1. 1 Linear Approximation and Differentials Lesson 3.8b

2. 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 x 2x

3. 3 Propagated Error We know that Then dy = 6x 2 dx = 6 * 3.5 2 * 0.105 = 7.7175 This is the approximate propagated error for the volume

4. 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. 5 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

6. 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 + 1 st unit the revenue gained for the x + 1 st unit

7. 7 Marginal Analysis in Economics Suppose C(q) = 0.1q 3 - 0.5q 2 + 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

8. 8 Newton-Raphson Method for Approximating Roots Given f(x) we seek a root If x n is an approximation for the root Then we claim is a better approximation x1x1 x n+1

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