The Gradient - Multivariable Calc + Calc III Notes

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42
Recall that Drf xo yo Du f x o yo fx xo yo u tfy Xo yo U2
I
Notice that this is fx Xoyo fyxoyo uh
If f hasfirstpartials at xo yo then thegradient
of f denoted Of Xo yo is the rector
cagraddf f xo yo fx Xo Yo Fy Xoyo
T is an operator acting on functions
at each point x y then I f x y gives the direction
ofthe maximum slope of the surface Z f x y and
themaximumslope is 11 If x y ll If x y gives the minimum slope
If f is diffrentiable at Xo yo and I f x o yo 0
then of x o yo is normal to the level curve of f xo y
Remember that If pointsinthedirectionthat f increasesmost rapidly
2 2000 2 2 492 and a hikeris at MountCalc at 20,51100
1 Steepestroute whatdirection 2 same height whatdirection
1 move in Pf x y fx 20,5 Fy 20,5
4x Fy 8y so I f x y L 80 40 401 21 1
2 orthogonal to TZ give level curves so 1 1,2 and 1 27

42
Lecture 42 Problems
1 x y Xcosytysinx.whichvect.perptolevelaerveflxiy70 o.o
orthogonal to Tf x y Tf fx o o fy o o
Excosytysinx cosy tycosx of 1,0 II
Excosytysinx xsingtsinx
2 same as example C
3 f 0,07 0 and for all x y w x y't l 11Of x y 1113
Of x y fx fy