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Linear Approximation and Differentials

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Title: Linear Approximation and Differentials


1
Linear Approximation and Differentials
  • Lesson 3.8b

2
Propagated Error
  • Consider a rectangular box with a square base
  • Height is 2 times length of sides of base
  • Given that x 3.5
  • You are able to measure with 3 accuracy
  • What is the error propagated for the volume?

x
3
Propagated Error
  • We know that
  • Then dy 6x2 dx 6 3.52
    0.105 7.7175This is the approximate
    propagated error for the volume

4
Propagated Error
  • The propagated error is the dy
  • sometimes called the df
  • The relative error is
  • The percentage of error
  • relative error 100

5
Marginal Analysis in Economics
  • C(x) cost to produce x unitsR(x) revenue
    gained by selling x units
  • C(x) called the marginal costR(x) called the
    marginal revenue
  • Consider the concept of the differential in this
    context

6
Marginal Analysis in Economics
  • We could say
  • where the dx the increase or decrease in sales
  • Assume ?x dx 1 unit
  • Then the differential for C(x) or R(x) is
  • the cost of producing the x 1st unit
  • the revenue gained for the x 1st unit

7
Marginal Analysis in Economics
  • SupposeC(q) 0.1q3 - 0.5q2 500q 200Current
    level is 4 units
  • What is the change of cost if we only produce 3.9
    units
  • dy C(q)dq
  • q 4 and dq 0.1

8
Newton-Raphson Method for Approximating Roots
  • Given f(x) we seek a root
  • If xn is an approximation for the root
  • Then we claimis a better approximation


xn1
x1
9
Newton-Raphson Method for Approximating Roots
  • We will create a spreadsheet which demonstrates
    this concept

10
Assignment
  • Lesson 3.8b
  • Page 173
  • Exercises 23 37 odd
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