Calculus AP  C materials (Draft 12-29-03)
CALENDAR SCHEDULE by week.
 1 Introduction & Review  overview of DE's Ch. IV More review.  Differential equations and Direction Fields IV.D Euler's Method  IV.E
 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 ex. 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 Taylor Theory I.IXA Taylor theory II IXA..  IXB  Applications: Definite integrals and DE's.
 5 Taylor theory III. IXB MacLaurin Polynomials Taylor theory III. IXB  Taylor Theory for remainder proven! IX.C  Taylor Theory derivatives, integrals, and ln(x).
 6 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.
 7 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
 8 Taylor Series convergence.X.B1_4  Theorem on RnSeries 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

Old schedule... for reference only!
 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 ex. 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 RnSeries 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.
 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

 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

 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) = ∫1t 1/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)