(c)’ = 0 | (sin u)’ = cos u * u’ | (arcsin u)’ = u’ / sqrt(1 – u^2) | (cu)’ = c * u’ | (1/u)’ = -u’ / u^2
(u^n)’ = n * u^(n-1) * u’ | (cos u)’ = -sin u * u’ | (arccos u)’ = -u’ / sqrt(1 – u^2) | (u ± v)’ = u’ ± v’ | (1/u^n)’ = -n * u’ / u^(n+1)
(e^u)’ = e^u * u’ | (tan u)’ = sec^2 u * u’ | (arctan u)’ = u’ / (1 + u^2) | (uv)’ = u’v + uv’
(a^u)’ = a^u * ln(a) * u’ | (cot u)’ = -csc^2 u * u’ | (arccot u)’ = -u’ / (1 + u^2) | (u/v)’ = (u’v – uv’) / v^2
(ln u)’ = u’/u | (sec u)’ = sec u * tan u * u’ | (arcsec u)’ = u’ / ((u) * sqrt(u^2 – 1)) | f(g(x))’ = f'(g(x)) * g'(x)
(log_a u)’ = u’ / (u * ln a) | (csc u)’ = -csc u * cot u * u’ | (arccsc u)’ = -u’ / ((u) * sqrt(u^2 – 1)) | (sqrt(u))’ = u’ / (2*sqrt(u))
Vector Gradiente:
∇f(P)=(∂f/∂x(P),∂f/∂y(P))
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Derivar Hx , Hy , Hz | si no hay g(x,y) Fx= ∇f(u,v,w) . (UxVx,Wx) … Fz | en v (direcc) es ∂f/∂v(P)
| en P (min crecí) es -∇f(P)
-Fun expl:
f(x,y)
| luego Hx(n , m , p(coord P)) = reemp en Hx derivada = ∇f(p)
| Derivada Direccional | 3
Direcc de crecí + rápido en P(max crecí)
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El valor de decre +rapi
calcula ∂f/∂x y ∂f/∂y, armar ∇f=(fx,fy).
| F(x,y)=g(x,y)f(⋯) (regla de la cadena) |
Duf(x,y)=∇f(x,y)⋅u | es ∇f(P). Y el valor es ||∇f(P)|| | es – ||∇f(P)||.
Si piden en P, sustituye x,y por las coord de P.
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Define ∇F(P)=(Fx,Fy,Fz) luego F(x,y,z)=g(x,y).F(u,v,w)
| Aplicaciones: | 4. v es direcc de decrecí en P | tasa de cam en y:
-cuando z=f(x,y)definido implí por F(x,y,z)=0 | poner u=fun1 ^ v=fun2 ^ w=fun3 |1. Tasa o razón de cambio en P en U(direcc)
| cuando ∂f/∂v(P)<0, | u =(0,1) en x: u=(1,0)
∇f = (fx,fy)= (-Hx/Hz , -Hy/Hz) | y las coordenadas de u,v,w con P | es def por ∂f/∂u(P), donde u=v / ||v|| | donde u = v/ ||v||. la razón de decrecí en v(direcc) es | tasa de camb forma
hallar z (si x=n,y=m….Reemp) | derivar Fx= g(x,y,z). ∇f(u,v,w) . (UxVx,Wx)…. Fz |2.
v es direcc de crecimi en P cuando ∂f/∂u(P)>0, | ∂f/∂u(P) | un ang: (cos(n),sen(n))
H(x,y,z) = 0(igualar a 0)
| luego Fx(n , m , p(coord P)) = reemp en Fx derivada = ∇f(p)=( , ,) | donde u=v / ||v||
.La razón de crecí | 5.
la direcc de decre + rapi |
1. Vector Gradien:halle∇f(2,1):
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DuT(1,-1)=∇T(1,-1).U | (8/6 , -16/6) .(a,b)=4/3 | ∇T(2,-1) = (Tx,Ty)=(-Hx/Hz, -Hy/Hz)
| (1/2, raiz3/2) – (2,-1)=(-3/2 ,1,86)
| (2x,6y,2z)=k(-1,3-2)
∇f(2,1)= (Fx,Fy)
| ∇T(1,-1) = (Tx,Ty)=(-Hx/Hz, -Hy/Hz) | a= 1-2b…(2) | Hx=.. H(2,-1)=9Hy=…H(2,-1)=-18 | ∇T(1,-1)U ->(9,-18).(-3/2 ,1,86) = -19,73C° |
Po= (-k/2 , k/2,-k)
F(x,y) =(2x-3)^2 . F(u,v) donde u=…^v=..
