClaurin Polynomials & Taylor theory.IX.D

Begin Sequences and series .

Geometric sequences
12.1 & X.A
Sequence properties.

Use of absolute values
Incr&bdd above implies convergent.

geometric series
Series Conv. I

Geometric and Taylor Series. The divergence test.
Harmonic Series.
Series Conv. II

Series Conv. III Positive series & Integral test.

Alternating Series [12.5]
Trig Integrals 8.2
I sin&cos

Taylor Series convergence.X.B1_4
Theorem on R_nSeries Conv. IV
Trig Integrals 8.2 II sec&tan

Series Conv. V
Positive comparison test [12.4 ++]?
Ratio test for Positive Series X.B5

Series Conv.VI Absolute conv. & conditional: The General ratio test:
Power Series I XI.A

L'Hospital's rule I [7.7]
Power Series II (Interval of convergence)XI.A

Week	Monday	Tuesday	Thursday	Friday
1	Introduction & Review	More review. Differential equations and Direction Fields IV.D	Euler's Method IV.E	More Euler's Method
2		More euler's method and Direction Fields.	estimate from (1+1/n)ⁿ. Begin 7.2 and Models for (Population) Growth and Decay: y' = k y; y(0)=1. k = 1. The exponential function.VI.A	Applications to graphing.
3	More on the relation between the DE y'=y with y(0)=1 and e^x.	Quick summary I of exponential function. The natural logarithm function:I.F.2	y = ln (x) and ln(2) ln\|x\| and integration of 1/x.	Models for learning. y' = k / x; y(1)=0. k =1 VI.B
4	Integration by parts I	Connections: 7.4* VI.C ln(exp(x)) = x exp(ln(y)) = y	The Big Picture Begin Arctan.VI.D	More on Arctan.
5	More on Arctan.	Integration by parts. II 8.1 and VII.C Separation of variables.	10.3 Growth/Decay Models.10.4 Improper Integrals I	More on improper integrals Breath
6	Numerical Integration.(Linear)	Numerical Integration. (quadratic) V.D	Numerical Integration. (quadratic) V.D	The Logistic Model 10.5
7	Integration of rational functions I.VII.F	Rational functions II.	More Rational functions VII.F	Darts Probability density, mean (9.5)
8	More darts	Darts Probability density, mean	One more Meany!? Rational functions III VII.F . .	End Rational Functions Begin Improper Integrals II comparison tests?
9	Improper Integrals II	comparison tests
			Taylor Theory I.IXA	Taylor theory II IXA.. IXB Applications: Definite integrals and DE's.
10	Taylor theory III. IXB MacLaurin Polynomials	Taylor theory III. IXB Taylor Theory for remainder proven!	IX.C Taylor Theory derivatives, integrals, and ln(x).
11	More on finding MacClaurin Polynomials & Taylor theory.IX.D	Begin Sequences and series .	Geometric sequences 12.1 & X.A Sequence properties.	Use of absolute values Incr&bdd above implies convergent.
12	geometric series Series Conv. I	Geometric and Taylor Series. The divergence test. Harmonic Series. Series Conv. II	Series Conv. III Positive series & Integral test.	Alternating Series [12.5] Trig Integrals 8.2 I sin&cos
13	Taylor Series convergence.X.B1_4 Theorem on R_nSeries Conv. IV Trig Integrals 8.2 II sec&tan	Series Conv. V Positive comparison test [12.4 ++]? Ratio test for Positive Series X.B5	Series Conv.VI Absolute conv. & conditional: The General ratio test: Power Series I XI.A	L'Hospital's rule I [7.7] Power Series II (Interval of convergence)XI.A
14	Power Series III (DE's) L'Hospital II.	Conics I Intro to loci-analytic geometry issues.(parabolae, ellipses)	L'Hospital III Conics II More on Ellipse and Parabola.	Trig substitution (begin- area of circle) I (sin) VII.E
15	Conics III The hyperbolae Other Inverse Functions (Arcsin) Trig substitution II (tan)	The conics IV Trig Substitution III (sec)	Arc Length VIII.B Taylor Series 12.10	How Newton used Geometric series to find ln(.9) Proof Of L'Hospital's Rule? Hyperbolic functions: DE's, Taylor Series, Algebra and Hyperbolas. exp(pi*i) = -1
16

Problem Assignments M.FLASHMAN

CALCULUS Stewart's Calculus 4th ed'n.

