PRECALCREVIEW:MODELINGwithFUNCTIONSII
    Modeling with Inverse Functions : needtofindwhenmodelpredictssomevalue.
      • Inverse Functions Basic Ideas: given y = f(x), inverse function is x = f−1(y).
       Issues for f−1(y): domain and range ?, uniqueness?, explicit form?
      • Modeling Example (1.6.36): if shrew population growth rate F as function of
       cocoon density C is F = .055(220C − 11000)/(320C), find C(F) and C(.02)?
      • Modeling Example (1.R.16): if pollution level P in Lake Bowegan as function of
       time t in days is P = 25 − 15cos(2πt/365), find t(P) and t(40)?
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                  PRECALCREVIEW:MODELINGwithFUNCTIONS
           • Log Functions as Inverse Functions: basic form y = logb(x), for b > 0.
            What is inverse function?
            Commonvs. Natural logs?
                                                          y
            Basic properties: logb(xy) =?, logb(x/y) =?, logb(x ) =?, base change?
            ModelingExample(1.6.38): ifMexicanpopulationmodelisP(t) = 67.38(1.026)t,
              find t(P) and doubling time.
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                 PRECALCREVIEW:MODELINGwithLOGFUNCTIONS
                                                                                              m
            • Log-LogPlotsandDataFitting: givenx,y data, find“best-fit” function y = kx .
              Example: River Object Diameter vs Transport Speed (text 1.6.44)
                               3.5
                                3
                               2.5
                                2
                               1.5
                                1
                               0.5
                                00   20   40   60   80   100  120  140  160   180
              Solution: take logs and find best fit line for “log-log” data
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                 PRECALCREVIEW:MODELINGwithLOGFUNCTIONS
              Example: River Object Diameter vs Speed (1.6.44) Log-Log Plot
                                                  River Object Data Log−Log Plot
                                 1.5
                                  1
                                 0.5
                                  0
                                 −0.5
                                Log of Current Speed−1
                                 −1.5
                                  −2
                                 −2.5
                                  −2    −1    0     1    2     3    4     5    6
                                                    Log of Object Diameter
              Matlab:
              x = [.2 1.3 5 11 20 45 80 180 ];
              y = [.1 .25 .5 .75 1 1.5 2.5 3.5 ];
              plot(log(x),log(y),’*’)
              title(’River Object Data Log-Log Plot’)