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What is Calculus?

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What is Calculus?

- The word Calculus comes from the Greek name for pebbles
- Pebbles were used for counting and doing simple algebra…

- “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.”

- “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”

- 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.

- Limits of sequences:
lim an = a

n

- Limits of sequences:
lim an = a

THAT’S CALCULUS!

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

n

- Instantaneous velocity

- Instantaneous velocity

- Instantaneous velocity

distance

time

= lim

both go to 0

- Instantaneous velocity

distance

time

= lim

both go to 0

- Instantaneous velocity
THAT’S CALCULUS TOO!

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

- Local slope

= lim

variation in F(x)

variation in x

both go to 0

- 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!

- 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!)

- 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

- 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)

- The function can converge

The function

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

We write

limx + f(x) = 1

- The function can converge

The function

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

We write

limx + f(x) = 0

- The function can diverge

The function doesn’t

converge to a single value but keeps growing.

It diverges.

We can write

limx + f(x) = +

- The function can diverge

The function doesn’t

converge to a single value but

its amplitude

keeps growing.

It diverges.

- The function may neither converge nor diverge!

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

The function converges at - and diverges at + .

We can write

limx + f(x) = +

limx - f(x) = 0

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

The function converges at + and diverges at -.

We can write

limx + f(x) = 0

- 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…

- 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.

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

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

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

- 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

- 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

f(x) = 1/x

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

0

2

x