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Re: Table to find lower and upper estimate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg68219] Re: Table to find lower and upper estimate
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Sat, 29 Jul 2006 01:00:53 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <ea72lp$k6p$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <ea72lp$k6p$1 at smc.vnet.net>,
 "T Harris" <tdh1967 at bellsouth.net> wrote:

> I hope someone could tell me where to look to or if easy enough, to advise 
> me on how to use Mathemetica for the type of problem I have below.  I 
> already have worked it and have the lower estimate to be -475 and the upper 
> estimate to be -85.  I was hoping to put Mathematica to a practical use here 
> and I really don't know how to do this.   I have typed the problem in 
> exactly as shown in James Stewart's, Calculus 5th Edition.

Another approach to solving this problem is interpolation.

> A table of values of an increasing function f is shown.  Use the table to 
> find lower and upper estimates for  f(x) dx.

 xdata = {0, 5, 10, 15, 20, 25}; 
 fdata = {-42, -37, -25, -6, 15, 36}; 

Lower estimate:

  lower = Interpolation[Transpose[{xdata, RotateRight[fdata]}],
     InterpolationOrder -> 0]


  Plot[lower[t], {t, 0, 25}]


  Integrate[lower[t], {t, 0, 25}]
  
  -475

Upper estimate:

  upper = Interpolation[Transpose[{xdata, fdata}],
     InterpolationOrder -> 0]


  Plot[upper[t], {t, 0, 25}]


  Integrate[upper[t], {t, 0, 25}]

  -85

Cubic interpolation:

  int = Interpolation[Transpose[{xdata, fdata}]]

  Plot[int[t], {t, 0, 25}]


  Integrate[int[t], {t, 0, 25}]

  -6905/24

with numerical value

  N[%]

  -287.7083333333333

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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