Interpolation_Foundation

# Interpolation_Foundation - MTHSC 360 – Introduction to...

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Unformatted text preview: MTHSC 360 – Introduction to Interpolation January 16, 2008 1 A Basic Example Suppose that we have an empirical function. We often encounter such functions in books or papers reporting an experiment that measures some physical property. Table 1 lists three properties of air as a function of temperature. The properties are: density ( ρ ), dynamic viscosity ( μ ), and kinematic viscosity ( v ). This data is taken from the text book Fluid Mechanics by Frank M. White, fifth edition, 2002. T ◦ C ρ, kg / m 3 μ, N · s / m 2 v, m 2 / s-40 1.52 1 . 51 × 10- 5 9 . 90 × 10- 6 1.29 1 . 71 × 10- 5 1 . 33 × 10- 5 20 1.20 1 . 80 × 10- 5 1 . 50 × 10- 5 50 1.09 1 . 95 × 10- 5 1 . 79 × 10- 5 100 0.946 2 . 17 × 10- 5 2 . 30 × 10- 5 150 0.835 2 . 38 × 10- 5 2 . 85 × 10- 5 200 0.746 2 . 57 × 10- 5 3 . 45 × 10- 5 250 0.675 2 . 75 × 10- 5 4 . 08 × 10- 5 300 0.616 2 . 93 × 10- 5 4 . 75 × 10- 5 400 0.525 3 . 25 × 10- 5 6 . 20 × 10- 5 500 0.457 3 . 55 × 10- 5 7 . 77 × 10- 5 Table 1: Properties of Air as a funciton of temperature at 1 atm Figures 1, 2, and 3, provide plots of this data, where the data points themselves are indicated with circles and then connected with straight lines. When we study the plots of the data we can observe several things. First, although the data points are only connected with straight lines, the data points are close enough so that the plots almost appear to be smooth curves. Second, the plot in Figure 1 indicates density ( ρ ) as a function of temperature is decreasing and concave up. Similarly, the plots in Figures 2 and 3 indicate that dynamic viscosity ( μ ), and kinematic viscosity ( v ) are both increasing functions of temperature. However, the dynamic viscosity ( μ ) is concave down, while the kinematic viscosity ( v ) is concave up. Suppose that we need to determine the density of air at 72 ◦ . The first step is to observe that 72 ◦ lies between the tabulated temperatures, 50 ◦ and 100 ◦ , for which the corresponding density values are 1 . 09 kg/m 3 and 0 . 946 kg/m 3 . Since 72 ◦ is between 50 ◦ and 100 ◦ the density must be between 1 . 09 kg/m 3 and 0 . 946 kg/m 3 . We can approximate the density funtion over this interval with a straight line that passes through the points (50 , 1 . 090) and (100 , . 946). Using the point-slope form for a straight line – a form that is used a lot in calculus – we can write the equation for the straight line as y = 1 . 09 + . 946- 1 . 09 100- 50 ( x- 50) = 1 . 09- . 002288 ( x- 50) We can now evaluate this straight line function at x = 72 and we get y = 1 . 09- . 00288 (72- 50) = 1 . 09- . 06336 = 1 . 03 That is, the density of air is 1 . 03 kg/m 3 at a temperature of 72 ◦ C. Note that we rounded the final result to three significant digits which was the precision used in the original table. We will have a lot more to say about accuracy, precision, and significant digits in later sections....
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Interpolation_Foundation - MTHSC 360 – Introduction to...

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