Calcul multivariable PDF

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Universidad de Pamplona Facultad de Ciencias Básicas Departamento de matemáticas Ejercicios propuestos Cálculo Multivariable 1. Funciones de más de una variable 1. Encuentre el dominio de las siguientes funciones y dibújelos como una región en el plano: x+y a ) f (x, y) = x−y xy b ) f (x, y) = 2 x −...

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❯♥✐✈❡rs✐❞❛❞ ❞❡ P❛♠♣❧♦♥❛ ❋❛❝✉❧t❛❞ ❞❡ ❈✐❡♥❝✐❛s ❇ás✐❝❛s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ♠❛t❡♠át✐❝❛s

❊❥❡r❝✐❝✐♦s ♣r♦♣✉❡st♦s ❈á❧❝✉❧♦ ▼✉❧t✐✈❛r✐❛❜❧❡

✶✳

❋✉♥❝✐♦♥❡s ❞❡ ♠ás ❞❡ ✉♥❛ ✈❛r✐❛❜❧❡

✶✳ ❊♥❝✉❡♥tr❡ ❡❧ ❞♦♠✐♥✐♦ ❞❡ ❧❛s s✐❣✉✐❡♥t❡s ❢✉♥❝✐♦♥❡s ② ❞✐❜ú❥❡❧♦s ❝♦♠♦ ✉♥❛ r❡❣✐ó♥ ❡♥ ❡❧ ♣❧❛♥♦✿ x+y x−y xy ❜ ✮ f (x, y) = 2 x − y2 p ❝ ✮ f (x, y) = 4x2 + 9y 2 − 36 1 ❞ ✮ f (x, y) = p x2 − y 2

❛ ✮ f (x, y) =

❡ ✮ f (x, y) = ln(1 + xy) exyz ❢ ✮ f (x, y, z) = √ xyz

✷✳ ❉✐❜✉❥❡✱ ❡♥ ❧❛ ♠✐s♠❛ ❣rá✜❝❛✱ ❝✐♥❝♦ ❝✉r✈❛s ❞❡ ♥✐✈❡❧ ❞❡ ❧❛s s✐❣✉✐❡♥t❡s ❢✉♥❝✐♦♥❡s✿ ❛ ✮ f (x, y) = x − y ❜ ✮ f (x, y) =

x2 y

y x2 + y 2 ❞ ✮ f (x, y) = xe−y r 1 − x2 ❡ ✮ f (x, y) = y

❝ ✮ f (x, y) =

✸✳ ▲❛s s✐❣✉✐❡♥t❡s ✜❣✉r❛s s♦♥ ❞♦s ❝♦♥❥✉♥t♦s ❞❡ ❝✉r✈❛s ❞❡ ♥✐✈❡❧✳ ❯♥❛s s♦♥ ❝✉r✈❛s ❞❡ ♥✐✈❡❧ ❞❡ ✉♥ ❝♦♥♦ ② ❧❛s ♦tr❛s s♦♥ ❝✉r✈❛s ❞❡ ♥✐✈❡❧ ❞❡ ✉♥ ♣❛r❛❜♦❧♦✐❞❡✳ ❊♥ ❛♠❜❛s ✜❣✉r❛s ✈✐❡♥❡♥ r❡♣r❡s❡♥t❛❞❛s ❧❛s ❝✉r✈❛s ❞❡ ♥✐✈❡❧ ♣❛r❛ ❧♦s ♠✐s♠♦s ✈❛❧♦r❡s ❝♦♥st❛♥t❡s✳ ➽❈✉á❧ ❡s ❝✉á❧ ② ♣♦r q✉é❄

✹✳ ❉❡s❝r✐❜❛ ❧❛s s✉♣❡r✜❝✐❡s ❞❡ ♥✐✈❡❧ ❞❡ ❧❛s s✐❣✉✐❡♥t❡s ❢✉♥❝✐♦♥❡s✿ ❛ ✮ f (x, y, z) = x2 + y 2 + z 2 ❜ ✮ f (x, y, z) = x + 2y + 3z ❝ ✮ f (x, y, z) = x2 + y 2 ❞ ✮ f (x, y, z) =

✶

x2 + y 2 z2

❡ ✮ f (x, y, z) = |x| + |y| + |z|

✺✳ ❙❡ ❡❧❛❜♦r❛ ✉♥❛ ❝❛❥❛ r❡❝t❛♥❣✉❧❛r s✐♥ t❛♣❛ ❝♦♥ ❝♦st♦ ❞❡ ♠❛t❡r✐❛❧ ❞❡ $10✳ ❊❧ ♠❛t❡r✐❛❧ ♣❛r❛ ❡❧ ❢♦♥❞♦ ❝✉❡st❛ $0,15 ♣♦r ♣✐❡ ❝✉❛❞r❛❞♦ ② ❡❧ ♠❛t❡r✐❛❧ ♣❛r❛ ❧♦s ❧❛❞♦s ❝✉❡st❛ $0,30 ♣♦r ♣✐❡ ❝✉❛❞r❛❞♦✳ ❛ ✮ ❖❜t❡♥❣❛ ✉♥ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ q✉❡ ❡①♣r❡s❡ ❡❧ ✈♦❧✉♠❡♥ ❞❡ ❧❛ ❝❛❥❛ ❝♦♠♦ ✉♥❛ ❢✉♥❝✐ó♥ ❞❡ ❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡❧

❢♦♥❞♦✳ ❉❡t❡r♠✐♥❡ ❡❧ ❞♦♠✐♥✐♦ ❞❡ ❧❛ ❢✉♥❝✐ó♥✳ ❜ ✮ ➽❈✉á❧ ❡s ❡❧ ✈♦❧✉♠❡♥ ❞❡ ❧❛ ❝❛❥❛ s✐ ❡❧ ❢♦♥❞♦ ❡s ✉♥ ❝✉❛❞r❛❞♦ ❝✉②♦ ❧❛❞♦ ♠✐❞❡ ✸ ♣✐❡s❄

✻✳ ❯♥ só❧✐❞♦ r❡❝t❛♥❣✉❧❛r ❞❡❧ ♣r✐♠❡r ♦❝t❛♥t❡✱ ❝♦♥ tr❡s ❝❛r❛s ❡♥ ❧♦s ❡❥❡s ❝♦♦r❞❡♥❛❞♦s✱ t✐❡♥❡ ✉♥ ✈ért✐❝❡ ❡♥ ❡❧ ♦r✐❣❡♥ ② ❡❧ ✈ért✐❝❡ ♦♣✉❡st♦ ❡♥ ❡❧ ♣✉♥t♦ (x, y, z) ❡♥ ❡❧ ♣❧❛♥♦ x + 3y + 2z = 6✳ ❛ ✮ ❖❜t❡♥❣❛ ✉♥ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ q✉❡ ❡①♣r❡s❡ ❡❧ ✈♦❧✉♠❡♥ ❞❡❧ só❧✐❞♦ ❝♦♠♦ ✉♥❛ ❢✉♥❝✐ó♥ ❞❡ ❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡

❧❛ ❜❛s❡✳ ❉❡t❡r♠✐♥❡ ❡❧ ✈♦❧✉♠❡♥ ❞❡ ❧❛ ❢✉♥❝✐ó♥✳ ❜ ✮ ➽❈✉á❧ ❡s ❡❧ ✈♦❧✉♠❡♥ s✐ ❧❛ ❜❛s❡ ❡s ✉♥ ❝✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ 1,25 ✉♥✐❞❛❞❡s❄

✼✳ ❊❧ ♣♦t❡♥❝✐❛❧ ❡❧é❝tr✐❝♦ ❡♥ ✉♥ ♣✉♥t♦ (x, y) ❡s V (x, y) ✈♦❧ts ② V (x, y) = p t❡♥❝✐❛❧❡s ❞❡ V ♣❛r❛ 16, 12, 8 ② 4✳

