DIFFERENTIALS

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DIFFERENTIALS

• Section 3.9

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When you are done with your homework, you should
be able to

• Understand the concept of a tangent line
  approximation
• Compare the value of the differential, dy, with
  the actual change in y,
• Estimate a propagated error using a differential
• Find the differential of a function using
  differentiation formulas

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My mentor was exiled from Athens and committed
suicide. I taught and clarified the Pythagorean
philososphy of nature. I taught that atoms were
in the shape of regular polyhedra. Who am I?

• Aristotle
• Plato
• Pythagoras
• Archimedes

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TANGENT LINE APPROXIMATIONS (AKA LINEAR
APPROXIMATION)

• In the last section, we used Newtons Method to
  use a tangent line to a graph to approximate the
  graph.
• In this section we will examine other situations
  where the graph of a function can be approximated
  by a straight line.

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What is the equation of the tangent line at

• Both A and C

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Find the equation of the tangent line T to the
graph of

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Find the equation of the tangent line T to the
graph

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Lets examine what occurs when x approaches c.
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DIFFERENTIALS

• When is small, is
  approximated by
• When we use the approximation above, the quantity
  is usually denoted by dx, and is called the
  differential of x.
• The expression is denoted by dy, and
  is called the differential of y, so we have

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ERROR PROPAGATION

• Estimation of errors propagated by physical
  measuring devices
• Consider x representing the measured value of a
  variable and representing the exact
  value, then is the error in measurement. If
  the measured value x is used to compute another
  value , the difference between
  and is the propagated error.

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DIFFERENTIAL FORMULAS

• Let u and v be differentiable functions of x.
• Constant Multiple
• Sum or Difference
• Product
• Quotient