Wind – Chill Index “A Calculus Approach”. By Felix Garcia. Rationale. In regions with severe winter weather, the wind – chill index is often used to describe the apparent severity of the cold.
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Instead, we use values of W compiled by the NOAA (National Oceanographic and Atmospheric Administration) and the Meteorological Service of Canada.
The following table is an excerpt from that data.
The values in the table is the perceived temperature W when the actual temperature is T and the wind speed is v.
For example, if T = -15˚C and v = 50 km/h the subjective temperature is -29˚C ( the intersection of the row that corresponds to -15˚C and the column that corresponds to 50 km/h.
Actual Temperature (˚C)
Which is the rate of the change of the dependent variable W with respect to change in one of the independent variables T or v ?
This rate is called the partial derivative of W with respect to T or v.
In order to make an estimate of those partial derivatives we consider two rates centered at the given point and we take the average of both.
For example …
and we take the average of both of them. In the vertical direction, h = 5 therefore
rates that we must average to estimate
How we can use the linearization of the function determined by the table to make approximations to values not shown in the table?
The linearization L(T,v) is by definition the tangent plane to the function at the given point, i.e.
Assume that we want to estimate W(-17,42) using the linearization. We have
Discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers.
The image at left shows a solution to a discretized partial differential equation, obtained with the finite element method.
This is an example of the so-called discretization of the continuous that started with the introduction of computers into everyday life.
Without the use of a continuous model and the traditional machinery of calculus we can derive good estimates that a few decades ago were unheard of.