7 25 − x2 − y 2

✳ ❉✐❜✉❥❡ ❧❛s ❝✉r✈❛s ❡q✉✐♣♦✲

✽✳ ❙✉♣♦♥❣❛ q✉❡ f ❡s ❧❛ ❢✉♥❝✐ó♥ ❞❡ ♣r♦❞✉❝❝✐ó♥ ❞❡ ❝✐❡rt♦ ❛rtí❝✉❧♦✱ ❞♦♥❞❡ f (x, y) ✉♥✐❞❛❞❡s s❡ ♣r♦❞✉❝❡♥ ❝✉❛♥❞♦ s❡ ❡♠♣❧❡❛♥ x ♠áq✉✐♥❛s ② y ❤♦r❛s✲♣❡rs♦♥❛ ❡stá♥ ❞✐s♣♦♥✐❜❧❡s✳ ❙✐ f (x, y) = 6xy ✱ ❞✐❜✉❥❡ ✉♥ ♠❛♣❛ ❞❡ ❝♦♥t♦r♥♦s ❞❡ f q✉❡ ♠✉❡str❡ ❧❛s ❝✉r✈❛s ❞❡ ♣r♦❞✉❝❝✐ó♥ ❝♦♥st❛♥t❡ ♣❛r❛ 30, 24, 18, 12 ② 6✳

✾✳ ❊❧ ♣♦t❡♥❝✐❛❧ ❡❧é❝tr✐❝♦ ❡♥ ✉♥ ♣✉♥t♦ (x, y, z) ❞❡❧ ❡s♣❛❝✐♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ❡s V (x, y, z) ✈♦❧ts✱ ❞♦♥❞❡ 8 . V (x, y, z) = p 2 16x + 4y 2 + z 2

1 2

▲❛s s✉♣❡r✜❝✐❡s ❞❡ ♥✐✈❡❧ ❞❡ V s❡ ❧❧❛♠❛♥ s✉♣❡r✜❝✐❡s ❡q✉✐♣♦t❡♥❝✐❛❧❡s✳ ❉❡s❝r✐❜❛ ❡st❛s s✉♣❡r✜❝✐❡s ♣❛r❛ 4, 2, 1 ② ✳ ✷✳

▲í♠✐t❡s ② ❝♦♥t✐♥✉✐❞❛❞

✶✳ ❊✈❛❧ú❡ ❧♦s s✐❣✉✐❡♥t❡s ❧í♠✐t❡s ♦ ❡①♣❧✐q✉❡ ♣♦r q✉é ♥♦ ❡①✐st❡♥ x2 + y 2 y (x,y)→(0,0) x ❜✮ l´ım (x,y)→(0,0) x2 + y 2

❛✮

l´ım

❝✮

l´ım

❞✮ ❡✮ ❢✮

x2 (y − 1)2 + (y − 1)2

(x,y)→(0,1) x2

x2 + y 2 y (x,y)→(0,0) l´ım

l´ım

(x,y)→(0,0)

sin(xy) x2 + y 2

x2 − y 2 (x,y)→(3,3) x − y l´ım

✷✳ ❉❡t❡r♠✐♥❡ ❡❧ ❝♦♥❥✉♥t♦ ❞❡ ♣✉♥t♦s ❡♥ ❧♦s ❝✉❛❧❡s ❧❛ ❢✉♥❝✐ó♥ ❡s ❝♦♥t✐♥✉❛✿ sin(xy) ex − y 2 √ ❜ ✮ f (x, y) = arctan(x + y)

❛ ✮ f (x, y) =

✷

❝ ✮ f (x, y) = ln(x2 + y 2 − 4) √ ❞ ✮ f (x, y) = x + y + z   x2 y 3 si (x, y) 6= (0, 0) ❡ ✮ f (x, y) = 2x2 + y 2  1 si (x, y) = (0, 0)

❢ ✮ f (x, y) =

✸✳

(

xy x2 + xy + y 2 0

si

(x, y) 6= (0, 0)

si (x, y) = (0, 0)  3 3  x +y si (x, y) 6= (0, 0) ❣ ✮ f (x, y) = x2 + y 2  1 si (x, y) = (0, 0) ( xy si (x, y) 6= (0, 0) |x| + |y| ❤ ✮ f (x, y) = 0 si (x, y) = (0, 0)   sin(x + y) si x + y 6= 0 ✐ ✮ f (x, y) = x+y  1 si x + y = 0

❉❡r✐✈❛❞❛s ♣❛r❝✐❛❧❡s

✶✳ ❊♥❝✉❡♥tr❡ t♦❞❛s ❧❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛❧❡s ❞❡ ♣r✐♠❡r ♦r❞❡♥ ❞❡ ❧❛s s✐❣✉✐❡♥t❡s ❢✉♥❝✐♦♥❡s ❡♥ ❡❧ ♣✉♥t♦ ❞❛❞♦✿ ❛ ✮ f (x, y, z) = x3 y 4 z 5 , (0, −1, −1) y , (−1, 1) ❜ ✮ f (x, y) = tan−1 x π  √ ❝ ✮ f (x, y) = sin(x y), ,4 3 1 , (−3, 4) ❞ ✮ f (x, y) = p 2 x + y2

❡ ✮ f (x, y, z) = xy ln z , (e, 2, e) x − y2 ❢ ✮ f (x, y, z, w) = , (3, 1, −1, −2) z + w2

✷✳ ❈❛❧❝✉❧❡ fx (0, 0) ② fy (0, 0) ✭s✐ ❡①✐st❡♥✮ ❞❡ ❧❛s s✐❣✉✐❡♥t❡s ❢✉♥❝✐♦♥❡s✿   2x3 − y 3 si (x, y) 6= (0, 0) ❛ ✮ f (x, y) = x2 + 3y 2  0  2  x − 2y 2 si x 6= y ❜ ✮ f (x, y) = x−y  0 si x = y  2  x − xy si (x, y) 6= (0, 0) ❝ ✮ f (x, y) = x+y  0 si (x, y) = (0, 0)

✸✳ ❈❛❧❝✉❧❡ fxy (0, 0) ② fyx (0, 0) ✭s✐ ❡①✐st❡✮ ❞❡ ❧❛s s✐❣✉✐❡♥t❡s ❢✉♥❝✐♦♥❡s  

2xy si (x, y) 6= (0, 0) x2 + y 2  0 si (x, y) = (0, 0)    y x  2 −1 −1 2 − y tan si x tan ❜ ✮ f (x, y) = x y  0 si   2xy(x2 − y 2 ) si (x, y) 6= (0, 0) ✹✳ ❙❡❛ f (x, y) = x2 + y 2  0 si (x, y) = (0, 0)

❛ ✮ f (x, y) =

xy 6= 0 xy = 0

❈❛❧❝✉❧❡ fx (x, y), fy (x, y), fxy (x, y) ② fyx (x, y) ♣❛r❛ (x, y) 6= (0, 0)✳ ❆❞✐❝✐♦♥❛❧♠❡♥t❡✱ ❝❛❧❝✉❧❡ ❡st❛s ❞❡r✐✈❛❞❛s ❡♥ (0, 0)✳ ❖❜s❡r✈❡ q✉❡ fxy (0, 0) 6= fyx (0, 0) ② ❡①♣❧✐q✉❡ ♣♦r q✉é ❡st❡ r❡s✉❧t❛❞♦ ♥♦ ❡s ❝♦♥tr❛❞✐❝t♦r✐♦✳

✺✳ ▲❛ t❡♠♣❡r❛t✉r❛ ❡♥ ✉♥ ♣✉♥t♦ (x, y) ❡♥ ✉♥❛ ♣❧❛♥❝❤❛ ❞❡ ♠❡t❛❧ ♣❧❛♥❛ ❡stá ❞❡✜♥✐❞❛ ♣♦r T (x, y) = 60/(1 + x2 + y 2 )✱ ❞♦♥❞❡ T s❡ ♠✐❞❡ ❡♥ ➦❈ ② x, y ❡♥ ♠❡tr♦s✳ ❈❛❧❝✉❧❡ ❧❛ r❛③ó♥ ❞❡ ❝❛♠❜✐♦ ❞❡ ❧❛ t❡♠♣❡r❛t✉r❛ ❝♦♥ r❡s♣❡❝t♦ ❛ ❧❛ ❞✐st❛♥❝✐❛ ❡♥ ❡❧ ♣✉♥t♦ (2, 1) ❡♥ ❧❛ ❞✐r❡❝❝✐ó♥ ❞❡ x ② ❧✉❡❣♦ ❡♥ ❧❛ ❞✐r❡❝❝✐ó♥ ❞❡ y ✳ ✸

