What is Calculus?

Origin of calculus

• The word Calculus comes from the Greek name for pebbles

• Pebbles were used for counting and doing simple algebra…

Google answer

• “A method of computation or calculation in a special notation (like logic or symbolic logic). (You'll see this at the end of high school or in college.)”

• “The hard deposit of mineralized plaque that forms on the crown and/or root of the tooth. Also referred to as tartar.”

Google answers

• “The branch of mathematics involving derivatives and integrals.”

• “The branch of mathematics that is concerned with limits and with the differentiation and integration of functions”

My definition

• The branch of mathematics that attempts to “do things” with very large numbers and very small numbers– Formalising the concept of very– Developing tools to work with very

large/small numbers– Solving interesting problems with these

tools.

Examples

• Limits of sequences:lim an = a

n

Examples

• Limits of sequences:lim an = a

THAT’S CALCULUS!

(the study of what happens when n gets very very large)

n

Examples

• Instantaneous velocity

Examples

• Instantaneous velocity

Examples

• Instantaneous velocity

Examples

• Instantaneous velocity both go to 0

distance time

= lim

Examples

• Instantaneous velocity

THAT’S CALCULUS TOO!

(the study of what happens when things get very very small)

both go to 0

distance time

= lim

Examples

• Local slope both go to 0

variation in F(x) variation in x

= lim

Important new concepts!

• So far, we have always dealt with actual numbers (variables)

• Example: f(x) = x2 + 1 is a rule for taking actual values of x, and getting out actual values f(x).

• Now we want to create a mathematical formalism to manipulate functions when x is no longer a number, but a concept of something very large, or very small!

Important new concepts!

• Leibnitz, followed by Newton (end of 17th century), created calculus to do that and much much more.

• Mathematical revolution! New notations and new tools facilitated further mathematical developments enormously.

• Similar advancements– The invention of the “0” (India, sometimes in 7th century)– The invention of negative numbers (same, invented for

banking purposes)– The invention of arithmetic symbols (+, -, x, = …) is very

recent (from 16th century!)

Plan

• Keep working with functions• Understand limits (for very small and very

large numbers)• Understand the concept of continuity• Learn how to find local slopes of functions

(derivatives)

= differential calculus• Learn how to use them in many applications

Chapter V: Limits and continuity

V.1: An informal introduction to limits

V.1.1: Introduction to limits at infinity.

• Similar concept to limits of sequences at infinity: what happens to a function f(x) when x becomes very large.

• This time, x can be either positive or negative so the limit is at both + infinity and - infinity:– lim x + f(x)

– limx - f(x)

Example of limits at infinity

• The function can converge

The function converges to a single value (1), called the limit of f.

We writelimx + f(x) = 1

Example of limits at infinity

• The function can converge

The function converges to a single value (0), called the limit of f.

We writelimx + f(x) = 0

Example of limits at infinity

• The function can diverge

The function doesn’tconverge to a single value but keeps growing.

It diverges. We can writelimx + f(x) = +

Example of limits at infinity

• The function can diverge

The function doesn’tconverge to a single value but its amplitudekeeps growing.

It diverges.

Example of limits at infinity

• The function may neither converge nor diverge!

Example of limits at infinity

• The function can do all this either at + infinity or - infinity

The function converges at - and diverges at + .

We can writelimx + f(x) = +limx - f(x) = 0

Example of limits at infinity

• The function can do all this either at + infinity or - infinity

The function converges at + and diverges at -.

We can writelimx + f(x) = 0

Calculus…

• Helps us understand what happens to a function when x is very large (either positive or negative)

• Will give us tools to study this without having to plot the function f(x) for all x!

• So we don’t fall into traps…

V.1.2: Introduction to limits at a point

• Limit of a function at a point:New concept!

• What happens to a function f(x) when x tends to a specific value.

• Be careful! A specific value can be approached from both sides so we have a limit from the left, and a limit from the right.

Examples of limits at x=0 (x becomes very small!)

• The function can have asymptotes (it diverges). The limit at 0 doesn’t exist…

Examples of limits at x=0

• The function can have a gap! The limit at 0 doesn’t exist…

Examples of limits at x=0

• The function can behave in a complicated (exciting) way.. (the limit at 0 doesn’t exist)

Examples of limits at x=0

• But most functions at most points behave in a simple (boring) way.

The function has a limit when x tends to 0 and that limit is 0.

We write

limx 0 f(x) = 0

Limits at a point

• All these behaviours also exist when x tends to another number

• Remember: if g(x) = f(x-c) then the graph of g is the same as the graph of f but shifted right by an amount c

Limits at a point

f(x) = 1/x

g(x) = f(x-2) = 1/(x-2)

x0 2