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NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA
1505 Mathematics
I
MidTerm (Review)
1F
o
r
m
u
l
a
This is the formula list that will appear in the midterm test, so besides these formulas, you need
to remember much more.
MA1505
MidTerm Test
Formulae List
1. The
Taylor series
of
f
at
a
is
∞
±
k
=0
f
(
k
)
(
a
)
k
!
(
x
−
a
)
k
=
f
(
a
)+
f
±
(
a
)(
x
−
a
···
+
f
(
n
)
(
a
)
n
!
(
x
−
a
)
n
+
2.
e
x
=
∞
±
n
=0
x
n
n
!
3.
sin
x
=
∞
±
n
=0
(
−
1)
n
x
2
n
+1
(2
n
+1)!
4.
cos
x
=
∞
±
n
=0
(
−
1)
n
x
2
n
(2
n
)!
5.
ln(1 +
x
)=
∞
±
n
=1
(
−
1)
n
−
1
x
n
n
6.
tan
−
1
x
=
∞
±
n
=0
(
−
1)
n
x
2
n
+1
2
n
+1
54
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Questions to Guide Your Review
Here are some questions from Thomas’ Calculus, they are not tested in the Midterm and
Final exam, but good for you to examine whether you understand the de±nition correctly.
If you are not sure about the answer, check your textbook. All the answers can be found
in Thomas’ Calculus.
2.1
Limits and Continuity
Remarks:
The textbook contains much more than your lecture notes. Based on your
lecture notes and syllabus, try to answer the following questions: Qn1Qn8; Qn15Qn28.
Chapter
2
Questions to Guide Your Review
1.
What is the average rate of change of the function
g
(
t
) over the in
terval from
to
How is it related to a secant line?
2.
What limit must be calculated to find the rate of change of a func
tion
g
(
t
) at
3.
What is an informal or intuitive definition of the limit
Why is the definition “informal”? Give examples.
lim
x
:
x
0
ƒ
s
x
d
=
L
?
t
=
t
0
?
t
=
b
?
t
=
a
4.
Does the existence and value of the limit of a function ƒ(
x
) as
x
approaches
ever depend on what happens at
Explain
and give examples.
5.
What function behaviors might occur for which the limit may fail
to exist? Give examples.
6.
What theorems are available for calculating limits? Give exam
ples of how the theorems are used.
x
=
x
0
?
x
0
7.
How are onesided limits related to limits? How can this relation
ship sometimes be used to calculate a limit or prove it does not
exist? Give examples.
8.
What is the value of
Does it matter whether
is measured in degrees or radians? Explain.
9.
What exactly does
mean? Give an example in
which you find a
for a given
and
in the pre
cise definition of limit.
10.
Give precise definitions of the following statements.
a.
b.
c.
d.
11.
What exactly do
and
mean?
Give examples.
12.
What are
(
k
a constant) and
How do
you extend these results to other functions? Give examples.
13.
How do you find the limit of a rational function as
Give examples.
14.
What are horizontal, vertical, and oblique asymptotes? Give ex
amples.
15.
What conditions must be satisfied by a function if it is to be con
tinuous at an interior point of its domain? At an endpoint?
16.
How can looking at the graph of a function help you tell where
the function is continuous?
17.
What does it mean for a function to be rightcontinuous at a
point? Leftcontinuous? How are continuity and onesided conti
nuity related?
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 Spring '09
 MA1505

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