**Assignments and recommended problems I** (*= interesting but optional)
DateDue:	Read:	Do:
	IV.D	Background Reality Check
	IV.D	1-11 odd [parts a and b only]	23	24
	4.10	43	45	47	48	51	52
	10.2 p 620-624 middle	(i) 2-6	9	11	*15
	10.2 p624-626	(ii) 21	23
	3.10	7, 21, 33
	IV.E	5-9 odd (a&b)
	IV.E	20	21	24
	10.2	(iii)13	14	19a
	*exponential functions* *I.F.2* *Stewart: pp 416-422*
	I.F.2	3	4
	VI.A
	7.2	(i) 29	33	34	37	47-51
	7.2	(ii)57	61	63	53
	7.2	(iii) 62	70	71-77odd	79	80	85	86
	7.3 pp 428-430 (review of logs)
	7.3 pp 428-430 (review of logs)	3-17 odd	31	33	35	41	47	59-61	*78
	VI.A	9	10	15	16
	VI.B.
	7.4 pp 435-439	(i) 3,7,9,13	25	28	8	22
	*7.2
	7.4	(ii) 15	13	35	53 (changed 9-12)
	(Log diff'n )	(iii) 45-47	52(changed 9-12)	58	*64
	(Integration )	(iv) 65 - 71 odd	74-76	80	81
	VI.B	13	14
	VI.C
	p468	19	23	33	37	51
	inverse tangent p472-3
	7.5	2a	3a	5b	16
	VI.D

**Assignments and recommended problems II** (*= interesting but optional)
DateDue	Read:	Do:
	VI.D	1-4	9-13	21	*(22&23)
	7.5 Examples 9&10	(i) 25-27	34	38	58	59
		(ii) 62	64	67	69	70	74*	75*
		(iii) 22	23	24	29	20	47	48	63	68
	Read VII.C
	8.1 (parts)	(i)1-11odd	29	30	51 (see page 390).
		(ii) 15 [oops: 21 removed 9-26]	23	25	33	41	42	45	46
	10.3 (sep'n of var's)(i)	(i) 1	3	4	7
		(ii) 9	10	15	29
	10.4 pp637-641 (growth/decay models)	(i) 1-7odd
	10.4 pp641-642	(ii)9-11
	10.4 pp642-643	(iii) 13	14	17
	10.5 (logistic model)
	Begin reading VII.F through Example VII.F.5 (rational functions)
	10.5 (oops- sorry if you thought this was VII.F)	1	5	*(11&12)
	8.7(num'l integr'n)	(i) 1	4	7a	11(a&b)	27 (n=4&8)	33a
	(simpson's method)	(ii) 7b	11c	31	32	35	36	*44
	*More help on* *Simpson's rule,etc* *can be found in V.D*
	8.8 (improper integrals) Type I only[omit ex. 2]
	8.8 type I	(i)3	5	7	8	9
		(ii)13	21	41
	8.8 type II	(iii) 27-30	33	34	37	38
	8.8 comparison	(iv) 49	51	55	*60	61	57	71
	IXA

Assignments and recommended problems III (*= interesting but optional)
DueDate	Read:	Do:
	Begin VII.F (rational functions)
	8.4	(i) 13	14	29
	8.4	(ii) 15	16	17	20	21
	VII.F	5	6	7	17
	10.5 (Again)	1	5
	8.4	(iii) 62	63	65
	8.4	(iv) 31	35	36	25
	Darts
	9.5 pp 603-607 andDarts
	9.5 pp 603-607 andDarts	1,3,4,5
	VII.F on-line check work on-line with Mathematica	1	3	7	10	14	15
	read 8.8 type II and comparison
	IXA	1	2
	IXA	3, 4	6	8	9	*10
	Read IX B
	IXB	(i)1	2	4	5	7
	IXB	(ii)	11	13	14	*23
	IX.C
	IX.C	(i) 1-4
	IX.C	(ii) 5-9
	IX.C	(iii) 12	14	16-18
	IX.D
	IX.D	1	3	5
	IX.D	8	10	14	15
	X.A
	X.A	1-3	5	7-9
	12.1 pp 727-729; examples 5-8 (sequences converge)	(i) 3-23 odd
	12.1	(ii)39-43 odd	51, 53-57	61	*63	*64
	12.2 pp 738 -741 (series- geometric series)
	12.2 pp 738 -741 (series- geometric series)	3	5	7	8
	X.B1_4
	12.2 (series- geometric series):	(i) 3	11-15		*51
	12.2 (geometric, etc)	(ii)41-45	49	50	51
	12.2 READ!!pp 742-745	(iii) 21-31odd
	12.3	(i) 1	3-6
	12.3	(ii) 9-15 odd
	X.B1_4
	12.5	(i) 2-5; 23	25	31
	12.5	(ii) 9-15 odd
	8.2 (trig integrals)	(i)1-5	7-15 odd
	8.2 More Trig Integrals	(ii) 21-25 odd	33	34	45	44	57	*59	*60	*61
	12.4 (comparison test)	(i) 3-7
		(ii) 9-17 odd
	X.B5 Ratio Test For Positive Series
	12.6 Use the ratio test for positive series to test for convergence.	2	17	23	20	29	*34
	7.7 p487 Note 3 (L'H)
	12.6	3-9 odd	19	*(31&32)	33	35

