PRECALCREVIEW:MODELINGwithFUNCTIONSI
         Modeling with Linear Functions
               • Basic Linear Function is y = mx + b, with slope m and y−intercept b.
                    Alternate (point-slope) form y = y +m(x−x ), given one data point (x ,y ).
                                                               1              1                                1   1
               • Important Application is Data Fitting (linear regression): given x,y data,
                    find the “best-fit” line for the data.
                 Example: Columbia River Depth vs Flow Velocity Data (Problem 1.2.45)
                                                     Problem 45 River Data Channel Flow Velocity as funciton of River Depth 
                                             1.6
                                             1.4
                                             1.2
                                              1
                                            Flow Velocity0.8
                                             0.6
                                             0.4
                                             0.20     2      4      6       8      10      12
                                                                  Depth in Feet
                 Data value lists for x and y
                 x = [ .7      2 2.6 3.3 4.6 5.9 7.3 8.6 9.9 10.6 11.2];
                 y = [1.55 1.11 1.42 1.39 1.39 1.14 .91 .59 .59 .41 .22];
                                                                                                                                    2
                   PRECALCREVIEW:MODELINGwithLINEARFUNCTIONS
                   Modeling problem: how to find good m and b values for y = mx + b.
                   Some possibilities:
                         a) use first two data values: slope m = (y2 − y1)/(x2 − x1) ≈ −.3385.
                         b) use first and last data values: slope m = (y                       −y )/(x −x )≈−.1267.
                                                                                           11      1      11       1
                         c) use all data values to find “best-fit” line:
                            using calculus and Matlab m = −.1165, b = 1.681.
                                                                    Problem 45 River Data Fit Lines 
                                               2
                                              1.5
                                               1
                                              0.5
                                               0
                                             −0.5
                                               −1
                                             −1.5
                                               −2
                                             −2.50        2        4         6        8         10       12
                   Estimated river velocity with 12 ft depth is ?
                                                                                                                                                                  3
                       Math formulas for best-fit line slope and intercept:
                                                          nPxy −Px Py                                        Py −mPx
                                                                     i  i             i         i                   i                  i
                                                 m=              P 2             P 2 , b=                                               ;
                                                              n       x −( x)                                            n
                                                                         i               i
                       Matlab instructions for previous graph
                          x = [.7 2 2.6 3.3 4.6 5.9 7.3 8.6 9.9 10.6 11.2];
                          y = [1.55 1.11 1.42 1.39 1.39 1.14 .91 .59 .59 .41 .22];
                          p = polyfit( x, y, 1 ); m = p(1), b = p(2)
                          m1 = (y(2)-y(1))/(x(2)-x(1));
                          m2 = (y(11)-y(1))/(x(11)-x(1));
                          plot(x,y,’*’,x,y(1)+m1*(x-x(1)),x,y(1)+m2*(x-x(1)),x,b+m*x)
                                                                                                                                    4
                  PRECALCREVIEW:MODELINGwithFUNCTIONSCONT.
          Modeling with Periodic Functions
                • Basic Periodic Function is y = acos(bx) or y = acos(bx − c)
                      with amplitude |a|, period 2π/b, and phase shift c/b;
                      could also use y = acos(bx − c) + d or sin instead of cos.
                • Model Application: need to find good values for a, b, and maybe c, d.
                   Example: some plotted data values
                                              6
                                              5
                                              4
                                              3
                                              2
                                              1
                                              0
                                             −1
                                             −20     1     2     3    4     5     6     7     8     9    10
                   a?, b?, c?, d?.