4.2 Maximum and Minimum Values

Mathematica script by Chris Parrish,
   cparrish@sewanee.edu

Problems from James Stewart,
  "Calculus: Concepts and Contexts,"
  Second Edition, Brooks/Cole, 2001

Space Shuttle Endeavour

Stewart, Exercise 4.2.54

We are keeping track of the timing of certain specific events that occur during the first 125 seconds of a launch of the Space Shuttle Endeavour.
The table shown below records the timing of those events, and the corresponding velocities of the Space Shuttle Endeavour for a specific launch.
See Stewart, Exercise 4.2.54, for details.

In[27]:=

Needs["Graphics`Colors`"] Clear[vs, ts, data, dots, dotPlot, interp, approximation,  ... p;  TableHeadings {{"t (sec)", "velocity (ft/sec)"}, None}]

Out[32]//TableForm=

t (sec) 0 10 15 20 32 59 62 125
velocity (ft/sec) 0 185 319 447 742 1325 1445 4151

Let's plot those points.

In[33]:=

dots = Transpose[data] ;  RowBox[{RowBox[{dotPlot,  , =,  , RowBox[{ListPlot, [, RowBo ... bsp; , AxesLabel  {"t (sec)", "velocity (ft/sec)"}}], ]}]}], ;}]

[Graphics:HTMLFiles/4.2_minMax_3.gif]

Now construct a cubic polynomial to approximate that data ...

In[35]:=

interp = Fit[dots, {1, x, x^2, x^3}, x]

Out[35]=

RowBox[{RowBox[{-, 21.2687}], +, RowBox[{24.9817,  , x}], -, RowBox[{0.115534,  , x^2}], +, RowBox[{0.00146137,  , x^3}]}]

In[36]:=

approximation[y_] := (interp /. x  y)  curvePlot = Plot[approximation[y], {y,  ... nbsp;   AxesLabel  {"t (sec)", "velocity (ft/sec)"}] ;

[Graphics:HTMLFiles/4.2_minMax_7.gif]

... and graph the polynomial and the data to verify that we have a reasonable correspondence.

In[38]:=

Show[curvePlot, dotPlot] ;

[Graphics:HTMLFiles/4.2_minMax_9.gif]

Now let's use the approximating poynomial to find a model for the acceleration of the shuttle.

In[39]:=

                                                                                               ...      AxesLabel  {"t (sec)", acceleration (ft/ sec )}] ;

[Graphics:HTMLFiles/4.2_minMax_11.gif]

Use this picture to estimate the maximum and minimum values of the acceleration during the first 125 seconds of the launch.


Created by Mathematica  (March 16, 2004)