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Math 264 Lecture 1 Notes
Biji Wong
September 2, 2019
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Lecture 1 Outline
1.
Go over course outline that’s also on myCourses
2.
Learning goals for the course
3.
Prerequisites
4.
Review of the gradient
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Why this course?
1.
If you want to model or analyze any
complex system
where
the variables take on a
continuum
of values, then you need
Calculus.
2.
Calculus gives you good training in abstract and quantitative
thinking.
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Advice
You should try
not
to view the material as simply
symbolic
manipulation
of mathematical symbols and objects in order to get
the right answer.
Instead you should focus on:
1.
an
understanding
of what these symbols and object mean
2.
how they relate to concrete physical or geometric problems
3.
why they are useful
4.
how you can couple your understanding with existing
technology to solve concrete problems.
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Concrete example of advice
Problem 1:
Solve this pair of equations for
x
and
y
:
x
+ 2
=
y

2
2
x

4
=
y
+ 2
.
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Concrete example of advice
Problem 2:
I
Laurel and Wren have just finished picking apples, and each
holds up their basket of apples.
I
Laurel looks at Wren and says: “If you give me two of your
apples, then we will have the same number of apples.”
I
Wren responds with: “No, if you give me two of your apples,
then I will have double the number of apples as you.”
I
How many apples did each pick?
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Concrete example of advice
I
These are equivalent problems.
I
If a student can do the first problem, but
not
the second
problem, then unfortunately in today’s modern world, they
have learned a useless skill.
I
Perhaps 100 years ago, equation solving could only be
performed by hand, and hence such a skill would’ve been
useful. Today, there are programs that will do it quickly and
accurately.
I
However, there aren’t programs that know how to analyze,
interpret, and manipulate answers to mathematical problems
(yet), so that’s still relevant.
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Learning Goals
6070% of the Course will be about integration.
Integration in Calculus 2 and 3:
I
single integrals
Z
b
a
f
(
x
)
dx
I
double integrals
ZZ
R
f
(
x
,
y
)
dxdy
,
where
R
is a either a rectangle or a more complicated domain
(region) in
R
2
.
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Learning Goals
I
triple integrals
ZZZ
S
f
(
x
,
y
,
z
)
dxdydz
,
where
S
is some domain (region) in
R
3
.