✻✳ ❊❧ ❢❛❝t♦r ❞❡ ❡♥❢r✐❛♠✐❡♥t♦ W ❡s ❧❛ t❡♠♣❡r❛t✉r❛ q✉❡ s❡ ♣❡r❝✐❜❡ ❝✉❛♥❞♦ ❧❛ t❡♠♣❡r❛t✉r❛ r❡❛❧ ❡s T ② ❧❛ ✈❡❧♦❝✐❞❛❞ ❞❡❧ ✈✐❡♥t♦ ❡s v ✱ ❞❡ ♠♦❞♦ q✉❡ W = f (T, v)✳ ❙✉♣♦♥❣❛ q✉❡ ❡❧ ❢❛❝t♦r ❞❡ ❡♥❢r✐❛♠✐❡♥t♦ s❡ ♠♦❞❡❧❛ ♠❡❞✐❛♥t❡ ❧❛ ❢✉♥❝✐ó♥ W = 13,12 + 0,6215T − 11,37v 0,16 + 0,3965T v 0,16

❞♦♥❞❡ T ❡s ❧❛ t❡♠♣❡r❛t✉r❛ ✭❡♥ ➦❈✮ ② v ❡s ❧❛ ✈❡❧♦❝✐❞❛❞ ❞❡❧ ✈✐❡♥t♦ ✭❡♥ km/h✮✳ ❈✉❛♥❞♦ T = −15➦❈ ② v = 30 km/h✱ ➽❝✉á♥t♦ ❡s♣❡r❛rí❛ ❝♦♥ ❝❡rt❡③❛ ✉st❡❞ q✉❡ ❝❛②❡r❛ ❡❧ ❢❛❝t♦r W s✐ ❧❛ t❡♠♣❡r❛t✉r❛ r❡❛❧ ❞✐s♠✐♥✉②❡ 1➦❈❄ ➽❨ s✐ ❧❛ ✈❡❧♦❝✐❞❛❞ ❞❡❧ ✈✐❡♥t♦ s❡ ✐♥❝r❡♠❡♥t❛ 1 km/h❄ ✼✳ ❙✐ V ❞ó❧❛r❡s ❡s ❡❧ ✈❛❧♦r ❛❝t✉❛❧ ❞❡ ✉♥❛ ❛♥✉❛❧✐❞❛❞ ♦r❞✐♥❛r✐❛ ❞❡ ♣❛❣♦s ✐❣✉❛❧❡s ❞❡ $100 ♣♦r ❛ñ♦ ♣❛r❛ t ❛ñ♦s ❛ ✉♥❛ t❛s❛ ❞❡ ✐♥t❡rés ❞❡ 100i ♣♦r❝✐❡♥t♦ ❛♥✉❛❧✱ ❡♥t♦♥❝❡s 1 − (1 + i)−t V = 100 i 



❛ ✮ ❈❛❧❝✉❧❡ ❧❛ t❛s❛ ❞❡ ✈❛r✐❛❝✐ó♥ ✐♥st❛♥tá♥❡❛ ❞❡ V ♣♦r ✉♥✐❞❛❞ ❞❡ ✈❛r✐❛❝✐ó♥ ❞❡ i s✐ t ♣❡r♠❛♥❡❝❡ ✜❥❛ ❡♥ 8✳ ❜ ✮ ❯t✐❧✐❝❡ ❡❧ r❡s✉❧t❛❞♦ ❞❡❧ ✐♥❝✐s♦ ❛✮ ♣❛r❛ ❝❛❧❝✉❧❛r ❧❛ ✈❛r✐❛❝✐ó♥ ❛♣r♦①✐♠❛❞❛ ❞❡❧ ✈❛❧♦r ❛❝t✉❛❧ s✐ ❧❛ t❛s❛ ❞❡ ✐♥t❡rés ✈❛rí❛ ❞❡ 6 ❛ 7 ♣♦r❝✐❡♥t♦ ② ❡❧ t✐❡♠♣♦ ♣❡r♠❛♥❡❝❡ ✜❥♦ ❡♥ 8 ❛ñ♦s✳ ❝ ✮ ❉❡t❡r♠✐♥❡ ❧❛ t❛s❛ ❞❡ ✈❛r✐❛❝✐ó♥ ✐♥st❛♥tá♥❡❛ ❞❡ V ♣♦r ✉♥✐❞❛❞ ❞❡ ✈❛r✐❛❝✐ó♥ ❞❡ t s✐ i ♣❡r♠❛♥❡❝❡ ✜❥❛ ❡♥ 0,06✳ ❞ ✮ ❯t✐❧✐❝❡ ❡❧ r❡s✉❧t❛❞♦ ❞❡❧ ✐♥❝✐s♦ ❝✮ ♣❛r❛ ❝❛❧❝✉❧❛r ❧❛ ✈❛r✐❛❝✐ó♥ ❛♣r♦①✐♠❛❞❛ ❞❡❧ ✈❛❧♦r ❛❝t✉❛❧ s✐ ❡❧ t✐❡♠♣♦ ❞✐s♠✐♥✉②❡ ❞❡ 8 ❛ 7 ❛ñ♦s ② ❧❛ t❛s❛ ❞❡ ✐♥t❡rés ♣❡r♠❛♥❡❝❡ ✜❥❛ ❡♥ 6 ♣♦r❝✐❡♥t♦✳ ✹✳

❘❡❣❧❛ ❞❡ ❧❛ ❝❛❞❡♥❛

✶✳ ❙✐ z = f (x, y)✱ ❞♦♥❞❡ x = 2s + 3t ② y = 3s − 2t✱ ❡♥❝✉❡♥tr❡ ✷✳ ❙✐ x = t sin s ② y = t cos s✱ ❡♥❝✉❡♥tr❡ ✸✳ ❊♥❝✉❡♥tr❡

∂2z ∂2z ∂2z ✱ ② 2✳ 2 ∂s ∂s∂t ∂t

∂2 f (x, y)✳ ∂s∂t

∂3 f (2x + 3y, xy) ❡♥ tér♠✐♥♦s ❞❡ ❧❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛❧❡s ❞❡ ❧❛ ❢✉♥❝✐ó♥ f ✳ ∂x∂y 2

✹✳ ▼❡❞✐❛♥t❡ ✉♥ ❞✐❛❣r❛♠❛ ❞❡ ár❜♦❧✱ ❡s❝r✐❜❛ ❧❛ r❡❣❧❛ ❞❡ ❧❛ ❝❛❞❡♥❛ ♣❛r❛ ❡❧ ❝❛s♦ ❞❛❞♦✳ ❛ ✮ u = f (x, y)✱ ❞♦♥❞❡ x = x(r, s, t), y = y(r, s, t)✳ ❜ ✮ R = f (x, y, z, t)✱ ❞♦♥❞❡ x = x(u, v, w), y = y(u, v, w), z = z(u, v, w), t = t(u, v, w)✳ ❝ ✮ t = f (u, v, w)✱ ❞♦♥❞❡ u = u(p, q, r, s), v = v(p, q, r, s), w = w(p, q, r, s)✳

✺✳ ❈❛❧❝✉❧❡ dy/dx ♠❡❞✐❛♥t❡ ❞❡r✐✈❛❝✐ó♥ ✐♠♣❧í❝✐t❛✳ ❛✮

√

xy = 1 + x2 y

❜ ✮ y 5 + x2 y 3 = 1 + yex ❝ ✮ cos(x − y) = xey

❞ ✮ sin x + cos y = sin x cos y

✻✳ ❈❛❧❝✉❧❡ ∂z/∂x ② ∂z/∂y ♠❡❞✐❛♥t❡ ❞❡r✐✈❛❝✐ó♥ ✐♠♣❧í❝✐t❛✳ ❛ ✮ xyz = cos(x + y + z) ❜ ✮ x − z = tan−1 (yz) ❝ ✮ yz = ln(x + z)

