Math 132A
Calculus for Management, Life and Social Sciences
Spring, 2010

Study Guide for Exam 1

Exam 1 will focus on material from Chapters 0, 1 and 2, ending with Section 2.7.

Functions

  1. How the concept is defined
  2. Related concepts: domain, range, independent/dependent variables, input/output (= the value of a function), zeros, intercepts
  3. Different ways of indicating a function
    1. by a verbal description
    2. by a formula
    3. by ordered pairs of points (or a table of values); useful, though, in most cases, one cannot specify the entire function (all of the inputs and corresponding outputs) this way
    4. by a graph
  4. Determining the domain
    1. from graph
    2. from formula (usually a matter of avoiding even roots of negative nos. and zero denominators)
  5. Some common types of functions
    1. constant functions: y = c
    2. linear functions: y = mx + b
      1. Be able to write an equation given
        1. a point and a slope
        2. two points
      2. Understand how to interpret the slope and y-intercept
    3. quadratic functions: y = ax2 + bx + c
      1. Be able to find exact zeros (quadratic formula; perhaps factoring)
    4. power functions: y = xr (many root functions fall into this category)
    5. polynomials
    6. rational functions: one polynomial divided by another
      1. polynomials are special cases of rational functions
      2. domain excludes zeros of denominator
      3. graphs often (but not always) have vertical asymptote at numbers excluded from domain
    7. absolute-value function
    8. piecewise-defined functions
  6. Combining functions through
    1. addition, subtraction, multiplication, division
    2. composition
      1. how to find expressions for things like f (g(x)), f (f(x)), etc., given expressions for f and g
      2. rewriting power-type expressions as compositions: for example, 5(3x2 - 7x)10 = f (g(x)), when f(x) = 5x10 and g(x) = 3x2 - 7x
      3. writing out and simplifying expressions like [f(x + h) - f(x)]/h, given a formula for f ; such a formula can be viewed as an average rate of change of f in going from one point P on the curve to another one Q
  7. Interpreting functions in various settings (like in Section 1.8)

Limits

  1. Notation used to describe a limit, and what it means
  2. Evaluating them given
    1. a graph (Be able, as well, to identify situations where a limit does not exist)
    2. an expression for the function involved (including situations in which the function is piecewise-defined)
  3. Continuity
    1. Be able to recognize points where a function is continuous; points where it is discontinuous
    2. When f is discontinuous at x = a, be able to identify which of the 3 criteria for continuity is violated

Derivatives

  1. Notation: f ′, f ′′, dy/dx, d2y/dx2, etc.
  2. How the concept is defined
    1. as the slope of a curve at a point (if such a slope exists)
      1. the meaning of “f is differentiable at x = a
      2. Be able to recognize points at which the slope of the curve is undefined (Recall: If f is discontinuous at x = a, then it is not differentiable at a.)
    2. as the limit of average rates of change (See point 3 under “composition”)
    3. as an instantaneous rate of change
  3. Finding derivatives for the various types of functions listed above in point V. under functions, except (possibly) rational functions
  4. Rules of differentiation: know how to apply them when necessary
    1. constant-multiple rule
    2. sum/difference rule
    3. power and generalized power rules
  5. Theorem: If f is differentiable at x = a, then f is continuous at a.
  6. Interpreting the meaning of a derivative in various contexts
Applications of derivatives
  1. Curve sketching
    1. describing increasing/decreasing behavior, relative extrema
      1. Main fact: If f ′(x) is positive (negative) for a < x < b, then f increases (decreases) on that interval
      2. finding extrema
        1. find places where f ′ is equal to zero
        2. test to see that f ′ changes sign as x values move by these numbers
    2. describing concavity, inflection points
      1. Main fact: If f ′′(x) is positive (negative) for a < x < b, then f is concave up (down) on that interval
      2. finding inflection points
        1. find places where f ′′ is equal to zero
        2. test to see that f ′′ changes sign as x values move by these numbers
    3. Be able to take information like that you gather in determining extrema, inflection points and sketch a reasonable graph.
  2. Optimization
    1. set things up
      1. define appropriate variables
      2. determine which variable is to be optimized (dependent variable)
      3. determine function for dependent variable
        1. must involve just one independent variable (if more than one, find an auxilliary equation(s) which may be used to eliminate the extra variable(s))
        2. domain of the function you've written?
        3. some specific scenarios/quantities which we have encountered: economic order quantity, economic lot size, revenue, demand (often linear in our problems), carrying costs, profit
    2. differentiate your function
    3. find critical numbers (places where derivative is zero)
    4. check value of the function at critical numbers and endpoints of the domain (unless you can provide reasons why you don't need to do the latter) to determine which one optimizes (maximizes or minimizes as appropriate for the problem) the dependent variable
    5. give answer(s) with appropriate units

Miscellany

  1. Be able to express intervals of numbers in the various standard ways
  2. Be able to find points of intersection between two functions, as in Section 0.4
  3. Interest formulas
    1. simple and compound interest
    2. know what variables stand for, how to use them
    3. Understand the derivation of these formulas well enough to handle settings like Exercise 94, p. 47
  4. Powers
    1. Know and be able to use the “Laws of Exponents” which make it possible to multiply/divide powers correctly. After studying some correct uses of these laws, you might have a look here and here to see some results when these laws are ignored or misapplied.
  5. Know and be able to use the quadratic formula

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This page maintained by: Thomas L. Scofield
Department of Mathematics and Statistics, Calvin College

Last Modified: Monday, 01-Mar-2010 08:20:59 EST