Midterm III Review for Math 126, Fall 2011

Topics we covered.

  1. Antiderivatives ["AD"s]:
    1. the definition (and idea)
    2. they all differ one from the other by only a constant
    3. when to write "+C" (if asked for "the most general AD") and when you don't have to (if asked for "an AD", "a particular AD", or "any AD").
    4. the CMR for ADs
    5. the +rule for ADs
    6. the power rule for ADs
  2. sigma notation
    1. basic meaning
    2. rules for manipulating sums (e.g., the CMR, the +Rule, ...)
    3. some general formulæ for sums which come up frequently:
      1. Σni=1 c = c n
      2. Σni=1 i = n(n+1)/2
  3. definite integrals
    1. basic idea: to compute the area under a curve (for a general integrand), or the distance traveled (if the integrand is a velocity function).
    2. notation
    3. a formula in terms of a limit of Riemann sums -- what the terms in the formula mean and how to calculate (in simple situations)
    4. consequences of the formula:
      1. computes signed area, not just area
      2. sum rule for definite integrals
      3. CMR for definite integrals
      4. formula for the definite integral of a constant function
      5. what happens to the definite integral when the bounds of integration are switched
      6. expression of an integral from a to b as a sum of one from a to c and one from c to b
      7. comparison properties of definite integrals
  4. the Fundamental Theorem of Calculus, in versions as FTC1 and FTC2.
    1. to compute values of a definite integral, when you know (or can figure out) an AD of the integrand
    2. to compute derivatives of functions built out of definite integrals to varying endpoints
  5. indefinite integrals — a table of such
  6. the substitution rule
    1. looking for a piece which is a function of a function, so the inside part can be called u
    2. computing du in terms of dx.
    3. changing the whole integrand to be in terms of u only, with no more x's
    4. changing the endpoints to their values for u if working on a definite integral

Things to be able to do:

  1. compute ADs for fairly simple functions from the AD rules
  2. solve Initial Value Problems, such as for position/velocity/acceleration of some object
  3. recognize, create, manipulate, and evaluate expressions using the Σni=1 ai notation
  4. interpret and create formulæ with Riemann sums
  5. calculate simple Riemann sums when n is fairly small, discuss if they give an over- or under-estimate of the actual integral.
  6. compute simple definite integrals from simple geometry and the "signed-area" interpretation
  7. use the FTC to compute more complex definite integrals
  8. use the FTC to compute derivatives of integrals
  9. use the substitution rule to compute definite and indefinite integrals