	7.7	(i) 5-11 odd
	7.7 examples 1-5	(ii) 21	27	29	15	23	18	33
	7.7 examples 6-8	(iii) 39-43 odd	47-51 odd
	7.7 examples 9-10	(iv)	55	57	63	*96	*97
	XI.A and 12.8	3-11 odd
	12.9 read Ex 1-3,5-8
	12.9	(i)3-9 odd	25	29	34	*39
	11.6 : pp 709-10 (thru ex.3)
	11.6 : pp 709-10 (thru ex.3)	(i) 1-7 odd	27	29
	12.9	(ii) 13	14	21	27
	8.3 (trig subs) (i) pp 517-519 middle
	8.3 (i) VII.E	2	4	7	11
	8.3 (ii) pp 519-520	3	6	19	9
	8.3 (iii) pp 521-522	1	5	21	23	27	29
	7.5 pp469-473	23	61
	Ch 8 review problems p568	1-11 odd	33	35

Date Due	Read:	Do:
	11.6: pp709-11
	11.6: pp 711-12	(ii) 11-14	31	33
	11.6	(iii) 19-22	37	39	47	*50
	12.10 Read only pp785-792
	12.7 Review of convergence tests	1-11 odd
	Read 12.11
	12.10	31	35	56	41	57	58
	9.1 through p 578
	9.1	1	3	19	21
	9.2	5	7	9
	9.5	1	3	7*

Final Topic Check List [This needs to be revised to match AP Syllabus ]Core Topics are italicized.

The Transcendental Functions.
The Natural Exponential Function.    Basic Properties
The Natural Logarithm Function.
    L(t) = ∫₁^t1/x dx:
         Basic properties of L(t) = ln(t) = LOG(t) .
        "inverse" relation between ln and exp.
        Applications of LN .
                --Logarithmic Differentiation.
                --Functions with exponents: a summary.
The Trigonometric Functions.
    The Inverse Trigonometric Functions and Their Derivatives.
    The Trigonometric Functions and Their Derivatives.
    Integration of Trigonometric Functions and Elementary Formulas.
Differential Equations and Integration
   Tangent Fields and Integral Curves.
   Numerical Approximations.
            Euler's Method.
            Midpoints.
            Trapezoidal Rule.
            Parabolic (Simpson's) Rule.
Integration by Parts.
   Integration of Trigonometric Functions.
   Trigonometric Substitutions.
   Integration of Rational Functions.
            Simple examples. Simple Partial fractions.
   Separation of Variables.

Applications: Probability: distributions, density, mean
Improper Integrals: Extending the Concepts of Integration.
Integrals with noncontinuous functions.
Integrals with unbounded intervals.

L'Hopital's Rule: 0/0 inf/inf inf - inf 0*inf 0^ 0 1^inf

Taylor's Theorem.
Taylor Polynomials. Calculus.
Using Taylor Polynomials to Approximate: Error Estimation.
      Derivative form of the remainder.
      Approximating known functions, integrals
      Approximating solutions to diff'l equations using Taylor's theorem.

Sequences and Series: Fundamental Properties.
Sequences.
Simple examples and definitions: visualizing sequences.
         How to find limits.
         Key theory of convergence.
             The algebra of convergence.
             Convergence for monotonic sequences.
Geometric series. Harmonic series. Taylor approximations.
Theory of convergence (series).
     The divergence test.
     Positive series.
          Bounded convergence tests.
           Integral tests.
           Ratio test (Part I).
           Absolute convergence.
             Absolute convergence implies convergence.
     Alternating Series Test.
     Ratio test (Part II).

Power Series: Polynomials and Series.
The radius and interval of convergence.
Functions and power series [derivatives and integrals].

Analytic Geometry, the Conic Sections
The Conic Sections as Loci.
Equations for conics centered at (0,0) and at (a,b)