of functions in the usual and in the polar coordinate system)
•
how to compute volumes of certain solids
•
how to compute integral averages
•
the statement of the Mean Value Theorem and how to use it to estimate integrals
Limits and continuity
You should know
•
the definition of the limit (and onesided limit) using the neighborhoods and also
ε

δ
.
•
the definition of infinite limits and limits at infinity
•
how to apply the definition to prove the existence of the limit (i.e how to do
ε

δ
proofs)
1
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•
all the basic limit laws
•
how to apply the limit laws to compute limits
•
the Squeezing Principle
•
the definition of continuity at a given point
•
how to apply the basic limit laws to show the continuity of certain functions (e.g. poly
nomials, rational functions in their domain, etc.)
•
the theorem about the composition of continuous functions
•
the limit of
sin
x
x
as
x
→
0 and how to use this to compute other limits
•
Bolzano’s theorem and the Intermediate Value Theorem and how to use them to show
that certain equations have solutions
•
the Extreme Value Theorem and how to apply it
•
the definition of uniform continuity
Sample practice problems
1. We know that
g
is integrable on [2
,
7] and
R
7
2
g
(
x
)
dx
= 5. Find
R
1
0
g
(2
x
+ 5)
dx
.
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 Fall '08
 Staff
 Math, Calculus, Continuous function, Value Theorem, basic limit laws

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