✼✳ ▲❛ ♣r♦❞✉❝❝✐ó♥ ❞❡ tr✐❣♦ ❡♥ ✉♥ ❛ñ♦ ❞❛❞♦✱ W ✱ ❞❡♣❡♥❞❡ ❞❡ ❧❛ t❡♠♣❡r❛t✉r❛ ♣r♦♠❡❞✐♦ T ② ❞❡ ❧❛ ♣r❡❝✐♣✐t❛❝✐ó♥ ♣❧✉✈✐❛❧ ❛♥✉❛❧ R✳ ▲♦s ❝✐❡♥tí✜❝♦s ❡st✐♠❛♥ q✉❡ ❧❛ t❡♠♣❡r❛t✉r❛ ♣r♦♠❡❞✐♦ s❡ ❡❧❡✈❛ ❛ r❛③ó♥ ❞❡ 0,15➦❈✴❛ñ♦✱ ② q✉❡ ❧❛ ♣r❡❝✐♣✐t❛❝✐ó♥ ❡stá ❞✐s♠✐♥✉②❡♥❞♦ ❛ r❛③ó♥ ❞❡ 0,1 cm/ao✳ ❚❛♠❜✐é♥ ❡st✐♠❛♥ q✉❡✱ ❛ ♥✐✈❡❧❡s ❞❡ ♣r♦❞✉❝❝✐ó♥ ❛♥✉❛❧❡s✱ ∂W/∂T = −2 ② ∂W/∂R = 8✳ ❛ ✮ ➽❈✉á❧ ❡s ❡❧ s✐❣♥✐✜❝❛❞♦ ❞❡ ❧♦s s✐❣♥♦s ❞❡ ❡st❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛❧❡s❄ ❜ ✮ ❊st✐♠❡ ❧❛ r❛③ó♥ ❞❡ ❝❛♠❜✐♦ ❛❝t✉❛❧ ❞❡ ❧❛ ♣r♦❞✉❝❝✐ó♥ ❞❡ tr✐❣♦✱ dW/dt.

✹

✽✳ ❙❡ ✐♥tr♦❞✉❝❡ ❛❣✉❛ ❡♥ ✉♥ t❛♥q✉❡ q✉❡ t✐❡♥❡ ❢♦r♠❛ ❞❡ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡❝t♦ ❛ ✉♥❛ t❛s❛ ❞❡ 45 π m3 /min✳ ❊❧ t❛♥q✉❡ s❡ ❡♥s❛♥❝❤❛ ❞❡ ♠♦❞♦ q✉❡✱ ❛✉♥ ❝✉❛♥❞♦ ❝♦♥s❡r✈❛ s✉ ❢♦r♠❛ ❝✐❧í♥❞r✐❝❛✱ s✉ r❛❞✐♦ s❡ ✐♥❝r❡♠❡♥t❛ ❛ ✉♥❛ t❛s❛ ❞❡ 0,2 cm/min✳ ➽◗✉é t❛♥ rá♣✐❞♦ s✉❜❡ ❧❛ s✉♣❡r✜❝✐❡ ❞❡❧ ❛❣✉❛ ❝✉❛♥❞♦ ❡❧ r❛❞✐♦ ❡s ❞❡ 2m ② ❡❧ ✈♦❧✉♠❡♥ ❞❡❧ ❛❣✉❛ ❡♥ ❡❧ t❛♥q✉❡ ❡s ❞❡ 20πm3 ❄ ✾✳ ❯♥❛ ♣❛r❡❞ ❞❡ r❡t❡♥❝✐ó♥ ❢♦r♠❛ ✉♥ á♥❣✉❧♦ ❞❡ 32 π rad ❝♦♥ ❡❧ s✉❡❧♦✳ ❯♥❛ ❡s❝❛❧❡r❛ ❞❡ 20 ♣✐❡s ❞❡ ❧♦♥❣✐t✉❞ ❡stá r❡❝❛r❣❛❞❛ ❝♦♥tr❛ ❧❛ ♣❛r❡❞ ❛ ✉♥❛ t❛s❛ ❞❡ 3 ♣✐❡s✴s✳ ➽◗✉é t❛♥ rá♣✐❞♦ ✈❛rí❛ ❡❧ ♣❛r❡❛ ❞❡❧ tr✐á♥❣✉❧♦ ❢♦r♠❛❞♦ ♣♦r ❧❛ ❡s❝❛❧❡r❛✱ ❧❛ ♣❛r❡❞ ② ❡❧ ♣✐s♦ ❝✉❛♥❞ ♦❧❛ ❡s❝❛❧❡r❛ ❢♦r♠❛ ✉♥ á♥❣✉❧♦ ❞❡ 61 π rad ❝♦♥ ❡❧ s✉❡❧♦❄ ✺✳

❉❡r✐✈❛❞❛s ❞✐r❡❝❝✐♦♥❛❧❡s ② ❣r❛❞✐❡♥t❡s

✶✳ ❊♥ ❧♦s ❡❥❡r❝✐❝✐♦s s✐❣✉✐❡♥t❡s✱ ❡♥❝✉❡♥tr❡ ❧❛ t❛s❛ ❞❡ ❝❛♠❜✐♦ ❞❡ ❧❛ ❢✉♥❝✐ó♥ ❞❛❞❛ ❡♥ ❡❧ ♣✉♥t♦ ❞❛❞♦ ② ❡♥ ❧❛ ❞✐r❡❝❝✐ó♥ ❡s♣❡❝✐✜❝❛❞❛✳ ❛ ✮ f (x, y) = 3x − 4y ❡♥ (0, 2) ❡♥ ❧❛ ❞✐r❡❝❝✐ó♥ ❞❡❧ ✈❡❝t♦r −2i✳

❜ ✮ f (x, y) = x2 y ❡♥ (−1, −1) ❡♥ ❧❛ ❞✐r❡❝❝✐ó♥ ❞❡❧ ✈❡❝t♦r i + 2j✳ x ❡♥ (0, 0) ❡♥ ❧❛ ❞✐r❡❝❝✐ó♥ ❞❡❧ ✈❡❝t♦r i − j✳ ❝ ✮ f (x, y) = 1+y ❞ ✮ f (x, y) = x2 + y 2 ❡♥ (1, −2) ❡♥ ❧❛ ❞✐r❡❝❝✐ó♥ q✉❡ ❢♦r♠❛ ✉♥ á♥❣✉❧♦ ♣♦s✐t✐✈♦ ❞❡ 60➦ ❝♦♥ ❡❧ s❡♠✐❡❥❡ ♣♦s✐t✐✈♦ x✳

✷✳ ➽❊♥ q✉é ❞✐r❡❝❝✐♦♥❡s ❡♥ ❡❧ ♣✉♥t♦ (2, 0) ❧❛ ❢✉♥❝✐ó♥ f (x, y) t✐❡♥❡ ✉♥❛ t❛s❛ ❞❡ ❝❛♠❜♦ ❞❡ −1❄ ➽❊①✐st❡♥ ❞✐r❡❝❝✐♦♥❡s ❝✉②❛ t❛s❛ ❞❡ ❝❛♠❜✐♦ s❡❛ ❞❡ −3❄ ➽❨ ❞❡ −2❄

✸✳ ➽❊♥ q✉é ❞✐r❡❝❝✐♦♥❡s ❡♥ ❡❧ ♣✉♥t♦ (a, b, c) ❧❛ ❢✉♥❝✐ó♥ f (x, y, z) = x2 + y 2 − z 2 ✐♥❝r❡♠❡♥t❛ ❛ ❧❛ ♠✐t❛❞ ❞❡ s✉ t❛s❛ ♠á①✐♠❛ ❡♥ ❞✐❝❤♦ ♣✉♥t♦❄ ✹✳ ▲❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✉♥❛ ♣❧❛❝❛ ❡♥ ❝❛❞❛ ♣✉♥t♦ ❞❡❧ ♣❧❛♥♦ xy ❡stá ❞❛❞❛ ♣♦r T (x, y) = x2 − 2y 2 ✳

