What is Calculus?

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What is Calculus?
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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
• THATS 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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Examples

• Instantaneous velocity

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Examples

• Instantaneous velocity
• THATS CALCULUS TOO!
• (the study of what happens when things get very
  very small)

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Examples

• Local slope

variation in F(x) variation in x
lim
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 doesnt 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 doesnt 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 dont 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 x0 (x becomes very small!)

• The function can have asymptotes (it diverges).
  The limit at 0 doesnt exist

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Examples of limits at x0

• The function can have a gap! The limit at 0
  doesnt exist

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Examples of limits at x0

• The function can behave in a complicated
  (exciting) way.. (the limit at 0 doesnt exist)

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Examples of limits at x0

• 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