Introduction to Calculus

1
Introduction to Calculus

• Barrett Walls
• November 18, 2004

2
Outline

• Rates of Change
• Areas
• How are the two ideas connected?

3
Rates of Change

• The average rate of change of a function f(x)
  from x a to x b is
• If f(x) x24 then the average rate of change on
  the interval (2,4) is

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• Let f(x) 65x5820 be the cost (in dollars) to
  produce x bicycles at a particular factory.
• The average change from x 50 to 100 is
• Our function f(x) is a line, so no matter what
  interval we use we will always get the same
  answer, 65.

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• We can think of the rate of change as
• What happens as point b gets closer and closer to
  point a?

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• Pick your two points as a x and b xh, so h
  is the distance between the two points.
• For example, if f(x) x24, we would get
• If h is very small this is very close to 2x.
• The derivative of f(x) x24 is f(x) 2x.

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Calculating Derivatives

• Derivatives are usually calculated with a few
  common techniques.
• For example, the derivative of a line is just the
  slope, so if f(x) m x b then f(x)m.
• If f(x) xn then f(x) nxn-1.
• If several parts are added, you can take the
  derivative of one piece at a time
• If f(x) x34x21 then f(x) 2x28x.

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Uses of Derivatives

• Derivatives tell us how a function is changing
  and by how much.
• Is the function getting bigger or smaller and by
  how much?
• If f(t) is your position at time t, then f(t) is
  gives your velocity.
• If f(t) is your velocity at time t, then f(t) is
  your acceleration.

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Maximums and Minimums

• At the maximums and minimums the function is not
  changing.
• If the function is not changing then f(x)0.
• We can find the largest (or smallest) value of a
  function by finding where the derivative is zero.

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Areas

• Given a function f(x), what is the area under
  f(x) over an interval?
• How can we find the area if the shape is not
  something we are familiar with?
• Take f(x) x2-x1, with x from 0 to 2.

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• To estimate the area we approximate it with
  rectangles.
• The height of each one is given by the function.
• The base of each is the length divided by the
  number of rectangles.

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• To get a better estimate we can use more
  rectangles.

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• The more rectangles we use the better an estimate
  we get.
• Using a computer to add together 2000 rectangles
  we find the area is approximately 2.667.

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Is There an Easier Way?

• Isaac Barrow (1630-1677) considered the idea of
  an area function F(x).
• What if we wanted the area under f(x) and could
  find a function F(x) with
• F(x) area under f(x) from some point a to x.

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• Barrow asked If there was such a function F(x)
  what would the derivative be?
• We see ?F is roughly a rectangle, so we get

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• A student of Barrow, Isaac Newton (1642-1727)
  found a way to use this idea to calculate area.
• To find the area under the curve f(x), we only
  need to find a function F(x) such that F(x)
  f(x).
• We call F(x) an antiderivative of f(x).
• In general to find the area under f(x) from xa
  to xb, we find F(x) then just calculate F(b)
  F(a).

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• What is the area under f(x) x2-x1 from 0 to 1?
• We saw the derivative of xn nx(n-1) so the
  antiderivative will be x(n-1)/(n-1).

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• Calculating area is only as difficult as finding
  an antiderivative.
• Unfortunately, finding an antiderivative is often
  difficult or impossible.
• For example, the area under f(x)e(-x2) is
  important for statistics but has no
  antiderivative.
• When you can though, it is usually much easier
  than adding up rectangles and gives you an exact
  answer, not an approximation.

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• When Newton originally discovered this approach
  he immediately made enormous advances in almost
  every field of science at the time.
• The idea was unknown to anyone else though for
  years until a German mathematician, Gottfried
  Leibniz (1646-1716) also discovered the idea and
  shared the technique with others.