4.1 Linear Approximations Thurs Jan 7

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4.1 Linear ApproximationsThurs Jan 7

• Do Now
• Find the slope of each function at
• 1) Y sinx
• 2) Y cosx

2
Quiz Review

• Quiz retakes until Fri

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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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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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Closure

• Use Linear Approximate to estimate f(3.02) - f(3)
  if f(x) x4
• HW p.213 1-13 odds, 17-21 odds

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4.1 LinearizationFri Jan 8

• Do Now
• Find the equation of the tangent line of
• at

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HW Review p.213 1-13 17-21

• 1) 0.12 19) -0.0005
• 3) -0.00222 21) 0.083333
• 5) 0.003333
• 7) 0.0074074
• 9) 0.04930
• 11) -0.03
• 13) -0.007
• 17) 0.1

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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

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Ex 1

• Compute the linearization of
• at a 1

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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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Ex 4

• Use linearization to approximate cos(1)

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More examples

• Use a linear approximation to approximate

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Closure

• Journal Entry Use Linear Approximation to
  estimate the square root of 26
• HW p.214 45-51 59-63 odds

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Linear Approximation PracticeMon Jan 11

• Do Now
• Use linear approximations to estimate

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HW Review p.214 45-51 59-63

• 45) L(x) 4x - 3
• 47) L(x) x - pi/4 1/2
• 49) L(x) -1/2 x 1
• 51) L(x) 1
• 59) L(17) 0.24219
• 61) L(10.03) 0.00994
• 63) L(64.1) 4.002083

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Linearization Review

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

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Practice

• (green book) Worksheet p.249 5-10, 17-22

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Closure

• Hand in Use linear approximation to estimate
• HW Finish worksheet p.249 5-10 17-22

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HW Review p.249 5-10

• 5)
• 6)
• 7)
• 8)
• 9)
• 10)

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HW Review p.249 17-22

• 17) .842
• 18) .788
• 19) 2.00125
• 20) 2.0025
• 21) 2.005
• 22) 1.030