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
 How the concept is defined
 Related concepts: domain, range, independent/dependent variables,
input/output (= the value of a function), zeros, intercepts
 Different ways of indicating a function
 by a verbal description
 by a formula
 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
 by a graph
 Determining the domain
 from graph
 from formula (usually a matter of avoiding even roots of negative
nos. and zero denominators)
 Some common types of functions
 constant functions: y = c
 linear functions: y = mx + b
 Be able to write an equation given
 a point and a slope
 two points
 Understand how to interpret the slope and yintercept
 quadratic functions: y = ax^{2} + bx + c
Be able to find exact zeros (quadratic formula;
perhaps factoring)
 power functions: y = x^{r} (many root
functions fall into this category)
 polynomials
 rational functions: one polynomial divided by another
 polynomials are special cases of rational functions
 domain excludes zeros of denominator
 graphs often (but not always) have vertical asymptote
at numbers excluded from domain
 absolutevalue function
 piecewisedefined functions
 Combining functions through
 addition, subtraction, multiplication, division
 composition
 how to find expressions for things like f (g(x)),
f (f(x)), etc., given expressions for f and
g
 rewriting powertype expressions as compositions: for example,
5(3x^{2}  7x)^{10} = f (g(x)), when
f(x) = 5x^{10} and g(x) = 3x^{2}  7x
 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
 Interpreting functions in various settings (like in Section 1.8)
Limits
 Notation used to describe a limit, and what it means
 Evaluating them given
 a graph (Be able, as well, to identify situations where a
limit does not exist)
 an expression for the function involved (including situations
in which the function is piecewisedefined)
 Continuity
 Be able to recognize points where a function is continuous;
points where it is discontinuous
 When f is discontinuous at x = a, be able
to identify which of the 3 criteria for continuity is violated
Derivatives
 Notation: f ′, f ′′,
dy/dx, d^{2}y/dx^{2}, etc.
 How the concept is defined
 as the slope of a curve at a point (if such a slope exists)
 the meaning of “f is differentiable at
x = a”
 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.)
 as the limit of average rates of change (See point
3 under “composition”)
 as an instantaneous rate of change
 Finding derivatives for the various types of functions listed above in
point V. under functions, except (possibly) rational functions
 Rules of differentiation: know how to apply them when necessary
 constantmultiple rule
 sum/difference rule
 power and generalized power rules
 Theorem: If f is differentiable at x = a, then
f is continuous at a.
 Interpreting the meaning of a derivative in various contexts
Applications of derivatives

Curve sketching
 describing increasing/decreasing behavior, relative extrema
 Main fact: If f ′(x) is positive (negative) for
a < x < b, then f increases (decreases)
on that interval
 finding extrema
 find places where f ′ is equal to zero
 test to see that f ′ changes sign as x values
move by these numbers
 describing concavity, inflection points
 Main fact: If f ′′(x) is positive (negative) for
a < x < b, then f is concave up (down)
on that interval
 finding inflection points
 find places where f ′′ is equal to zero
 test to see that f ′′ changes sign as x
values move by these numbers
 Be able to take information like that you gather
in determining extrema, inflection points and sketch
a reasonable graph.

Optimization
 set things up
 define appropriate variables
 determine which variable is to be optimized (dependent variable)
 determine function for dependent variable
 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))
 domain of the function you've written?
 some specific scenarios/quantities which we have encountered:
economic order quantity, economic lot size, revenue,
demand (often linear in our problems),
carrying costs, profit
 differentiate your function
 find critical numbers (places where derivative is zero)
 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
 give answer(s) with appropriate units
Miscellany
 Be able to express intervals of numbers in the various standard ways
 Be able to find points of intersection between two functions, as in
Section 0.4
 Interest formulas
 simple and compound interest
 know what variables stand for, how to use them
 Understand the derivation of these formulas well enough to
handle settings like Exercise 94, p. 47
 Powers
 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.
 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, 01Mar2010 08:20:59 EST