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Hx=…H(1,-1,2)=-8 | remp 2 en 1 | DuT(2,-1) =(9 , -18) . (1,0) = 9C°/cm | 4.
TpoE;(P-Po).N=0 |
Reemp en la ec S -> k=+-2
enP(2,1)u= -3, v= 1 |
Hy=…H(1,-1,2)=-16 | b=0 ^ b=4/5 | 3.B.
u=(a,b) ->a^2 + b^2 = 1 …(1)
| dhallar ns= ∇G(1,1,-1)
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Si k=2 -> Po(-1,1,2)
Fx= …. Fx(2,1) = 14 |Hz=…H(1,-1,2)=6 | Si b=0 -> en(1) a=+-1 |
DuT(2,-1)=0 ->∇T(1,-1)U=0 ->(9,-18)(a,b)=0 |
S:…∇G(Sx,Sy,Sz)=ns |
TpE: (x+1,y-1,z+2)(-1,3-2)=0->rpta
Fy=….Fy(2,1)=17 |
DuT(1,-1) =(8/6 , -16/6) . (0,1) =16/6 C°/cm | Si b=4/5->a=3/5 | b=1/raiz5 , a=2/raiz5 | (z^2-2xy,-x^2+4y^3,2zx)
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Si k=2 -> Po(1,-1,2)
2. A Variación H(2,-1)
| 2.B
DuT(1,-1)= 4/3, ∇T(1,-1)U = 4/3 | v=(0,1),(0,-1)(4/5,3/5)
| 3.C u=(cos(pi/3),sin(pi/3))
|∇G(1,1,-1)=(-1 , 3, -2) = ns |
TpE: (x-1,y+1,z-2)(-1,3-2)=0->rpta
reemp z=2 , z>0 | a^2 + b^2 = 1 …(1)
| 3.A
DuT(2,-1)=∇T(2,-1).U | u = (1/2, raiz3/2) -> (1/2, raiz3/2) ||u||=2,38 | Halla Po(x,y,z) ->∇F(x,y,z)//ns |
5.TpoS=? S:…Ec | (2x+2z,2y+x,2z+2x)=k(1,1,1)
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Tpos = (x-6,y,z+3)(1,1,1)=0->rpta |
2.Matriz HessianaH(x,y)=fxx,fxy,fyx,fyy |
Hxyz=[4x+8, 0 ,0][0, -4, 0][0 , 0,-2]
TpoS//L: x=1-2t;y=3t;z=1-t | 2x+2z+y=k ->y=0 | Tpos = (x+6,y,z-3)(1,1,1)=0->rpta | {12x+2y , 2x}{2x,-2} |
H2,3,-3= [16,0.0][0,-4,0][0. 0.-2] 1△16>0,2△-64<0,3△128>0 ->punto silla
TpoS⊥TT= x-2y+2=-3 | 2y+x=k -> x=k | 6.
1.Encontrar puntos críticos P(a,b)
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H(0,4){8 , 0}{0,-2} ->△H(0,4)=-16<0 Ǝ punta silla | H-6,3,-3) =1△-16<0,2△64>0
n=Vxn1 | 2z+2x=k -> z =-k/2 |
1. Fx=6x^2 +2xy=0…(1) 2.Fy=x^2 -2y+8=0…(2)
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H(-4,12){-24 , -8}{-8,-2} ->△H(-4,12)=-16<0 Ǝ punta silla | ,3△-128<0-> max local
V=(-2,3,-1) | reem x,y,z en ec Po(0,k,-k/2) | 2x(3x+4)=0-> x=0 V y=-3x |
H(-2,6){-12 , -4}{-4,-2} ->△H(-2,6)=-8>0 ∧ Fxx < 0 Ǝ MaxLocal |
SeaPo(x,y,z) | k=+-6, si k=6-> Po(6,0,-3)
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Si x=0 2y+8=0->y=4 P1(0,4)
| 7
1. Puntos Críticos Fx: … =0..(1), Fy: … =0..(2) , Fz: … =0..(1)
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∇F(Po)//n->∇F(Po) =Kn | si k=-6->Po(-6,0,3)
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Si y=-3x -> x=-4, x=-2 | z =-3, y=3, x=2 ∧ -6 -> Po(2,3,2) P1(-6,3,-3)
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F(x,y,z)=..Ec = 0 | TpoS=(P-Po)n=0 | Po(-4,12)P1(-2,6) | 2.Matriz hessian [fxx, fxy, fxz][fyx,fyy,fyz][fzx,fzy,fzz] |