Math 006 - Introduction to Calculus

Lecture 1

Old Stuff

We will go over the some basic algebra that is needed for this course.

Sample Quiz

In this class there will be a diagnostic test. I would like you to hand this in so I know what things to cover in the tutorials, but the mark will not contribute to your final mark.

Please write any other relevant comments on the sheet, such as areas of basic algebra you feel you need some extra work on.

New Stuff

Calculus is about analysis of functions. This class we will look at some examples of functions.

• Specific Examples
  • Example of a function: Speed.
  • Example of a function: Prices of stock, shares, etc.
  • Volume of a gas as a function of it's temperature.
• General Properties
  • What is a function?
  • Some Qualitative properties of functions
    • slope
    • Continuous functions
    • Smooth functions

We are going to look at functions and the behaviour of functions, so you can make some prediction about what they will do, and answer the following kind of questions:

• What does "speed" mean?
• How can you represent the movement of something by a graph?
• How do you intrepret what the graph means, and predict what might happen?

More questions to think about:

• (1) On a car journey, what is the average speed over the whole jouney?
• (2) What is the speedometer measuring?
• (3) Will the speedometer ever measure the same as the average speed of the car?

What is a function?

Basically, a function is a rule that takes one number and gives you another.

In real life, functions are usually not very easy to describe, but to make things easy, we find approximations that are easier to work with. Here are some examples of the kind of functions we will consider in this course:

• f(x)=x²+4
• f(x)=sin(x)
• f(x)=e^x

If we want to examine functions, we need to be able to compare them, look at what's going on; we need some vocabulary.

We need a more precise way of knowing about these things, how to give explicit numbers to speed, how to calculate these things for given functions, etc. This is where calculus comes in.

Tutorial and Examples

The following questions from the text book should be done for practice, but are not to be handed in for marking.

• Ex 1, page 3: 1b, 2e, 3d, 4f
• Ex 2, page 4: 1f, 2c
• Ex 1.2, page 9: Ae, B2, B6, C11

The following questions from the text book should be handed in Wednesday 28th January (or before if possible)

• Ex 1, page 3: 1h
• Ex 2, page 4: 2d
• Ex 1.2, page 9: B7, B10

Show complete working to get marks.

Please send corrections to spelling mistakes, and any other comments, to me.