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?

xx

2x

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

32 2

3% 3.5 0.105

V x x x x

dx

4

Propagated Error

The propagated error is the dy sometimes called the df

The relative error is

The percentage of error relative error * 100%

7.71750.09

( ) 85.75

dy

f x

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

'( )

'( )

C C x dx

R R x dx

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 claim

is a better approximation

1

( )

'( )n

n nn

f xx x

f x

x1

•xn+1

9

Newton-Raphson Method for Approximating Roots

We will create a spreadsheet which demonstrates this concept

1

( )

'( )n

n nn

f xx x

f x