THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH1010 University Mathematics (Fall 2019)
PreCoursework Exercise 5
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2
Dear Students of MATH1010,
The main goal of this note is to provide a list of the facts/results from the teaching
materials that you have learned and seen in the past week of lectures. There are a few
ways of getting ready for and preparing to do any coursework problems in the tutorial
class this week:
•
WeBWork: There are a few assigned problems that can be found in WeBWork.
We sincerely encourage you to do them multiple times without any time constraint
until you get a correct answer for each question.
•
Worked Examples:
–
Additional Exercise Set: There are six additional exercise sets posted every
two weeks that fit in your study plan. Please try to work out all the problems.
–
Extra Exercise Set: There are numerous worked exercises to go along with
each lecture topic.
These drilltype problems will help you understand the
course materials. Many challenge problems and their suggested solutions can
be found in the extra problem sets. (You need the username and password)
–
PreCoursework Exercise Set: There are many worked examples contained
in this file with their suggested solutions.
Please work out every step and
try to understand the line of reasoning behind this exercise. (You need the
username and password)
•
MathGym: If you have any questions, please visit MathGym and click the link
below:
Kindly reminder
: You should expect a few problems that you might never have seen/worked
out/thought of before in the coursework problems. That is exactly what we planned to
do  we want you to solve some challenging questions in order to enhance your problem
solving skills and test how much you understand the MATH1010 teaching materials every
week!
Yours truly,
The MATH1010 Lecturer
Department of Mathematics, CUHK
The MATH1010 Course Website
3
Here are a few useful observations and notes on the topic of limits:
•
Let Δ
f
=
f
(
a
+ Δ
x
)

f
(
a
). The Linear Approximation is the estimate
Δ
f
≈
f
0
(
a
)Δ
x
(for Δ
x
small)
.
•
Differential notation:
dx
is the change in
x
,
df
=
f
0
(
a
)
dx
, Δ
y
=
f
(
a
+
dx
)

f
(
a
)
.
In this notation, the Linear Approximation reads
Δ
y
≈
dy,
(for Δ
x
small)
.
•
The linearization of
f
(
x
) at
x
=
a
is the function
L
(
x
) =
f
0
(
a
)(
x

a
) +
f
(
a
)
.
•
The Linear Approximation is equivalent to the approximation
f
(
x
)
≈
L
(
x
)
,
(for
x
close to
a
)
.
•
The error in the Linear Approximation is the quantity
Error =

Δ
f

f
0
(
a
)Δ
x

.
In many cases, the percentage error is more important than the error itself:
Percentage error =
error
actual value
×
100%
.
•
Extreme Values on an Interval
Definition 1.
Let
f
(
x
) be a function on an interval
I
and let
a
∈
I.
We say that
f
(
a
) is
the
–
Absolute minimum of
f
(
x
) on
I
if
f
(
a
)
≤
f
(
x
) for all
x
∈
I
.
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 Fall '13