Linear Approximation and Differentials

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Linear Approximation and Differentials

• Lesson 3.8a

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Tangent Line Approximation

• Consider a tangent to a function at a point x a
• Close to the point, the tangent line is an
  approximation for f(x)

yf(x)

• The equation of the tangent liney f(a) f
  (a)(x a)

f(a)
a
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Tangent Line Approximation

• We claim that
• This is called linearization of the function at
  the point a.
• Recall that when we zoom in on an interval of a
  function far enough, it looks like a line

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New Look at
?y
dy



x ?x
x
?x dx

• dy rise of tangent relative to ?x dx
• ?y change in y that occurs relative to ?x dx

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New Look at

• We know that
• then
• Recall that dy/dx is NOT a quotient
• it is the notation for the derivative
• However sometimes it is useful to use dy and dx
  as actual quantities

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The Differential of y

• Consider
• Then we can say
• this is called the differential of y
• the notation is d(f(x)) f (x) dx
• it is an approximation of the actual change of y
  for a small change of x

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Try It Out

• Note the rules for differentialsPage 200
• Find the differential of3 5x2x e-2x

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Differentials for Approximations
This will be a little bit different from cos(?/2)

• Consider
• Think of this as cos(x)d(cos(x)) dy -sin(x)
  dx
• Then

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Differentials for Approximations

• Solution

Note how close the approximation is
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Assignment

• Lesson 3.8a
• Page 173
• 1 - 21 odd