Transcript CalculusIntro

Calculus
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4 sks, 15 weeks, 28-30 sessions
Class: Wed 10:00 – 11:40,
Fri 08:00 – 09:40
Tutorials:
Texbook: Calculus 6 ed., Edwards &
Penney, ch. 1 – ch. 15.
Assignments: 20%
Midtest : week 5 and 11 (tentative) @ 20%
Final: 40%
Useful link: Visual Calculus
Course Outline
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11.
Function Graphs:
Ch: 1, 2; session: 1,2.
Derivatives:
Ch: 3, 4; session: 3, 4, 5.
Transcendental Functions: Ch 7; session: 6, 7
====== Test 1: session 8 or 9.
Integrals:
Ch: 5, 6, 8 ; session: 9 – 14.
Polar coordinates:
Ch 10; session : 15, 16.
Series:
Ch 11; session: 17, 18.
====== Mid-exam : session 19 or 20.
Vectors, curves:
Ch 12; session 20, 21.
Partial Differentiation:
Ch 13; session 22, 23.
Multiple Integrals:
Ch 14; session 24, 25.
Vector Calculus:
Ch 15; session 26, 27.
Revision: session 28
Why study Calculus?
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What calculus does for you.
It gives you a vantage point on the world that you
cannot have any other way. It teaches you the language
you must know to understand how the wind blows, how
the waters flow, how the sun shines, how music
reaches your ear, how the planets cycle through the
heavens, and much more. Even the ebb and flow of
such human activities as population dynamics and
economics are better viewed from calculus' highlands.
 Principal objectiveof calculus is the analysis of
problems of change (eg.motion) and of content (eg.
Volume, area).
Study Tips
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Tip 1: Do the homework exercises.
The homeworks are for your benefit, not the
lecturer's. The exercises will train your mind
and sharpen your intuition. So do the work. It
will pay off in the end.
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Tip 2: Math books are meant to be read
slowly.
You cannot speed read it and expect to get any
benefit out of it at all. When you encounter a new
concept in a math book, do not expect to understand
it on the first reading, no matter how carefully your
read it. You should go over each difficult paragraph
several times. If you are still uncomfortable with it,
read ahead a page or so, then come back to the
difficult passage. And remember that math books are
meant to be read with paper and pencil in hand. Use
the paper and pencil to work through any steps that
the book skips over.
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Tip 4: Your greatest assets are in the class
with you. Your classmates are in the same
boat as you. Organize a study group.
 Tip 5: In your group activity, take turns.
Have one person get up and do a problem on the
board, explaining what he or she is doing as the
problem unfolds. If the person at the board gets stuck,
the others in the group should try to provide hints or ask
the person at the board telling questions. If the person
at the board is doing fine, the others in the group should
challenge him or her. Make the problem-doer justify
each step orally. If anybody in the group does not
understand a step, the person at the board ought to be
able to explain it to his or her satisfaction.
Real Numbers
N = set of all natural numbers: 1, 2, 3, …
 Z = set of all integers: 0, ±1, ±2, ±3, …
 Q = set of all rational numbers:
real numbers of form p where p and q
are integers and q  0. q
 Irrational numbers: real numbers which are not
fractions:
2 ,  , e  2.718
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R = set of all real number – represented by a
decimal expansion. Q  irrational
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Ordering on R. Geometrically a<b means a lies
to left of b on the real number line.
We write a  b if a  b and a  b
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Absolute value (or modulus) of x is given by
 x, if x  0
x 
 x, if x  0
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; so x  0
Examples
1
 1 1
      , 4  4
2
 2 2
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The distance between a and b  R
b  a, if b  a
ba  ab  
 a  b, if b  a
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Properties
(i) . d
2
 d 2 , thus d  d 2 (positive square root)
(ii). ab  a  b
(iii). a  b  a  b (triangle inequality )
eg. 2  (3)   1  1,
2   3  2  3  5.
Line intervals
 Closed interval:
a, b  {x  R | a  x  b}
 set of x  R such that a  x  b.
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Open interval:
a, b  {x  R | a  x  b}
 set of x  R such that a  x  b
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Half open, half closed:
a, b   {x  R | a  x  b}
a, b  {x  R | a  x  b}
Inequalities
Properties:
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If a < b and b < c, then a < c.
If a < b, then a+c < b+c.
If a < b and c > 0, then ac < bc.
If a < b and c < 0, then ac > bc.
If a > 0, then
x  a  a  x  a  x  (a, a)
x  a  x  a or x  a  x   ,a   (a, )
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Similarly for  (greater than equal to) and 
Functions and Graphs
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Definition : A real-valued function f defined on a
set D of real numbers is a rule that assigns to every
x in D exactly one real number denoted by f(x).
f(x) is the value of f at x.
D is the domain of the function f.
The set of all values y = f(x) is called the range of f.
Examples
f : R  R, f ( x)  x 2 , for all x  R.
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Range
f  {x 2 | x  R}  [0, )