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mws_civ_inp_txt_ndd_examples

mws_civ_inp_txt_ndd_examples - Chapter 05.03 Newtons...

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05.03.1 Chapter 05.03 Newton’s Divided Difference Interpolation – More Examples Civil Engineering Example 1 To maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth data for a lake in Table 1. Table 1 Temperature vs. depth for a lake. Temperature, C T Depth, m z 19.1 0 19.1 –1 19 –2 18.8 –3 18.7 –4 18.3 –5 18.2 –6 17.6 –7 11.7 –8 9.9 –9 9.1 –10

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05.03.2 Chapter 05.03 Figure 1 Temperature vs. depth of a lake. Using the given data, we see the largest change in temperature is between m 8 z and m 7 z . Determine the value of the temperature at m 5 . 7 z using Newton’s divided difference method of interpolation and a first order polynomial. Solution For linear interpolation, the temperature is given by ) ( ) ( 0 1 0 z z b b z T Since we want to find the temperature at m 5 . 7 z , and we are using a first order polynomial, we need to choose the two data points that are closest to m 5 . 7 z that also bracket m 5 . 7 z to evaluate it. The two points are 8 0 z and 7 1 z . Then , 8 0 z 7 . 11 ) ( 0 z T , 7 1 z 6 . 17 ) ( 1 z T gives ) ( 0 0 z T b 7 . 11 0 1 0 1 1 ) ( ) ( z z z T z T b 8 7 7 . 11 6 . 17
Newton’s Divided Difference Interpolation – More Examples: Civil Engineering 05.03.3 9 . 5 Hence ) ( ) ( 0 1 0 z z b b z T ), 8 ( 9 . 5 7 . 11 z 7 8 z At 5 . 7 z ) 8 5 . 7 ( 9 . 5 7 . 11 ) 5 . 7 ( T C 65 . 14 If we expand ), 8 ( 9 . 5 7 . 11 ) ( z z T 7 8 z we get , 9 . 5 9 . 58 ) ( z z T 7 8 z This is the same expression as obtained in the direct method. Example 2 To maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth data for a lake in Table 2.

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mws_civ_inp_txt_ndd_examples - Chapter 05.03 Newtons...

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