The lecture refers to the following chapters in the textbook:
1. Introduction: Chapter 1
2. Mathematical Foundations: Appendix A
3. Probability Theory: Appendix B
4. Estimation and Hypothesis Testing: Appendix C
5. Introduction to Causal Analysis: not in
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Griff Ware (ANU)
MATH1116 Analysis
Lecture 18, 2016
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Directional Derivatives
Definition
Let v Rn with kvk = 1, and f : Rn R. We say that f admits a derivative in
the direction of v at x Rn , if
f (x + hv) f (x)
h0
h
lim
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MATH1014 Calculus
September 19, 2016
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Justifying the formula for the length of a curve
We have used the Mean Value Theorem to derive the following formula for
Math1116 Analysis Lecture Notes
Australian National University, Semester 2, 2016
Pierre Portal and Gri Ware
(last update: 15 June 2016)
Preface
These notes provide a summary of the material that will be presented in MATH1116
analysis lectures in 2016. The
Solution to Analysis Questions 7 and 8
7. Suppose, in order to gain a contradiction, that the desired result is not true, that is:
8n 2 N 9x 2 I
8a 2 I
1
,x
n
(x
+ n1 ) 6 (a
Use this to choose a sequence (xn ) in I such that
1
, xn
n
+ n1 ) 6 (a
8a 2 I
(x
Solution to Problem 6 on the Week 3 and Week 4 Tutorial Worksheets
MATH1116, Semester 2, 2016
What condition must nonnegative integers m, n, and p satisfy so that
xm y n
lim
(x,y)!(0,0) (x2 + y 2 )p
exists? Prove your answer.
[Note: if m = n = p = 0, then
Advanced Mathematics and Applications
Series of Real Numbers Module
Dr Griffith Ware
Mathematical Sciences Institute
c 2015
Any application of the ANU logo on
a coloured background is subject
to approval by the Marketing Office.
Please send to [email protected]
Advanced Mathematics and Applications
Series of Real Numbers Module
Dr Griffith Ware
Mathematical Sciences Institute
c 2015
Any application of the ANU logo on
a coloured background is subject
to approval by the Marketing Office.
Please send to [email protected]
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Griff Ware (ANU)
MATH1116 Analysis
Lecture 2, 2016
1 / 12
Tutorials
If you have emailed me about tutorial selection because the tutorial you want is
full
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Copyright Bill Amend
Griff Ware (ANU)
MATH1116 Analysis
Lecture 3, 2016
1/8
Norms and Metrics
In the previous lecture we introduced two abstract ideas, concerning (a) the
length of a vector in a vector space and (b) the dist
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Griff Ware (ANU)
MATH1116 Analysis
Lecture 5, 2016
1/9
Review: Normed Spaces and Metric Spaces
Definition
Let (xn ) be a sequence in a metric space (X, d
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http:/www.smbc- comics.com/?id=2675
Griff Ware (ANU)
MATH1116 Analysis
Lecture 4, 2016
1/7
Review: Normed Spaces and Metric Spaces
Recall the following two definitions we discussed previously:
Defi
41
5.3. Dierentiability
References: Adams 12.6, Trench 5.3.
5.3.1. Remark. Let f : R2 ! R be given by
(
xy
, if (x, y) 6= (0, 0)
x2 +y 2
f (x, y) =
0,
if (x, y) = (0, 0) .
We have previously seen that f is not continuous at 0 (Example 5.1.2).
However,
f (