Midterm I Review for Math 126, Fall 2009

Topics we covered.

1. Limits
   1. Motivation: the problems of defining instantaneous velocity and tangent lines
   2. Idea of a limit
      1. what the function is "tending towards"
      2. but the limit does not care what the function does at exactly that point!
      3. left- and right-hand limits: idea and the fact that both must exist and be equal for the two-sided limit to exist.
   3. Strategy for computing limits:
      1. try plugging in the endpoint, see what you get:
         1. if it is 0/0, must do more work
         2. if it is a well-defined number, that is the limit if the function is a polynomial, rational function, and possibly including trig functions and non-integral powers
         3. if it is x/0, where x≠0, then the answer might be ±∞, if both one-sided limits are equal
      2. if necessary, do algebra to make plugging in give a nice answer, typically using commong denominators and/or multiplying some part by its conjugate
      3. quote the appropriate Limit Laws during your work
   4. the ε-δ definition of a limit
      1. identifying (creating) all the usual pieces in a picture showing a limit
      2. finding the δ from such a picture
      3. proving the limit with this definition, at least for simple (linear) functions
2. Continuous functions
   1. the definition
   2. the (three) things which can go wrong to make a function discontinuous
   3. the Intermediate Value Theorem -- also, using it e.g., to find roots
3. Derivatives
   1. the definition(s), different notations
   2. instantaneous velocity — speed
   3. slope of the TL, finding the equation of a TL
   4. computing the derivative from the limit definition
   5. the (three) things that can go wrong to make a derivative not exist
   6. graphing a function and its derivative, the relationship
   7. computing the derivative with differentiation formulæ
      1. the sum and difference rules
      2. the constant multiple rule
      3. derivatives of constants
      4. the power rule
      5. the product rule
      6. the quotient rule
      7. derivatives of sin and cos (and, possibly, the other four trig functions)