Math 131 Spring 2013
CHAPTER 1 EXAM (PRACTICE PROBLEMS - SOLUTIONS)
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Problem 1. Many apartment complexes check your income and credit history before letting you
rent an apartment. Let I = f (r ) be the minimum annual gross income, in thousands of dolla

Applications of Greens Theorem
Let us suppose that we are starting with a path C
and a vector valued function F in the plane. Then as
we traverse along C there are two important (unit)
dy
vectors, namely T, the unit tangent vector h dx
ds , ds i,
dy
dx
an

Coordinate systems
There are three main ways to describe a point in
space. We have already discussed the Cartesian coordinate system (also known as rectangular coordinates). The other two are generalizations of polar coordinates in the plane, so before ju

Cross product
Cross product, denoted with , gives a way to
multiply vectors together and get a new vector, i.e.,
(vector)(vector)=(vector). But there is a big catch,
namely it only works in three dimensions (conveniently though we live in three dimensions

Algebra
Algebra is the foundation of calculus. The basic
idea behind algebra is rewriting equations and simplifying expressions; this includes such things as factoring, FOILing (i.e., (a+b)(c+d) = ac+ad+bc+bd),
adding fractions (remember to get a common d

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Relative maxima, relative minima and saddle points
The developments of the previous section (Multivariate Calculus (part 1) are helpful in studying
maxima and minima of functions of several variables. We restrict our attention here to functions
f (x, y)

Second Derivative Test
1. The Second Derivative Test
We begin by recalling the situation for twice dierentiable functions f (x) of one variable.
To nd their local (or relative) maxima and minima, we
1. nd the critical points, i.e., the solutions of f (x)

Calculus 1
Lia Vas
Concavity and Inflection Points. Extreme Values and The
Second Derivative Test.
Consider the following two increasing functions. While they are both increasing, their concavity
distinguishes them.
The first function is said to be concav