MA115 Mathematical Analysis I
Test 1
Instructions
The test consists of two sections,
Part 1: Questions 1 – 6
Part 2: Questions 7 – 10
You are to do all of Part 1. For Part 2, select and complete 2 of the 4 problems
given. Please clearly indicate which two problems you choose. The two problems
you indicate will be the only problems from Part 2 graded.
Show your work. This does not mean your answers should be long—we just need
you to tell us
how
you obtained your answer in addition to
what
you obtained. Any
response to a question consisting of just an answer will be marked incorrect.
Please type your answers up as you do your homework, and submit your completed
test via email by 9PM Wednesday, July 7, to the course instructor and the two
teaching assistants.
You are allowed to use your textbook and all of the notes from the course. Good
luck!
1
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Part 1
1.
(10 points)
(a) For
f
(
x
) =
x
2
we have
f
(
x
+
h
)

f
(
x
)
h
=
(
x
+
h
)
2

x
2
h
=
x
2
+ 2
xh
+
h
2

x
2
h
=
2
xh
+
h
2
h
=
2
x
+
h
(b) From part (a),
lim
h
→
0
f
(
x
+
h
)

f
(
x
)
h
=
lim
h
→
0
(2
x
+
h
)
=
2
x
2.
(10 points)
(a) The domain of
H
(
x
) =
√
1

ln
x
is
{
x

0
< x
≤
e
}
. Two parts of the
function
H
(
x
) influence the domain: the square root and the natural log.
Since we can’t take the square root of a negative number we must have
1

ln
x
≥
0
Solving this inequality gives
x
≤
e
.
But also, since we can’t take the
logarithm of any number less than or equal to zero, we must have
x >
0.
Since both
x
≤
e
and
x >
0 must be true the domain is
{
x

0
< x
≤
e
}
.
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 Fall '08
 Mahalanobis
 Math, Limit, lim, Continuous function, lim F

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