4.1 Linear Approximations Mon Dec 21

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4.1 Linear ApproximationsMon Dec 21

• Do Now
• Find the equation of the tangent line of each
  function at
• 1) Y sinx
• 2) Y cosx

2
Quiz Review

• Retakes this week

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Differentials

• We define the values
• as the difference between 2 values
• These are known as differentials, and can also
  be written as dx and dy

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Linear Approximations

• The tangent line at a point of a function can be
  used to approximate complicated functions
• Note The further away from the point of
  tangency, the worse the approximation

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Linear Approximation of df

• If were interested in the change of f(x) at 2
  different points, we want
• If the change in x is small, we can use
  derivatives so that

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Steps

• 1) Identify the function f(x)
• 2) Identify the values a and
• 3) Use the linear approximation of

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Ex 1

• Use Linear Approximation to estimate

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Ex 2

• How much larger is the cube root of 8.1 than the
  cube root of 8?

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Ex 3,4

• In the book bc lots to type

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You try

• 1) Estimate the change in f(3.02) - f(3) if f(x)
  x3
• 2) Estimate using Linear Approximation

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Linearization

• Again, the tangent line is great for
  approximating near the point of tangency.
• Linearization is the method of using that tangent
  line to approximate a function

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Linearization

• The general method of linearization
• Find the tangent line at x a
• Solve for y or f(x)
• If necessary, estimate the function by plugging
  in for x
• The linearization of f(x) at x a is

13
Ex 1

• Compute the linearization of
• at a 1

14
Ex 2

• Find the linearization of f(x) sin x, at a 0

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Ex 3

• Find the linear approximation to f(x) cos x
  at and approximate cos(1)

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Closure

• Journal Entry Use Linearization to estimate the
  square root of 37
• HW p.214 1, 5, 9, 13, 17-33 odds,37 45 47 49,
  53-63 odds

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Linear Approximation ReviewTues Dec 22

• Do Now
• Use linear approximations to estimate

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HW Review p.214
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Linearization Review

• We can use linear approximation (tangent line
  equations) for 2 uses
• 1) Find the difference between two values of f(x)
• 2) Estimate the value of f(x) at specific points

20
Closure

• Find the linearization of f(x) 1/x at a 2
• HW Retry hw problems