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Uses and Limitations of Dynamic Geometry and Computer Algebra in the Analysis of the Shrinking Circle Problem
Dougla= s B. Meade
Department of Mathematics
Univer= sity of South Carolina
ACMS Biennial Conference, Messiah College
1 June 2007
------=_NextPart_01C7A94C.ACFDEEC0 Content-Location: file:///C:/ECBAB22B/ACMS2007-Shrink_files/slide0002.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Uses and Limitations of Dynamic Geometry and Computer Algebra in the Analysis of the Shrinking Circle Problem
Shrinking Circle Problem
•Let <= /div>
•<= /span>C be t= he unit circle with center (1,0) <= /span>
•<= /span>Cr be the circle with radius <= span style=3D'font-size:88%'>r and = center (0,0)
•<= /span>P be t= he point (0, r)
•<= /span>Q be t= he upper point of intersection of C and C= r
•<= /span>R be t= he intersection of the line PQ and the x= -axis.
•
•What happens to <= span style=3D'font-size:88%'>R as <= /span>Cr shrinks to the origin? =
•= 3;
<= span style=3D'mso-special-format:nobullet;display:none'>•Stewart, E= ssential Calculus: Early Transcendentals,<= br> Thomson Brooks/Cole, 2007, p. 45, Exercise 56.
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------=_NextPart_01C7A94C.ACFDEEC0 Content-Location: file:///C:/ECBAB22B/ACMS2007-Shrink_files/slide0005.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Uses and Limitations of Dynamic Geometry and Computer Algebra in the Analysis of the Shrinking Circle Problem
Proofs
•Numeric
–Extremely sensitive to floating-point cancellation
–= 3;
•Symbolic
–Indeterminate form
(l’Hopital, or simply rationalize)
–= 3;
•Geometric <= /div>
–Tom Banchoff (Brown University)
–
–
<= ![endif]> <= ![endif]> <= ![endif]>
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Computer Algebra in the Analysis of the Shrinking Circle Problem
OP =3D OQ
         is isosceles
SR=3DSQ < OS
OR =3D OS + SR < 2 OS
P à O
Q à O
QS à OS <= /div>
OR =3D OS+SR =3D OS+QS 2OS
Geometric Proof
•
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