Part 3 of Statistics within Capacity Management and Performance Engineering Series.

Having spent the past 2 weeks within the world of probability and distributions a change was necessary. Herein we will be discussing linearization. Let’s assume you have plotted a trend and it is not linear, how can you interpret this and make a prediction from this?

Let’s assume we have a function of one variable (these principles can be expanded for multivariable systems):

This is easily manipulated within our example. Let us consider a point on the this line *(x _{0},f(x_{0})*. We know this point and we want the approximation at

*(x,f(x))*. With this little information we can make a linearization of the curve.

With observation it is clear that the linear approximation then follows the form of:

This is rather convenient as it matches the form of a linear function. The approximation is said to be exact at the point at which it is defined *(x _{0},f(x_{0}))*. The further from this point the approximation needs to be treated with caution.

#### Example

Using (2) and substituting the values:

And now we can simplify:

Thus we now have a linear approximation at the point 1,1* *for* y=x ^{2}* using this we can now make an approximation. Let’s assume we want to know what the value of

*y*will be when

*x*is 1.1. The approximation gives 1.2, while the actual value is 1.21. Our approximation falls within 1% of the actual value. This is a powerful tool for making approximations close to a known value. And can provide a good estimate before much more complicated analysis is undertaken.