❛ ✮ ➽❊♥ q✉é ❞✐r❡❝❝✐ó♥ ❞❡❜❡ ♠♦✈❡rs❡ ✉♥❛ ❤♦r♠✐❣❛ q✉❡ s❡ ❡♥❝✉❡♥tr❛ ❡♥ ❧❛ ♣♦s✐❝✐ó♥ (2, −1) s✐ ❞❡s❡❛ ❜❛❥❛r s✉

t❡♠♣❡r❛t✉r❛ ❧♦ ♠ás rá♣✐❞♦ ♣♦s✐❜❧❡❄ ❜ ✮ ❙✐ ❧❛ ❤♦r♠✐❣❛ s❡ ♠✉❡✈❡ ❡♥ ❞✐❝❤❛ ❞✐r❡❝❝✐ó♥ ❛ ✉♥❛ ✈❡❧♦❝✐❞❛❞ k ✭✉♥✐❞❛❞❡s ❞❡ ❞✐st❛♥❝✐❛ ♣♦r ✉♥✐❞❛❞ ❞❡ t✐❡♠♣♦✮✱ ➽❛ q✉é t❛s❛ ❧❛ ❤♦r♠✐❣❛ s✐❡♥t❡ ❧❛ ❜❛❥❛❞❛ ❞❡ t❡♠♣❡r❛t✉r❛❄ ❝ ✮ ➽❆ q✉é t❛s❛ s✐❡♥t❡ ❧❛ ❤♦r♠✐❣❛ ❧❛ ❜❛❥❛❞❛ ❞❡ t❡♠♣❡r❛t✉r❛ s✐ s❡ ♠✉❡✈❡ ❞❡ (2, −1) ❛ ✉♥❛ ✈❡❧♦❝✐❞❛❞ k ❡♥ ❧❛ ❞✐r❡❝❝✐ó♥ ❞❡❧ ✈❡❝t♦r −i − 2j❄

✺✳ ❙✉♣♦♥❣❛ q✉❡ ✉♥❛ ❝✐❡rt❛ r❡❣✐ó♥ ❞❡❧ ❡s♣❛❝✐♦ ❡❧ ♣♦t❡♥❝✐❛❧ ❡❧é❝tr✐❝♦ V ❡stá ❞❡✜♥✐❞♦ ♣♦r V (x, y, z) = 5x2 − 3xy + xyz ✳ ❛ ✮ ❉❡t❡r♠✐♥❡ ❧❛ r❛③ó♥ ❞❡ ❝❛♠❜✐♦ ❞❡❧ ♣♦t❡♥❝✐❛❧ ❡♥ P (3, 4, 5) ❡♥ ❧❛ ❞✐r❡❝❝✐ó♥ ❞❡❧ ✈❡❝t♦r v = i + j − k✳ ❜ ✮ ➽❊♥ q✉é ❞✐r❡❝❝✐ó♥ ❝❛♠❜✐❛ V ❝♦♥ ♠❛②♦r r❛♣✐❞❡③ ❡♥ P ❄ ❝ ✮ ➽❈✉á❧ ❡s ❧❛ r❛③ó♥ ♠á①✐♠❛ ❞❡ ❝❛♠❜✐♦ ❡♥ P ❄

✻✳ ❙✉♣♦♥❣❛ q✉❡ ❡s❝❛❧❛ ✉♥❛ ♠♦♥t❛ñ❛ ❝✉②❛ ❢♦r♠❛ ❧❛ ❞❛ ❧❛ ❡❝✉❛❝✐ó♥ z = 1000 − 0,005x2 − 0,01y 2 ✱ ❞♦♥❞❡ x, y, z s❡ ❞❛♥ ❡♥ ♠❡tr♦s✱ ② ✉st❡❞ ❡stá ♣❛r❛❞♦ ❡♥ ✉♥ ♣✉♥t♦ ❝✉②❛s ❝♦♦r❞❡♥❛❞❛s s♦♥ (60, 40, 966)✳ ❊❧ ❡❥❡ ❞❡ ❧❛s x ♣♦s✐t✐✈❛s ✈❛ ❤❛❝✐❛ ❡❧ ❡st❡ ② ❡❧ ❡❥❡ ❞❡ ❧❛s y ♣♦s✐t✐✈❛s ✈❛ ❤❛❝✐❛ ❡❧ ♥♦rt❡✳ ❛ ✮ ❙✐ ❝❛♠✐♥❛ ❞✐r❡❝t♦ ❤❛❝✐❛ ❡❧ s✉r✱ ➽❡♠♣❡③❛rá ❛ ❛s❝❡♥❞❡r ♦ ❞❡s❝❡♥❞❡r❄ ❜ ✮ ❙✐ ❝❛♠✐♥❛ ❤❛❝✐❛ ❡❧ ♥♦r♦❡st❡✱ ➽❡♠♣❡③❛rá ❛ ❛s❝❡♥❞❡r ♦ ❞❡s❝❡♥❞❡r❄ ❝ ✮ ➽❊♥ q✉é ❞✐r❡❝❝✐ó♥ ❡s ❧❛ ♠á①✐♠❛ ♣❡♥❞✐❡♥t❡❄ ➽❈✉á❧ ❡s ❧❛ r❛③ó♥ ❞❡ ❝❛♠❜✐♦ ❡♥ ❡s❛ ❞✐r❡❝❝✐ó♥❄ ➽❊♥ q✉é á♥❣✉❧♦

♣♦r ❛rr✐❜❛ ❞❡ ❧❛ ❤♦r✐③♦♥t❛❧ ❧❛ tr❛②❡❝t♦r✐❛ ✐♥✐❝✐❛ ❡♥ ❡s❛ ❞✐r❡❝❝✐ó♥❄

  psin(xy) ✼✳ ❙❡❛ f (x, y) = x2 + y 2  0

❛ ✮ ❈❛❧❝✉❧❡ ∇f (0, 0)✳

si

(x, y) 6= (0, 0)

si

(x, y) = (0, 0)

❜ ✮ ❯s❡ ❧❛ ❞❡✜♥✐❝✐ó♥ ❞❡ ❞❡r✐✈❛❞❛ ❞✐r❡❝❝✐♦♥❛❧ ♣❛r❛ ❝❛❧❝✉❧❛r Du f (0, 0)✱ ❞♦♥❞❡ u =

√1 (i 2

+ j)✳

✽✳ ❉❡t❡r♠✐♥❡ ❧❛ ❞✐r❡❝❝✐ó♥ ❛ ♣❛rt✐r ❞❡❧ ♣✉♥t♦ (1, 3) ♣❛r❛ ❧❛ ❝✉❛❧ ❡❧ ✈❛❧♦r ❞❡ f ♥♦ ❝❛♠❜✐❛ s✐ f (x, y) = e2y tan−1 ✺

y ✳ 3x

✻✳ P❧❛♥♦s t❛♥❣❡♥t❡s ② r❡❝t❛s ♥♦r♠❛❧❡s ❛ s✉♣❡r✜❝✐❡s ✶✳ ❖❜t❡♥❣❛ ✉♥❛ ❡❝✉❛❝✐ó♥ ❞❡ ❧❛ r❡❝t❛ ♥♦r♠❛❧ ② ❛ ❧❛ s✉♣❡r✜❝✐❡ ❡♥ ❡❧ ♣✉♥t♦ ✐♥❞✐❝❛❞♦✳ ❛✮ ❜✮ ❝✮ ❞✮ ❡✮

z z z z z

= 4x2 − y 2 + 2y ✱ (−1, 2, 4) = 3(x − 1)2 + 2(y + 3)2 + 7✱ (2, −2, 12) √ = xy ✱ (1, 1, 1) = y ln x✱ (1, 4, 0) = y cos(x − y)✱ (2, 2, 2)

❢ ✮ z = ex

2

−y 2

✱ (1, −1, 1)

✷✳ ❙✐ ❧❛s ❞♦s s✉♣❡r✜❝✐❡s s❡ ✐♥t❡rs❡❝t❛♥ ❡♥ ✉♥❛ ❝✉r✈❛✱ ❞❡t❡r♠✐♥❡ ❡❝✉❛❝✐♦♥❡s ❞❡ ❧❛ r❡❝t❛ t❛♥❣❡♥t❡ ❛ ❧❛ ❝✉r✈❛ ❞❡ ✐♥t❡rs❡❝❝✐ó♥ ❡♥ ❡❧ ♣✉♥t♦ ✐♥❞✐❝❛❞♦❀ s✐ ❧❛s ❞♦s s✉♣❡r✜❝✐❡s s♦♥ t❛♥❣❡♥t❡s ❡♥ ❡❧ ♣✉♥t♦ ❞❛❞♦✱ ❞❡♠✉éstr❡❧♦✳ ❛✮ ❜✮ ❝✮ ❞✮

x2 + y 2 − z = 8✱ x − y 2 + z 2 = −2❀ (2, −2, 0) y = x2 ✱ y = 16 − z 2 ❀ (4, 16, 0) y = ex sin(2πz) + 2✱ z = y 2 − ln(x + 1) − 3❀ (0, 2, 1) x2 + z 2 + 4y = 0✱ x2 + y 2 + z 2 − 6z + 7 = 0❀ (0, −1, 2)

