Midterm I Review for Math 126, Fall 2009
Topics we covered.
- Limits
- Motivation: the problems of defining instantaneous velocity and
tangent lines
- Idea of a limit
- what the function is "tending towards"
- but the limit does not care what the function does at
exactly that point!
- left- and right-hand limits: idea and the fact that both
must exist and be equal for the two-sided limit to exist.
- Strategy for computing limits:
- try plugging in the endpoint, see what you get:
- if it is 0/0, must do more work
- 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
- if it is x/0, where x≠0, then the answer
might be ±∞, if both one-sided limits are equal
- if necessary, do algebra to make plugging in give a nice
answer, typically using commong denominators and/or multiplying
some part by its conjugate
- quote the appropriate Limit Laws during your work
- the ε-δ definition of a limit
- identifying (creating) all the usual pieces in a picture
showing a limit
- finding the δ from such a picture
- proving the limit with this definition, at least for
simple (linear) functions
- Continuous functions
- the definition
- the (three) things which can go wrong to make a function
discontinuous
- the Intermediate Value Theorem -- also, using it e.g., to
find roots
- Derivatives
- the definition(s), different notations
- instantaneous velocity — speed
- slope of the TL, finding the equation of a TL
- computing the derivative from the limit definition
- the (three) things that can go wrong to make a derivative not
exist
- graphing a function and its derivative, the relationship
- computing the derivative with differentiation formulæ
- the sum and difference rules
- the constant multiple rule
- derivatives of constants
- the power rule
- the product rule
- the quotient rule
- derivatives of sin and cos (and, possibly,
the other four trig functions)
Jonathan Poritz
(jonathan.poritz@gmail.com)