Calculus AP  C materials (Draft 12-29-03)
1 Introduction & Review  overview of DE's Ch. IV More review. 
Differential equations and Direction Fields IV.D
Euler's Method  IV.E
2More 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. 
3More 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 
Taylor Theory I.IXA
Taylor theory II IXA.
Applications: Definite integrals and DE's.
5 Taylor theory III. IXB MacLaurin PolynomialsTaylor theory III. IXB 
Taylor Theory for remainder proven! 
Taylor Theory derivatives, integrals, and ln(x).
6 More on finding MacClaurin Polynomials & Taylor theory.IX.DBegin Sequences and series .  Geometric sequences 
12.1 & X.A 
Sequence properties. 
Use of absolute values 
Incr&bdd above implies convergent.
7geometric 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 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