76 LECTURE NOTES [TEL AVIV, 2009)
Similarly,
arccot i = .
iii. Let f($) = am. Then 9(y) = logi1 y and
1 _ 1
amloga _ yloge'
(10a; 90 =
(Weve knovvn already the ansvver in advance, of course).
Th
7i LECTURE NOTES [TEL AVIV, 2009)
are also continuous on [a, 6].
Exercise 14.2.4.
i. Let the function f be uniformly continuous on a bounded set E. Prove that f is
bounded.
ii. Let f E C(a. b) where (
DIFFERENTIAL AND INTEGRAL CALCULUS, I 73
and
_ ,_ _ sin(e/2) E _
(slna) l13]( 6/2 )ces (33+ 2) ces;r.
In a similar way, one nds the derivative of the cosine function
(cos 3:)" = sin 3:.
(iv) Next, c
DIFFERENTIAL AND INTEGRAL CALCULUS, I 79
16.2. The tangent line. Given a curve 7 in the (:r, y)plane and a peint Mg(a:g, ya)
en 7/, we want te draw threugh Mg a tangent line te 7. Fer that, we censide
Bi] LECTURE NOTES [TEL AVIV, 2009)
Example 16.2.1. Let x) 2 a2 sin; for a: 75 0 and f(0) = 0. This function is
differentiable at the origin, and f(0) = limE_,U s sin% = 0. We see that the saxis is the
74 LECTURE NOTES [TEL AVIV. 2009)
15.2. Some rules. In this section we show several simple rules which help us to corn
pute derivatives.
Theorem 15.2.1. Let the functions f and g be dened on an tnteru
72 LECTURE NOTES [TEL AVIV. 2009)
15. THE DERIvATIvE
15.1. Denition and some examples.
Denition 15.1.1 (The derivative). f be a function dened in an open neighbourhood
U of a point :1: E R. The functi
DIFFERENTIAL AND INTEGRAL CALCULUS, I 81
It is also worth to mention another form of the formula (c):
13(3?) Hits)
(63) E i ,
(22(13) (55' :1?le (33s)
which provides the partial fraction decompositi
DIFFERENTIAL AND INTEGRAL CALCULUS, I
iii. If f(:r:) = u($)($), than
I .I|I
fJr = (elgu) = evlguw 10g u) = u (Uflgu +19%).
F01 example,
1
(3:53)! = :13: (Inga: + :r: E) = as: (logsc + 1).
77
DIFFERENTIAL AND INTEGRAL CALCULUS, I 75
Example 15.2.3. If
P(:I:) = Z jj"?
j:
is a polynomial of degree n, then
111
on) = Zn + 1).1m1
i=0
is a polynomial of degree n 1.
15.3. Derivative of the invers
DIFFERENTIAL AND INTEGRAL CALCULUS, I 71
In the other direction, suppose f[a,b] be a closed interval and suppose that f is
discontinuous at c E [n,b]. We assume that c 6 (1,3), the cases or = a, and c
78 LECTURE NOTES [TEL AVIV, 2009)
16. APPLICATIONS OF THE EEEIvATIVE
The differential calculus was systematically developed by Newton and Leibnitz, how
ever Archimedes, Fermat, Barrow and many ether g
CHAPTER 2.1
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Week 23 notes
Reading: Ch 2 addendum
I gave you everything in the trade game. i told you to ignore where the goods came from for the
purposes of that game. How do we take that a step back and talk abo
The Science of Geography .
And who cares?
What and Why
How big is the Universe?
Hubble Deep Field View
http:/hubblesite.org/newscenter/newsdesk/archive/rele
ases/2004/07/image/a
The Hubble Telescope
Week 2 notes
Reading: Ch 2
Thinking like an economist
0 Economics is a social science deals with how societies and individuals interact in order to divvy
up scarce resources
0 Economics deals with peo
Production if entire day is spent on just one good
good 1
good 2
Kate Dave Sarah
4
2
16
6
10
4
4
0
0
6
2
0
0
10
16
0
0
4
0
16
20
16
16
20
16
10
20
22
10
0
Individual PPFs
12
10
8
good 2
Sarah
Kate
Dav
Graphing and Algebra Review
The re are some quick math topics to review at the beginning of the semester that will make your time in
E0252 (Principles of Microeconomics) much easier if you grasp them
GE 112
Physical
Geography:Landfor
Landscapes and the forces that
ms
shape them.
THE SYLLABUS
According to research every
student enters a class with 5
questions:
Am
I in the right place?
If you are n
EC-252: Principles of Microeconomics
Fall 2016
Homework #1
Due Date: Wednesday, September 14, 2016
Name: _
1. Write down the ten principles of economics.
2. In a brief sentence or two, please explain