Math 132A
Calculus for Management,
Life and Social Sciences
Spring, 2010
Study Guide for Exam 3
Exam 3 will focus on material from Chapters 6-7: Sections
6.1-6.5 and 7.1-7.5, specifically. Of course, we have continued
in these chapters to call upon knowledge that came earlier.
Antiderivatives
- Some rules
- the antiderivative of a sum (resp. difference) is the sum
(resp. difference) of antiderivatives
- the antiderivative of a constant multiple of some function
is that constant times the antiderivative of the function
- given one antiderivative of a function, any constant added
to that antiderivative produces another antiderivative
- Notation of antidifferentiation
looks very much like that of the definite integral (Can
you guess why?)
- specific antiderivatives to know
- antiderivative of x^{n} (Note the difference
in result when n equals (-1) vs. when it does not.)
- antiderivative of e^{mx+b}
(see Exercises 13-16, p. 315)
- antiderivative of (mx+b)^{n}
(see Exercises 17-22, p. 315; note the case n=-1 is special,
yielding antiderivatives involving the natural log fn.)
Definite integrals
- Polygonal approximation
- regions above/under “polygonal” f may be
divided easily into triangles and rectangles
to get exact “area” (see class handout; also Example 1,
p. 328)
- when f not polygonal, rectangles are
what we used (see Riemann sum approximations)
- Riemann sum (rectangle) approximations (uniform partition case)
- take the form [f(x_{1}) + f(x_{2}) + ... +
f(x_{n})](Δx),
where f is the integrand
- “area” (geometric) interpretation of integral
- truly approximates area if function being integrated
is nonnegative
- interpretation must be adapted to a “difference
of areas” when function is negative
- Fundamental Theorem of Calculus
- gives rise to non-geometric interpretation of definite
integral:
If the function being integrated is continuous and is
the rate of change (derivative) of some quantity on
a ≤ x ≤ b then its definite
integral from a to b gives the total
change in that quantity between a and b.
- allows for exact computation of definite integrals
in cases where an antiderivative may be found
- Applications of definite integrals
- gives total change in “the” antiderivative
doesn't matter which antiderivative
- areas in the plane: regions bounded by various
“curves”
- sketch region, determining which function serves
as upper boundary, which as lower
- solve for limits of integration if not specified
- set up integral and evaluate (remember to subtract
lower-boundary function from upper)
- average value of the function being integrated
remember that it is not the definite integral itself, but
this integral divided by (b-a), where
a and b are the left and right endpoints
respectively of the interval in which the average value
is desired
- consumers' surplus
integrate f(x) - B from 0 to A,
where A is a specified sales level, f(x)
is the demand function (sales level is input, price
is output), and B = f(A).
- volumes of rotation
note similarity to the area-of-circle formula
A = π r^{2}
Functions of multiple variables
- General knowledge
- evaluating them (require multiple inputs to produce
ones output)
- domain and range (How do these ideas change from functions
of 1 variable?)
- viewing them
- graph requires one more dimension than the number of inputs
(so a function of 3 or more variables cannot be graphed)
- level sets
- also called level curves when there are 2 inputs; level
surfaces when there are 3
- a different one for each possible value in the range of
the function
- for a fixed output, a level set consists of all input
pairs/tuples that produce that output
- use one less dimension than the graph, yet help us
see the “terrain” (analogy of a
topographical map)
- Be able to associate collections of level sets with
corresponding graphs (as in Exercises 23-26, p. 391)
- Partial derivatives
- Notations used to denote them
- Definition as limits: f_{x}(x,y) =
lim_{h → 0} [f(x+h, y) - f(x,y)] / h, and
f_{y}(x,y) =
lim_{h → 0} [f(x, y+h) - f(x,y)] / h
- Understand what they tell you about the surface z = f(x,y)
- Know how to find them for various f using the usual
derivative rules
- How are these derivatives different than implicit differentiation?
- Optimization
- Finding possible sites of relative extrema (i.e., critical points)
in unconstrained problems
An application: Least squares regression (Be able to carry out
exercises like those assigned in Section 7.5)
- Methods for finding absolute extrema under a constraint
- Solve the constraint equation for one of its variables, then
substitute into objective function
- Lagrange multiplier method
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This page maintained by:
Thomas L. Scofield
Department of Mathematics and Statistics,
Calvin College
Last Modified:
Thursday, 06-May-2010 08:38:57 EDT