Mathematical Ideas that Shaped the World Calculus

Dr. Yuzhao Wang yuzhaow@mun.ca\ HH2026

Plan for this class 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 Arguments for Arguments against 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. Arguments against 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? 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? limit Differential Integral Infinite series Others…

What is calculus? Differentiation – calculating how much a quantity changes in response to changes in another quantity. E.g. the speed of a cannonball.

What is calculus? 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) 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 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 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 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 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 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 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 How to define a limit How to compute a limit, like

Course goals II How to define the derivative How to derivative every function, like

Course goals II 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: 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: 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 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: 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: 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: 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: HH-2026. 15:30-16:30 Tuesday/Wednesday/Thursday/Friday, or by appointment.

Math help center HH-3015. 10:00-16:00 Monday/Tuesday/Wednesday/Thursday; 9:00-13:00 Friday

Course website: 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 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!