Classification of Functions

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Classification of Functions

• We may classify functions by their formula as
  follows
• Polynomials
• Linear Functions, Quadratic Functions.
  Cubic Functions.
• Piecewise Defined Functions
• Absolute Value Functions, Step Functions
• Rational Functions
• Algebraic Functions
• Trigonometric and Inverse trigonometric
  Functions
• Exponential Functions
• Logarithmic Functions

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Functions Properties

• We may classify functions by some of their
  properties as follows
• Injective (One to One) Functions
• Surjective (Onto) Functions
• Odd or Even Functions
• Periodic Functions
• Increasing and Decreasing Functions
• Continuous Functions
• Differentiable Functions

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Power Functions
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Combinations of Functions
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Composition of Functions
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Inverse Functions
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Exponential Functions
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Logarithmic Functions
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The logarithm with base e is called the natural
logarithm and has a special notation
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Correspondence between degree and radian
The Trigonometric Functions
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Some values of and
Trigonometric Identities
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Graphs of the Trigonometric Functions
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When we try to find the inverse trigonometric
functions, we have a slight difficulty. Because
the trigonometric functions are not one-to-one,
they dont have inverse functions. The difficulty
is overcome by restricting the domains of these
functions so that hey become one-to-one.
Inverse Trigonometric Functions
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The Limit of a Function
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Calculating Limits Using the Limit Laws
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Infinite Limits Vertical Asymptotes
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Limits at Infinity Horizontal Asymptotes
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Tangents

• The word tangent is derived from the Latin word
  tangens, which means touching.
• Thus, a tangent to a curve is a line that touches
  the curve. In other words, a tangent line should
  have the same direction as the curve at the point
  of contact. How can his idea be made precise?
• For a circle we could simply follow Euclid and
  say that a tangent is a line that intersects the
  circle once and only once. For more complicated
  curves this definition is inadequate.

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Instantaneous Velocity Average Velocity

• If you watch the speedometer of a car as you
  travel in city traffic, you see that the needle
  doesnt stay still for very long that is, the
  velocity of the car is not constant. We assume
  from watching the speedometer that the car has a
  definite velocity at each moment, but how is the
  instantaneous velocity defined?
• In general, suppose an object moves along a
  straight line according to an equation of motion
  , where is the displacement
  (directed distance) of the object from the origin
  at time . The function that describes the
  motion is called the position function of the
  object. In the time interval from to
• the change in position is
  . The average velocity
  over this time interval is

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• Now suppose we compute the average velocities
  over shorter and shorter time intervals
  . In other words, we let approach . We define
  the velocity or instantaneous velocity at
  time to be the limit of these average
  velocities
• This means that the velocity at time is
  equal to the slope of the tangent line at .

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The Derivative of a Function
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Differentiable Functions
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The Derivative as a Function
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What Does the First Derivative Function Say about
the Original Function?
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What Does the Second Derivative Function Say
about the Original Function?
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Indeterminate Forms and LHospitals Rule
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Antiderivatives
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