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13 Cards in this Set
- Front
- Back
Parametrization |
r(t) = <x(t), y(t), z(t)> |
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Vector equation |
r(t) = a r(t)=<t^2 +1, 3t+4, e^t-1> a=<2,7,1> 1 vector equation = 3 regular equations. |
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Speed |
(||r(t+h)-r(t)||)/h Distance/time elapsed |
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Vector equation of tangent line |
T(t)=r(t) + t*r'(t) T(t)= a +tv |
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Length of curve |
Integral from 0-t of ||r'(t)||dt |
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Partial derivatives |
Fx and Fy of f(x,y) |
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Tangent plane approx/linear approx |
z = L(x,y) = f(a,b) + Fx(a,b)(x-a) + Fy(a,b)(y-b) z = D + A(x-a) + B(y-b) (Aa + Bb -D) = Ax + By -Z Normal = <A, B, -1> Tan plane = <Fx, Fy, -1> |
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Gradient vector |
Vf(a,b) = <Fx, Fy> |
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Chain rule |
Vf * r'(t) |
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Directional derivative |
Z = f(x,y) u = <h, k> unit vector Duf(a,b) = Vf(a,b) * u |
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Critical points |
Fx(a,b) = 0 or does not exist Fy(a,b) = 0 or does not exist |
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Local min and max |
Max : f(x,y) <= f(a,b); D>0, Fxx(a,b)<0 Min : f(x,y) >= f(a,b); D>0, Fxx(a,b)>0 If D<0 then f has a saddle point |
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Hessian |
Fxx(x,y), Fxy(x,y) Fyx(x,y), Fyy(x,y) |