✸✳ ❯t✐❧✐❝❡ ❡❧ ❣r❛❞✐❡♥t❡ ♣❛r❛ ♦❜t❡♥❡r ✉♥❛ ❡❝✉❛❝✐ó♥ ❞❡ ❧❛ r❡❝t❛ t❛♥❣❡♥t❡ ❛ ❧❛ ❝✉r✈❛ ❞❛❞❛ ❡♥ ❡❧ ♣✉♥t♦ ✐♥❞✐❝❛❞♦✳ ❛✮ ❜✮ ❝✮ ❞✮

x3 − y 3 = 1❀ (1, 2) 16x4 + y 4 = 32❀ 1, 2 2x3 + 2y 3 − 9xy = 0❀ (1, 2) x4 + 2xy − y 2 = 4❀ (2, −2)

✼✳ ❊①tr❡♠♦s ❞❡ ❢✉♥❝✐♦♥❡s ❞❡ ❞♦s ✈❛r✐❛❜❧❡s ✶✳ ❊♥❝✉❡♥tr❡ ② ❝❧❛s✐✜q✉❡ ❧♦s ♣✉♥t♦s ❝rít✐❝♦s ❞❡ ❧❛s s✐❣✉✐❡♥t❡s ❢✉♥❝✐♦♥❡s ❛ ✮ f (x, y) = x2 + 2y 2 − 4x + 4y x 8 ❜ ✮ f (x, y) = + − y y x ❝ ✮ f (x, y) = x3 + y 3 − 3xy ❞ ✮ f (x, y) = x sin y ❡ ✮ f (x, y) = x2 ye−(x +y ) xy ❢ ✮ f (x, y) = 2 + x4 + y 4 1 ❣ ✮ f (x, y) = 1 − x + y + x2 + y 2     1 1 1 1 1+ + ❤ ✮ f (x, y) = 1 + x y x y 2

2

✷✳ ▲❛s r❡❣✉❧❛❝✐♦♥❡s ♣♦st❛❧❡s r❡q✉✐❡r❡♥ q✉❡ ❧❛ s✉♠❛ ❡♥tr❡ ❧❛ ❛❧t✉r❛ ② ❡❧ ♣❡rí♠❡tr♦ ❞❡ ❧❛ ❜❛s❡ ❞❡ ✉♥ ♣❛q✉❡t❡ ♥♦ ❡①❝❡❞❛ L ✉♥✐❞❛❞❡s✳ ❊♥❝✉❡♥tr❡ ❡❧ ✈♦❧✉♠❡♥ ♠ás ❣r❛♥❞❡ ❞❡ ✉♥❛ ❝❛❥❛ r❡❝t❛♥❣✉❧❛r q✉❡ ♣✉❡❞❛ s❛t✐s❢❛❝❡r ❡st❡ r❡q✉✐s✐t♦✳ ✸✳ ❊❧ ♠❛t❡r✐❛❧ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❤❛❝❡r ❧❛ ♣❛rt❡ ✐♥❢❡r✐♦r ❞❡ ✉♥❛ ❝❛❥❛ r❡❝t❛♥❣✉❧❛r ❡s ❞♦s ✈❡❝❡s ♠ás ❝❛r♦ ♣♦r ✉♥✐❞❛❞ ❞❡ ár❡❛ q✉❡ ❡❧ ♠❛t❡r✐❛❧ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❤❛❝❡r ❧❛ ♣❛rt❡ s✉♣❡r✐♦r ❞❡ ❧❛s ♣❛r❡❞❡s ❧❛t❡r❛❧❡s✳ ❊♥❝✉❡♥tr❡ ❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡ ❧❛ ❝❛❥❛ ❞❡ ✈♦❧✉♠❡♥ V ❞❛❞❛✱ ❝✉②♦ ❝♦st♦ ❞❡ ♠❛t❡r✐❛❧❡s ❡s ♠í♥✐♠♦✳ ✹✳ ❊♥❝✉❡♥tr❡ ❡❧ ✈♦❧✉♠❡♥ ❞❡ ❧❛ ❝❛❥❛ r❡❝t❛♥❣✉❧❛r ♠ás ❣r❛♥❞❡ ✭❝♦♥ ❝❛r❛s ♣❛r❛❧❡❧❛s ❛ ❧♦s ♣❧❛♥♦s ❝♦♦r❞❡♥❛❞♦s✮ q✉❡ ♣✉❡❞❡ s❡r ✐♥s❝r✐t❛ ❡♥ ❧❛ ❡❧✐♣s♦✐❞❡ s✐❡♥❞♦ a, b, c ❝♦♥st❛♥t❡s✳

y2 z2 x2 + + = 1, a2 b2 x2

✺✳ ❊♥❝✉❡♥tr❡ tr❡s ♥ú♠❡r♦s ♣♦s✐t✐✈♦s a, b ② c ❝✉②❛ s✉♠❛ s❡❛ 30 ② ❧❛ ❡①♣r❡s✐ó♥ ab2 c3 s❡❛ ♠á①✐♠♦✳ ✻✳ ❊♥❝✉❡♥tr❡ ❧♦s ♣✉♥t♦s ❝rít✐❝♦s ❞❡ ❧❛ ❢✉♥❝✐ó♥ z = g(x, y) q✉❡ s❛t✐s❢❛❝❡ ❧❛ ❡❝✉❛❝✐ó♥ e2zx−x − 3e2zy+y = 2✳ ➽❊s ♣♦s✐❜❧❡ ❝❧❛s✐✜❝❛r❧♦s❄ 2

✻

2

✽✳

▼✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡

✶✳ ❊♥❝✉❡♥tr❡✱ ✉s❛♥❞♦ ❡❧ ♠ét♦❞♦ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡✱ ❡❧ ✈♦❧✉♠❡♥ ❞❡ ❧❛ ❝❛❥❛ r❡❝t❛♥❣✉❧❛r ♠ás ❣r❛♥❞❡ ✭❝♦♥ ❝❛r❛s ♣❛r❛❧❡❧❛s ❛ ❧♦s ♣❧❛♥♦s ❝♦♦r❞❡♥❛❞♦s✮ q✉❡ ♣✉❡❞❡ s❡r ✐♥s❝r✐t❛ ❡♥ ❧❛ ❡❧✐♣s♦✐❞❡ y2 z2 x2 + + = 1, a2 b2 x2

s✐❡♥❞♦ a, b, c ❝♦♥st❛♥t❡s✳ ✷✳ ❯s❡ ❡❧ ♠ét♦❞♦ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ♣❛r❛ ♠❛①✐♠✐③❛r x3 y 5 s✉❥❡t♦ ❛ ❧❛ r❡str✐❝❝✐ó♥ x + y = 8✳ ✸✳ ❊♥❝✉❡♥tr❡ ❧❛ ❞✐st❛♥❝✐❛ ♠ás ❝♦rt❛ ❞❡❧ ♣✉♥t♦ (3, 0) ❛ ❧❛ ♣❛rá❜♦❧❛ y = x2 ♣♦r ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡✳ ✹✳ ❊♥❝✉❡♥tr❡ ❧❛ ❞✐st❛♥❝✐❛ ♠ás ❝♦rt❛ ❞❡❧ ♦r✐❣❡♥ ❛❧ ♣❧❛♥♦ x + 2y + 2z = 3✳ ✺✳ ❊♥❝✉❡♥tr❡ ❧♦s ✈❛❧♦r❡s ♠á①✐♠♦ ② ♠í♥✐♠♦ ❞❡ ❧❛ ❢✉♥❝✐ó♥ f (x, y, z) = x + y + z s♦❜r❡ ❧❛ ❡s❢❡r❛ x2 + y 2 + z 2 = 1✳ ✻✳ ❯s❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ♣❛r❛ ❡♥❝♦♥tr❛r ❧❛ ❞✐st❛♥❝✐❛ ♠ás ❣r❛♥❞❡ ② ❧❛ ♠ás ♣❡q✉❡ñ❛ ❞❡❧ ♣✉♥t♦ (2, 1, −2) ❛ ❧❛ ❡s❢❡r❛ x2 + y 2 + z 2 = 1✳ ✼✳ ❊♥❝✉❡♥tr❡ ❧❛ ❞✐st❛♥❝✐❛ ♠ás ❝♦rt❛ ❞❡❧ ♦r✐❣❡♥ ❛ ❧❛ s✉♣❡r✜❝✐❡ xyz 2 = 2✳ x2

