Linear Approximation and Differentials

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Linear Approximation and Differentials

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Presentation transcript:

Linear Approximation and Differentials Lesson 3.8b

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 2x x

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

Propagated Error The propagated error is the dy The relative error is sometimes called the df The relative error is The percentage of error relative error * 100%

Marginal Analysis in Economics C(x) = cost to produce x units R(x) = revenue gained by selling x units C’(x) called the marginal cost R’(x) called the marginal revenue Consider the concept of the differential in this context

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

Marginal Analysis in Economics Suppose C(q) = 0.1q3 - 0.5q2 + 500q + 200 Current 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

Newton-Raphson Method for Approximating Roots Given f(x) we seek a root If xn is an approximation for the root Then we claim is a better approximation • xn+1 x1

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

Assignment Lesson 3.8b Page 173 Exercises 23 – 37 odd