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Exercise 29, Section 13.1

> restart;

> with( plots ): 

Warning, the name changecoords has been redefined

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Parametric curve for Exercise 29 (Section 13.1)

> x := 1/(1+t^2);

> y := 1/(t*(1-t));

> PARAM := t = 0..1;

x := 1/(1+t^2)

y := 1/(t*(1-t))

PARAM := t = 0 .. 1

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Initial plot of curve

> plot( [x,y, PARAM] );

[Maple Plot]

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A slightly more refined view of the curve

> plot( [x,y, PARAM], view=[0..1,0..100], labels=['x','y'] );

[Maple Plot]

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Animated plot of curve

> animatecurve( [x,y, PARAM], view=[0..1,0..100], frames=75, numpoints=200 );

[Maple Plot]

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Note how the curve moves quickly down the right side, slowly across the bottom, then rapidly up the left side.

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The problem in the text is to find dy/dx and d^2*y/(dx^2) . The Maple command for differentiation is diff .

> dxdt := diff( x, t );

dxdt := -2/(1+t^2)^2*t

> dydt := diff( y, t );

dydt := -1/(t^2*(1-t))+1/(t*(1-t)^2)

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and so

> dydx := dydt / dxdt;

dydx := -1/2*(-1/(t^2*(1-t))+1/(t*(1-t)^2))*(1+t^2)...

> yp := simplify( dydx );

yp := -1/2*(-1+2*t)*(1+t^2)^2/t^3/(-1+t)^2

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The second derivative is computed using

> dypdt := diff( yp, t );

dypdt := -(1+t^2)^2/t^3/(-1+t)^2-2*(-1+2*t)*(1+t^2)...

> dy2dx2 := simplify( dypdt / dxdt );

dy2dx2 := -1/4*(1+t^2)^3*(-9*t+7*t^2+3*t^3+3)/t^5/(...

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