y2

z2

✽✳ ❊♥❝✉❡♥tr❡ a✱ b ② c ❞❡ ♠♦❞♦ q✉❡ ❡❧ ✈♦❧✉♠❡♥ V = 4πabc/3 ❞❡ ✉♥❛ ❡❧✐♣s♦✐❞❡ 2 + 2 + 2 = 1 q✉❡ ♣❛s❛ ♣♦r ❡❧ a b c ♣✉♥t♦ (1, 2, 1) s❡❛ ❧♦ ♠ás ♣❡q✉❡ñ♦ ♣♦s✐❜❧❡✳ ✾✳ ❊♥❝✉❡♥tr❡ ❧♦s ✈❛❧♦r❡s ♠á①✐♠♦s ② ♠í♥✐♠♦s ❞❡ ❧❛ ❢✉♥❝✐ó♥ ❞❡ n ✈❛r✐❛❜❧❡s x1 + x2 + ... + xn s✉❥❡t❛ ❛ ❧❛ r❡str✐❝❝✐ó♥ x21 + x22 + ... + x2n = 1✳ ✶✵✳ ❊♥❝✉❡♥tr❡ ❡❧ ✈♦❧✉♠❡♥ ♠á①✐♠♦ ❞❡ ✉♥❛ ❝❛❥❛ r❡❝t❛♥❣✉❧❛r ❝✉②❛s ❝❛r❛s s♦♥ ♣❛r❛❧❡❧❛s ❛ ❧♦s ♣❧❛♥♦s ❝♦♦r❞❡♥❛❞♦s s✐ ✉♥❛ ❞❡ ❧❛s ❡sq✉✐♥❛s ❡stá ❡♥ ❡❧ ♦r✐❣❡♥ ② ❧❛ ❡sq✉✐♥❛ ❞✐❛❣♦♥❛❧♠❡♥t❡ ♦♣✉❡st❛ ❡stá ❡♥ ❡❧ ♣❧❛♥♦ 4x + 2y + z = 2✳ ✾✳

■♥t❡❣r❛❧❡s ❞♦❜❧❡s

✶✳ ❖❜t❡♥❡r ❡❧ ✈♦❧✉♠❡♥ ❞❡❧ só❧✐❞♦ ✐♥❞✐❝❛❞♦✳ ❛ ✮ ❆❜❛❥♦ ❞❡ z = 1 − x2 ② ❡♥❝✐♠❛ ❞❡ ❧❛ r❡❣✐ó♥ 0 ≤ x ≤ 1✱ 0 ≤ y ≤ x✳

❜ ✮ ❆❜❛❥♦ ❞❡ z = 1 − x2 − y 2 ② ❛rr✐❜❛ ❞❡ ❧❛ r❡❣✐ó♥ ❞❡❧ ♣r✐♠❡r ❝✉❛❞r❛♥t❡ ❧✐♠✐t❛❞❛ ♣♦r ❧♦s ❡❥❡s ❝♦♦r❞❡♥❛❞♦s ② ❧❛ r❡❝t❛ x + y = 1✳

❝ ✮ ❆❜❛❥♦ ❞❡ z = 1 − y 2 ② ❛rr✐❜❛ ❞❡ z = x2 ✳ 1 ❞ ✮ ❆❜❛❥♦ ❞❡ ❧❛ s✉♣❡r✜❝✐❡ z = ② ❛rr✐❜❛ ❞❡ ❧❛ r❡❣✐ó♥ ❡♥ ❡❧ ♣❧❛♥♦ xy ❛❝♦t❛❞❛ ♣♦r x = 1✱ x = 2✱ y = 0 ② x+y y = x✳ ❡ ✮ ❆❜❛❥♦ ❞❡ ❧❛ s✉♣❡r✜❝✐❡ x2 sin(y 4 ) ② ❛rr✐❜❛ ❞❡❧ tr✐á♥❣✉❧♦ ❡♥ ❡❧ ♣❧❛♥♦ xy ❝♦♥ ✈ért✐❝❡s (0, 0)✱ (0, π 1/4 ) ② (π 1/4 , π 1/4 )✳ ❢ ✮ ❆rr✐❜❛ ❞❡❧ ♣❧❛♥♦ xy ② ❛❜❛❥♦ ❞❡ z = 1 − x2 − 2y 2 ✳

❣ ✮ ❆rr✐❜❛ ❞❡❧ tr✐á♥❣✉❧♦ ❝♦♥ ✈ért✐❝❡s (0, 0)✱ (a, 0) ② (0, b) ② ❛❜❛❥♦ ❞❡❧ ♣❧❛♥♦ z = 2 −

❤ ✮ ❊♥tr❡ ❧♦s ❞♦s ❝✐❧✐♥❞r♦s x2 + y 2 = a2 ② y 2 + z 2 = a2 ✳

x y − ✳ a b

✐ ✮ ❉❡♥tr♦ ❞❡❧ ❝✐❧✐♥❞r♦ x2 + 2y 2 = 8✱ ❛rr✐❜❛ ❞❡❧ ♣❧❛♥♦ z = y − 4 ② ❛❜❛❥♦ ❞❡❧ ♣❧❛♥♦ z = 8 − x✳

✷✳ ❈❛♠❜✐❡ ❡❧ ♦r❞❡♥ ❞❡ ✐♥t❡❣r❛❝✐ó♥ ❞❡ ❧❛s s✐❣✉✐❡♥t❡s ✐♥t❡❣r❛❧❡s ❞♦❜❧❡s✳ ❙✉❣❡r❡♥❝✐❛✿ ❛♣ó②❡s❡ ❡♥ ❧❛ ❣rá✜❝❛ ❞❡ ❧❛ r❡❣✐ó♥✳ ❛✮

✼

Z

❜✮

Z

❝✮

Z

1 0 1 0

Z

Z

ey 1 1

x 5π/2

π/2

f (x, y)dxdy ✳

f (x, y)dydx✳

Z

1 sin x

f (x, y)dydx✳

❞✮

Z

4 0

Z

2 √

x

f (x, y)dydx✳

✸✳ ❯t✐❧✐❝❡ ✐♥t❡❣r❛❧❡s ❞♦❜❧❡s ♣❛r❛ ❝❛❧❝✉❧❛r ❡❧ ár❡❛ ❞❡ ❧❛ r❡❣✐ó♥ ❧✐♠✐t❛❞❛ ♣♦r ❧❛s ❝✉r✈❛s ❞❡❧ ♣❧❛♥♦ xy ✳ ❉✐❜✉❥❡ ❧❛ r❡❣✐ó♥✳ ❛ ✮ y = x3 ② y = x2 ❜ ✮ y 2 = 4x ② x2 = 4y ❝ ✮ y = x2 − 9 ② y = 9 − x2

❞ ✮ x2 + y 2 = 16 ② y 2 = 6x ✶✵✳

❆♣❧✐❝❛❝✐♦♥❡s ❞❡ ❧❛s ✐♥t❡❣r❛❧❡s ❞♦❜❧❡s

✶✳ ❊♥❝✉❡♥tr❡ ❧❛ ♠❛s❛ ② ❡❧ ❝❡♥tr♦ ❞❡ ♠❛s❛ ❞❡ ❧❛ ❧á♠✐♥❛ q✉❡ ♦❝✉♣❛ ❧❛ r❡❣✐ó♥ D ② t✐❡♥❡ ❧❛ ❢✉♥❝✐ó♥ ❞❡ ❞❡♥s✐❞❛❞ ρ ❞❛❞❛✳ ❛ ✮ D ❡s ❧❛ r❡❣✐ó♥ tr✐❛♥❣✉❧❛r ❝♦♥ ✈ért✐❝❡s (0, 0)✱ (2, 1) ② (0, 3)❀ ρ(x, y) = x + y ✳ ❜ ✮ D ❡stá ❛❝♦t❛❞❛ ♣♦r ❧❛s ♣❛rá❜♦❧❛s y = x2 ② x = y 2 ✳

