Math 161 D
Calculus I
Fall 2000

Test 1 Review Sheet

Note: Depending on your brouser, some of these symbols may not appear quite right. This review sheet is also available as a pdf file and as a postscript file.

1. Limits

   1. Interpreting
      lim
      x ® a f(x) = L,
      lim
      x ® a^- f(x) = L,
      lim
      x ® a⁺ f(x) = ¥,
      lim
      x ® -¥ f(x) = L, etc. in words.
   2. Definition

      1. be able to find a d for e with simple functions
      2. how definition make "close to" a precise notion
   3. Infinite limits and limits at infinity

      1. Relationship to horizontal/vertical asymptotes
      2. Know:
         lim
         x ® ±¥ x^-r = 0 if r > 0.
      3. Know:
         lim
         x ® -¥ a^x = 0,
         lim
         x ® ¥ a^x = ¥ if a > 1 (and vice versa if 0 < a < 1).
      4. Know: If
         lim
         x ® a f(x) ¹ 0,
         lim
         x ® a g(x) = 0 (both limits existing), then
         lim
         x ® a^- f(x)/g(x) (also the right-hand limit) equals either ¥ or -¥. Be able to reason out which it is.
   4. Finding a limit (one-sided, two-sided)

      1. Numerically (assuming it exists)
      2. From the graph
      3. Proving existence and finding the value

         1. Via limit laws (and common algebraic techniques)
         2. Via continuity
         3. Via the Squeeze Theorem
2. Continuity

   1. Definition of continuity; definition of continuity from the left/right
   2. Various functions that are continuous (see Thms. 4, 5, 7 and 9, pp. 125, 127, 128)
   3. Intermediate Value Theorem

      1. Be able to state it, draw pictures to represent it, use it
      2. Understand its purposed as an ``Existence Theorem" (i.e., provides a basis for faith in a solution but doesn't show how to find one)
   4. Differentiability implies continuity (Thm. p. 169)
3. Derivatives

   1. Relationship between slope of secant line and slope of tangent line
   2. Relationship between average rate of change and instantaneous rates of change
   3. Formal definitions of

      1. Derivative (see boxes 2, 3 on p. 156) [Be able to both state it and explain it.]
      2. Differentiability at a number a and on an interval (a,b)
   4. Using graph of f to sketch graph of f¢
   5. Applications of derivative

      1. Interpreting

         1. As the limit of average rates of change
         2. As (instantaneous) rate of change and finding units
      2. If s(t) gives position, then s¢(t) is velocity
      3. Finding equations of tangent lines
4. Important (classes of) functions

   1. Polynomials

      1. all polynomials are continuous at every point on real line (why?)
      2. if p(a) = 0 then (x-a) will factor out
   2. Rational functions (quotients of polynomials)

      1. continous on their domains (Why?)
      2. techniques for evaluating limits of

         1. factoring (see note on polynomials) and simplifying
         2. dividing by largest power in denominator (for limits at infinity).
   3. Logarithmic and exponential functions

      1. inverse relationship (and inverses generally)
      2. models for growth and decay situations
      3. rules for working with
      4. rough sketches of graphs
   4. Trigonometric functions

      1. definitions in terms of unit circle
      2. relationship to right triangles (soh-cah-toa)
      3. Pythagorean Theorem
      4. evaluation at special angles (multiples of p/6 and p/4)
      5. inverses (including domain used for sin, cos, and tan).