[ sketchy schedule for 21-122 ]

===== Week 1 =====

5.5 Review substitution rule
  From composition or chain rule
    d (f o g) = d_x f(g(x))
              = df (g(x)) dg(x)
  integrate to
    f*g = Int f(g(x)) dx
        = Int df(g(x)) dg(x)
        = Int df(g) dg
  or
    Int F(u) du = F(u(x)) |_x

7 Techniques of integration

7.1 Parts.
    d(u v) = du v + u dv.
  integrate to
    u v = Int v du + Int u dv
  or
    Int u dv = Int v du

Also: review limits; preview improper integrals

7.2 Trig identities
  * the unit circle:
    sin-cos
    tan-sec
    cot-csc

7.3 Trig substitution.
  #2 days
  * more unit circles
  * forward and backward substitution

===== Week 2 =====

7.4 Partial fractions.
  #2 days
  * polynomials, rational functions
  * long division and remainder theorem
  * decomposing linear factors: disctinct or repeated
  * quadratic factors:  disctinct or repeated
  * rationalizing substitutions

#7.5 Strategy. covered as review
#7.6 Tables and Computer algebra systems
7.7 Approximation.
  #2 days
  Midpoint rule.
  Trapezoid rule.
  Simpson's rule.
  Error bounds.
  * how to bound derivatives
  Summation formulae.

7.8 Improper integrals.
  #2 days

===== Week 3 =====

8 Applictions

8.1 Arclength.
#all else requires 3D calculus and vectors
  once you view everything as an integral, it all falls into place.

9 Differential Equations

9.1 Intro: what is a diff.equ.  Population models.

9.2 Direction fields, Euler's method.

9.3 Separable equations. #quickly
  Factor
    dy/dx = f(x)/g(y)
  into
    g(y) dy = f(x) dx
  Then integrate to
    g(t) = f(t) + const

9.4 Exponential growth & decay. #quickly

9.5 The Logistic equation.
    P' = k P(1 - P/P0)
  * direction fields
  * steady-state solution
  * stability

9.6 Linear equations
    dy/dx + P(x) y - Q(x) = 0
  * integrating factors

===== Week 4 =====

9.7 Predator-prey models: coupled diff.equ.'s
  * phase portraits
  * steady-state solution

17.1 homogeneous 2nd-order linear equations. 
  #2 days
    P(x) y_xx + Q(x) y_x + R(x) y = 0
  * initial conditions
  * boundary conditions

11 Sequences and series

11.1 Sequences
  #2 days
  * convergence and divergence, limits, boundedness
  Theorem: Monotone convergence.

11.2 Series
  #2 days
  * geometric, harmonic
  Theorem: divergence test

11.3 Integral test, estimates of sums

11.4 Comparison test
  #2 days
  Theorem: comparison test
  Theorem: limit comparison test

===== Week 5 =====

11.5 Alternating series
  Theorem: alternating series test
  Theorem: approximating monotone alternating series

11.6 Absolute convergence, ratio test, root test
  #2 days
  * what to do when a test fails

11.7 Testing strategies

11.8 Power series
  Theorem: Convergence on disk of radius in [0,infty]

===== Week 6 =====

11.9 Power series representation of functions
  Theorem: representation

11.10 Taylor and Maclauren series
  #2 days
  Theorem: coefficients from derivatives
  Theorem: Taylor representation
  Theorem: Taylor's inequality (for error bounds)
  Lots of examples.
  Operations on power series.

#11.11 binomial series

11.12 Applications of taylor polynomials
  * how calculators work
  * lower-order terms

REVIEW

FINAL