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14.6
4-6 Determine la derivada direccional de f en el punto dado en la dirección que indica el ángulo θ :

4.- f (x,y)= x2 y3-y4 , (2,1) , θ= π4
F(x)= 2xy3
F(y)=3y2x2-4y3
Duf(x,y) = 2xy3cosπ4 + 3y2 x2-4y3 sen π4
Duf(2,1) = 1.41xy3+2.12xy3-2.88y3
Duf(2,1) = 1.41(2)(1)3+ 2.12(1)2(2)2-2.82(1)3
Duf(2,1) = 2.82 +8.48-2.82
Duf(2,1)=8.48


5.- f (x,y)= ye-x , (0,4) , θ= 2π3F(x)= - ye-x
F(y)= e-x
Duf(x,y) = -ye-x cos2π3 + e-x sen 2π3
Duf(x,y) ye-x2 + .86e-x
Duf(0,4) = 4e-02 + .86e-0
Duf(0,4)=2.86

6.- f (x,y)= xsenxy, (2,0), θ= π3
F(x)= cos(xy)F(y)=xcos(xy)
Duf(x,y) = cos⁡(xy) cosπ3 + xcos(xy) sen π3
Duf(x,y) = cos⁡(xy)2 + .866xcos(xy)
Duf(2,0) = cos⁡(20)2 + .8662cos((2)(0))
Duf(2,1) = .5+1.7
Duf(2,0)=2.2

11-17 calcule la derivadadireccional de la función en el punto en la dirección del vector

11.- f (x,y)= 1+2xy, (3,4) , v=4,-3
F(x)= 2y
F(y)=xy
∇ f (x,y)= 2y i + xy j
∇ f (3,4)= 4 i + 32 j
U=425 i-325 j
Duf(3,4)=4i+ 32j425 i-325 j
Duf(3,4)= 1625-9225=2.3

13.- g(p,q) = p4 -p2q3, (2,1), v= i+3j
g(p) = 4p3 -2pq3
g(q) =-3 p2 q2
∇g(p,q)= (4p3 -2pq3) i- 3p2 q2j
∇g(2,1)= 28i-12j
U= 110i + 310j
Dug (2,1)= 28i-12j 110i + 310j
Dug (2,1)= 2810 + 3610 = 8.8-11.3= -2.58

14.- g(r,s)= tan-1(rs), (1,2), v=5i+10j
g(r)= s1+(rs)2
g(s)= r1+(rs)2
∇g(r,s)= 51+(rs)2 i + r1+(rs)2 j
∇g(1,2)= 25 i + 15 jU=5125 i+ 10125j
Dug(1,2) =25 i + 15 j 5125 i+ 10125j
Dug(1,2) = 105125 i+ 105125j = .17+.17 = .35

15.- f (x,y,2)= xey +yex + zex , (0,0,0) , v= 5,1,-2
f (x)= ey +yex + zex
f (y)= xey + ex
f(z)= ex
∇f(x,y,z)= ey +yex + zex i+ xey + ex j+ exk
∇f(0,0,0)= i + j + k
U= 530 i+ 130 j - 230 k
Duf (x,y,z) = i + j + k 530 i+ 130 j - 230 k
Duf (0,0,0) = 530 + 130 - 230 = .9 + .18 - .3 =.71

16.- f(x,y,z)= xyz , (3, 2, 6) , v=-1,-2, 2
f(x)= 12 x,y,z-12
f(y)= 12 x,y,z-12
f(z)= 12 x,y,z-12
∇f(x,y,z)= 12 x,y,z-12 i + 12 x,y,z-12 j + 12 x,y,z-12 k
∇f(3,2,6)= .08i...
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