Midterm 1: Limits & Derivatives

1.    Knowhow to compute limits algebraically.

First, try substitution and see what limit type you get. It may be:

o A determinate type of limit, which you can compute directly, by substitution.

(Recall that the following types of limits are determinate:  "  " = 0, "  "  = +∞, "  " = −∞ )

To do this step correctly, you’ll need to know all your basic functions (powers,rational, exp, log, trig and inverse trig fcts), in particular their special values, asymptotes, and graphs (including behavior    when x → ±∞)

o An indeterminate type, such as  0    ∞    ∞ − ∞  0 ∙ ∞ . In this case try an appropriate algebraic

technique to compute the limit. (Show all steps and don’tdrop the “lim” until you’ve evaluated the limit!)

o Or, the limit may not exist. Say so, and explain why the limit does not exist. For instance, compute the one-sided limits and show they are not equal.

Some techniques we employed to compute indeterminate limits (for what types of limits do each work?)

o Rationalize and /or simplify algebraically. Depending on type: factor, rationalize, bring to common denominator, make a product into a quotient, etc.

o Divide by highest power in the denominator (when exactly is this technique useful?)

o The Squeeze Theorem (when is this technique useful?)

o Know thatx(li)  = 1. Be able to use this to compute limits like those in §3.3.

2.    Knowhow to determine the horizontal and vertical asymptotes of a function, if any. What’s the difference?

Make sure to compute the appropriate limits and to know what conclusion to draw, based on value of the limits.

3.    Understand and be able to determine the continuity of a function at a point. What ingredients do you need? Know the discontinuity points of basic functions (domains, asymptotes, etc)

Be able to determine if a function is continuous and where it is discontinuous (and what type of discontinuity):

o from its graph

o from its formula

4.    Understand the limit definition of the derivative. Be able to use it to compute derivatives

(only if specifically asked to do so -- otherwise you can use the rules of differentiation)

5.    When is a function differentiable at a point (or on an open interval)? When is it not? For what reasons?

6.    Understand and be able to apply the interpretations of the derivative f ′ (a) as:

o  Instantaneous rate of change off(x) atx  = a. Units?

(Iff(t) is distance or displacement vs. time, how do you compute velocity? Acceleration?)

o  Slope of tangent line to the graph off(x) atx  = a.

Know how to compute the slope of a secant line.

Understand that it corresponds to an average rate of change, and how it relates to the slope of a tangent.