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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.

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Origin of calculus

  • The word Calculus comes from the Greek name for pebbles

  • Pebbles were used for counting and doing simple algebra…


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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.”


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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”


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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.


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Examples

  • Limits of sequences:

    lim an = a

n 


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Examples

  • Limits of sequences:

    lim an = a

    THAT’S CALCULUS!

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

n 


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Examples

  • Instantaneous velocity


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Examples

  • Instantaneous velocity


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Examples

  • Instantaneous velocity


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distance

time

= lim

both go to 0

Examples

  • Instantaneous velocity


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distance

time

= lim

both go to 0

Examples

  • Instantaneous velocity

    THAT’S CALCULUS TOO!

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


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Examples

  • Local slope

= lim

variation in F(x)

variation in x

both go to 0


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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!


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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!)


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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


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Chapter V: Limits and continuityV.1: An informal introduction to limits


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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)


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Example of limits at infinity

  • The function can converge

The function

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

We write

limx + f(x) = 1


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Example of limits at infinity

  • The function can converge

The function

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

We write

limx + f(x) = 0


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Example of limits at infinity

  • The function can diverge

The function doesn’t

converge to a single value but keeps growing.

It diverges.

We can write

limx + f(x) = +


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Example of limits at infinity

  • The function can diverge

The function doesn’t

converge to a single value but

its amplitude

keeps growing.

It diverges.


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Example of limits at infinity

  • The function may neither converge nor diverge!


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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 write

limx + f(x) = +

limx - f(x) = 0


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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 write

limx + f(x) = 0


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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…


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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.


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

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


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Examples of limits at x=0

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


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Examples of limits at x=0

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


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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


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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


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Limits at a point

f(x) = 1/x

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

0

2

x


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