For f ( t ) = ( ( ln t ) 2 t , t 3 ) what is the distance between f(2) and f(4)?

Question

For             f              (        t        )              =          (                                    (                          ln              t                        )                    2                t            ,              t        3            )       what is the distance between f(2) and f(4)?

Freddy Doyle · Accepted Answer

Step 1We have a parametric function giving us cartesian coordinates for the specified parameter:            f              (        t        )              =          (                                    (                          ln              t                        )                    2                t            ,              t        3            )      This with       t    =    2   we get:            f              (        2        )              =          (                                    (                          ln              2                        )                    2                2            ,      8      )      And with       t    =    4   we get:            f              (        4        )              =          (                                    (                          ln              4                        )                    2                4            ,      64      )                                          =          (                                    (                                          ln                2                            2                        )                    2                4            ,      64      )                                          =          (                                    (            2                          ln              2                        )                    2                4            ,      64      )                                          =          (                        (                      ln            2                    )                2            ,      64      )      Step 2The distance. d, between these two cartesian coordinates is given by pythagoras:            d      2        =                  (                              (                          ln              2                        )                    2                -                                            (                              ln                2                            )                        2                    2                )            2        +                  (        64        -        8        )            2                                       =                  (                                            (                              ln                2                            )                        2                    2                )            2        +                  (        56        )            2                                       =                            (                      ln            2                    )                4            4        +    3136  Giving:       d    =                                                      (                              ln                2                            )                        4                    4                +        3136                            ≈    56.000515    ...

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