Term
| Partial derivative (with respect to x for example) |
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Definition
δz/δx = fx
Taken by holding y constant and taking the derivative of x
f = xy + x2+y2
fx= y + 2x fy= x + 2y
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Term
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Definition
| The function f is continuous |
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Term
| gradient of g(x,y,z) at the point (xo,yo,zo) |
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Definition
| gx(xo,yo,zo)i + gy(xo,yo,zo)j + gz(xo,yo,zo)k |
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Term
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Definition
set z = to a constant, evaluate
the shape of x and y
if it is a set of more variables, could be a level surface, level hypesurface, etc... |
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Term
| Equation of a plane tangent to a surface g(x,y,z) at (xo,yo,zo) |
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Definition
| fx(xo,yo,zo)(x-xo) + fy(xo,yo,zo)(y-yo) + fz(xo,yo,zo)(z-zo) = 0 |
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Term
Tangent plane to z = f(x,y)
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Definition
| fx(xo,yo)(x-xo)+fy(xo,yo)-(z-zo) = 0 |
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Term
| Plane approximation for z = f(x,y) |
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Definition
Find xo,yo such that zo = f(xo,yo) is easy to compute
z = zo +fx(xo,yo)(x-xo) +fy(xo,yo)(y-yo) |
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Term
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Definition
Used to find the boundary extrema of a function f(x,y) bounded by another function g(x,y) = c
fx(x,y) = λgx(x,y)
fy(x,y) = λfy(x,y)
g(x,y) = c
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Term
| Direction derivative of g(x,y,z) in direction of some vector v |
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Definition
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Term
| Critical points of g(x,y,z) |
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Definition
Critical points exist when
gx=0
gy=0
gz=0
all at once |
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Term
| Evaluate what a critical value is |
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Definition
plug in critical point to D(x,y) (hope to god original function is in form z = f(x,y))
D(x,y) = fxx(x,y)fyy(x,y) - (fxy(x,y))2
if D(x,y) > 0 and fxx(x,y) > 0 (x,y) = rel min
if D(x,y) > 0 and fxx(x,y) < 0 (x,y) = rel max
if D(x,y) < 0 (x,y) = saddle point
if D(x,y) = 0 you know nothing more |
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