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 2x x

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

4. 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%

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. 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
Suppose C(q) = 0.1q q q 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. 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

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