Mathematical Ideas that
Shaped the World
Calculus
Dr. Yuzhao Wang
yuzhaow@mun.ca\
HH2026
Plan for this class
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Does 0.999… = 1?
What is calculus?
How has it been important?
Who invented it: Newton or Leibniz?
Course goals.
Handout.
A modern paradox
Do you think that
0.9999…. = 1?
The evidence
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Arguments for
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We believe that
⅓ = 0.3333…
so 3 times this number
should equal 1.
What is 1-0.9999…?
If x=0.9999…. then
10x = 9.9999…
and so
10x – x = 9x = 9,
so x = 1.
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Arguments against
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It’s a number that’s
always less than 1. It
approaches 1 but never
reaches it.
There shouldn’t be two
ways to write down the
same number.
How to decide?
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To answer the question, we need to really
define what 0.99999… is.
We will do this using the theory of limits.
The definition was first made
by the French mathematician
Cauchy (1789 – 1857).
Born in Paris in the year of
the French revolution!
What is calculus?
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Differential
Integral
Infinite series
Others…
 limit
What is calculus?
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Differentiation – calculating how much a
quantity changes in response to changes in
another quantity.
E.g. the speed of a cannonball.
What is calculus?
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Integration – finding areas and volumes of
curved shapes.
E.g. a deep-sea diver needs to know the force
of the water on their body.
Archimedes (287- 212 BC)
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Once upon a time, a man
called Archimedes wanted to
find the area of a circle.
How do you work out the
exact area of a curved shape?
?
Finding π
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Archimedes used the method of exhaustion
to find a good approximation for π.
He found the areas of polygons in and
around the circle.
His best estimate used a 96-sided polygon!
Modern integration
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Our current method of integration was
developed by Newton & Leibniz, and made
rigorous by Riemann.
Idea: split the area under a graph into tiny
rectangles.
Speed at an instant
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Suppose a car is accelerating from 30mph to
50mph.
At some point it hits the speed of 40mph, but
when?
Speed = (distance travelled)/(time passed)
How is it possible to define the speed at a
single point of time?
(We’re back to Zeno’s paradox!)
Tangent line
Newton and Leibniz
Newton: 1643 – 1727
Leibniz: 1646 - 1716
Fundamental theorem of calculus
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Newton and Leibniz are credited with
inventing calculus, but the main ideas were
developed long before them.
Their main contribution was to discover the
fundamental theorem of calculus:
Differentiation and integration are the
opposites of each other.
The great dispute
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Newton and Leibniz also did a lot to unify
and make precise the work of others.
Unfortunately their work was clouded in a
bitter dispute over who wrote down the ideas
first and whether there had
been plagiarism.
Consequences
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As a result of the quarrel, English
mathematicians became isolated from the
continent for a century.
Their refusal to use Leibniz’s notation and the
new methods of analysis held back British
mathematical advances for a long time.
Course goals I
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How to define a limit
How to compute a limit, like
sin( x   )  ( x  1)
lim
2
3
x 0
cos( x )  (1  x)
4
5
Course goals II
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How to define the derivative
How to derivative every function, like


d
x 2
arctan 

dx
 exp(arcsin x  3) 
1/3
Course goals II
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How use differential calculus to solve
problems, like
•A hole is punched near the bottom of
a tank
•How long does it take for the tank to
drain?
Textbook:
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1. Calculus: Early
Transcendentals by Jon
Rogawski (printed version)
with WebAssign Access Card
(bundle), OR
2. Standalone WebAssign
Access Card. The WebAssign
access card now includes
access to the e-book of Jon
Rogawski (2nd edition).
Evaluation:
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55% Final exam
30% Two in-class tests (the first worth 20%,
and the second worth 10%)
7.5% Assignments
7.5% WebAssign.
Tests and Final exam
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No notes, textbooks, calculators or any electronic
communications device are permitted during any
test or final exam.
There will be no make-up for a missed in-class
test. Upon presentation of documentation of a
valid excuse, the corresponding percentage of
the final mark will be added to the final exam.
With no presentation of such documentation a
grade of zero will be entered for the missed test.
Supplementary exams:
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Students who have a passing term mark but a
final grade of between 45-49F (inclusive) may
apply to write a supplementary exam.
If you pass the supplementary exam, your
final grade will be recalculated using the
supplementary exam mark in place of the
original final exam mark.
However, your new final mark cannot exceed
your term mark.
Assignments:
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Homework assignment must be handed in no later than 17:00 on
the due date. They may be submitted in class or placed in the
marking box (number: ??) near the general office of math
department.
Late assignment will NOT be marked. Assignments should be
submitted on loose leaf, 8:5"×11" paper, stapled in the upper left
corner, written on one side only, with problems done in order. They
should not be placed in folders or any other container. Assignments
should include a cover page with your name and student number,
the name of the course and the assignment number.
If you are unable to submit an assignment due to illness,
bereavement or other acceptable reason, please provide me with
appropriate documentation and I will average together your
remaining assignments to create a mark out of 7.5.
WebAssign:
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The website http://www.math.mun.ca/webassign/ contains
everything you should know. You are expected to read this website,
in full, as soon as possible.
You need to purchase a WebAssign access code from the
University Bookstore by January 23 (Thursday). If you purchased an
access code in a previous semester, you will need to purchase a new
access code for this semester.
You MUST use your MUN email address when signing up for
WebAssign. A notification that a new problem set has been posted
will be sent to your MUN email address.
It's your responsibility to be aware of WebAssign deadlines.
If you have further questions with WebAssign, please contact the
WebAssign administrator, Dr. Sullivan (shannon@mun.ca, 864-8073,
HH-3037).
Office hours:
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HH-2026.
15:30-16:30
Tuesday/Wednesday/Thursday/Friday,
or by appointment.
Math help center
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HH-3015.
10:00-16:00
Monday/Tuesday/Wednesday/Thursday;
9:00-13:00 Friday
Course website:
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Class information, assignments and select
solutions will be available on my home page.
http://www.math.mun.ca/~yuzhaow/
Email: yuzhaow@mun.ca
Phone: 864-8417
Email:yuzhaow@mun.ca
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I will usually respond to email within one day.
If you have not received a reply after 24
hours, you should re-send your email.
Please include Mtah 1000 as part of your
subject line.
Begin with “Dr. Wang” or similar.
Include your full name and student number
in the body of the message.
End.
 Thanks
and Have fun!