✷✳ ❯♥❛ ❧á♠✐♥❛ ♦❝✉♣❛ ❧❛ ♣❛rt❡ ❞❡❧ ❞✐s❝♦ x2 +y 2 ≤ 1 ❡♥ ❡❧ ♣r✐♠❡r ❝✉❛❞r❛♥t❡✳ ❊♥❝✉❡♥tr❡ s✉ ❝❡♥tr♦ ❞❡ ♠❛s❛ s✐ ❧❛ ❞❡♥s✐❞❛❞ ❡♥ ❝✉❛❧q✉✐❡r ♣✉♥t♦ ❡s ♣r♦♣♦r❝✐♦♥❛❧ ❛ s✉ ❞✐st❛♥❝✐❛ ❞❡s❞❡ ❡❧ ❡❥❡ x✳ ❆❞✐❝✐♦♥❛❧♠❡♥t❡✱ ❡♥❝✉❡♥tr❡ ❧♦s ♠♦♠❡♥t♦s ❞❡ ✐♥❡r❝✐❛ Ix ✱ Iy ❝♦♥ r❡s♣❡❝t♦ ❛ ❧♦s ❡❥❡s x ② y r❡s♣❡❝t✐✈❛♠❡♥t❡✳ √

√

✸✳ ▲❛ ❢r♦♥t❡r❛ ❞❡ ✉♥❛ ❧á♠✐♥❛ ❡stá ❢♦r♠❛❞❛ ♣♦r ❧♦s s❡♠✐❝ír❝✉❧♦s y = 1 − x2 ② y = 4 − x2 ❥✉♥t♦ ❝♦♥ ❧❛s ♣♦r❝✐♦♥❡s ❞❡❧ ❡❥❡ x q✉❡ ❧❛s ✉♥❡✳ ❊♥❝✉❡♥tr❡ ❡❧ ❝❡♥tr♦ ❞❡ ♠❛s❛ ❞❡ ❧❛ ❧á♠✐♥❛ s✐ ❧❛ ❞❡♥s✐❞❛❞ ❡♥ ❝✉❛❧q✉✐❡r ♣✉♥t♦ ❡s ♣r♦♣♦r❝✐♦♥❛❧ ❛ s✉ ❞✐st❛♥❝✐❛ ❞❡s❞❡ ❡❧ ♦r✐❣❡♥✳ ✹✳ ❍❛❧❧❡ ❡❧ ❝❡♥tr♦ ❞❡ ♠❛s❛ ❞❡ ✉♥❛ ❧á♠✐♥❛ ❡♥ ❧❛ ❢♦r♠❛ ❞❡ ✉♥ tr✐á♥❣✉❧♦ ✐sós❝❡❧❡s r❡❝t♦ ❝♦♥ ❧❛❞♦s ✐❣✉❛❧❡s ❞❡ ❧♦♥❣✐t✉❞ a s✐ ❧❛ ❞❡♥s✐❞❛❞ ❡♥ ❝✉❛❧q✉✐❡r ♣✉♥t♦ ❡s ♣r♦♣♦r❝✐♦♥❛❧ ❛❧ ❝✉❛❞r❛❞♦ ❞❡ ❧❛ ❞✐st❛♥❝✐❛ ❞❡s❞❡ ❡❧ ✈ért✐❝❡ ♦♣✉❡st♦ ❛ ❧❛ ❤✐♣♦t❡♥✉s❛✳ ❆❞✐❝✐♦♥❛❧♠❡♥t❡✱ ❡♥❝✉❡♥tr❡ ❧♦s ♠♦♠❡♥t♦s ❞❡ ✐♥❡r❝✐❛ Ix ✱ Iy ❝♦♥ r❡s♣❡❝t♦ ❛ ❧♦s ❡❥❡s x ② y r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ✺✳ ❉❡t❡r♠✐♥❡ ❡❧ ár❡❛ ❞❡ ❧❛ s✉♣❡r✜❝✐❡ ❝♦rt❛❞❛ ❡♥ ❡❧ ♣❧❛♥♦ 2x + y + z = 4 ♣♦r ❧♦s ♣❧❛♥♦s x = 0✱ x = 1✱ y = 0 ② y = 1✳ ✻✳ ❈❛❧❝✉❧❡ ❡❧ ár❡❛ ❞❡ ❧❛ s✉♣❡r✜❝✐❡ ❝♦rt❛❞❛ ❡♥ ❡❧ ♣❧❛♥♦ z − 2x − y = 5 ♣♦r ❧♦s ♣❧❛♥♦s x = 0✱ x = 2✱ y = 0 ② y = 4✳

✼✳ ❖❜t❡♥❣❛ ❡❧ ár❡❛ ❞❡ ❧❛ ♣♦r❝✐ó♥ ❞❡ s✉♣❡r✜❝✐❡ ❞❡❧ ♣❧❛♥♦ 36x + 16y + 9z = 144 ❝♦rt❛❞❛ ♣♦r ❧♦s ♣❧❛♥♦s ❝♦♦r❞❡♥❛❞♦s✳ ✽✳ ❉❡t❡r♠✐♥❡ ❡❧ ár❡❛ ❞❡ ❧❛ s✉♣❡r✜❝✐❡ ❝♦rt❛❞❛ ❡♥ ❡❧ ♣❧❛♥♦ z = ax + by ♣♦r ❧♦s ♣❧❛♥♦s x = 0✱ x = a✱ y = 0 ② y = b✱ ❞♦♥❞❡ a > 0 ② b > 0✳ ✾✳ ❈❛❧❝✉❧❡ ❡❧ ár❡❛ ❞❡ ❧❛ s✉♣❡r✜❝✐❡ ❞❡❧ ♣r✐♠❡r ♦❝t❛♥t❡ ❝♦rt❛❞❛ ❡♥ ❡❧ ❝✐❧✐♥❞r♦ x2 + y 2 = 0 ♣♦r ❡❧ ♣❧❛♥♦ x = z ✳

✶✵✳ ❖❜t❡♥❣❛ ❡❧ ár❡❛ ❞❡ ❧❛ ♣♦r❝✐ó♥ ❞❡❧ ♣❧❛♥♦ x = z q✉❡ ❡stá ❡♥tr❡ ❧♦s ♣❧❛♥♦s y = 0 ② y = 6 ② ❞❡♥tr♦ ❞❡❧ ❤✐♣❡r❜♦❧♦✐❞❡ 9x2 − 4y 2 + 16z 2 = 144✳ ✶✶✳

■♥t❡❣r❛❧❡s ❞♦❜❧❡s ❡♥ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s

✶✳ ❊✈❛❧ú❡ ❧❛ ✐♥t❡❣r❛❧ ❞♦❜❧❡ ❝❛♠❜✐❛♥❞♦ ❛ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s✳ ❛✮ ❜✮ ❝✮ ❞✮ ❡✮ ❢✮

✽

ZZ

ZZ

R

R

xy dA✱ ❞♦♥❞❡ R ❡s ❡❧ ❞✐s❝♦ ❝♦♥ ❝❡♥tr♦ ❡♥ ❡❧ ♦r✐❣❡♥ ② r❛❞✐♦ 3✳ (x + y) dA✱ ❞♦♥❞❡ R ❡s ❧❛ r❡❣✐ó♥ q✉❡ ②❛❝❡ ❛ ❧❛ ✐③q✉✐❡r❞❛ ❞❡❧ ❡❥❡ y ② ❡♥tr❡ ❧♦s ❝ír❝✉❧♦s x2 + y 2 = 1 ②

x2 + y 2 = 4 ZZ ln(x2 + y 2 ) dA✱ ❞♦♥❞❡ R ❡...