I f u = x y t h e n ∂ u ∂ y i s e q u a l t o If\ u\ =\ x^y\ then\ \frac{\partial u}{\partial y}\ is\ equal\ to\ I f u = x y t h e n ∂ y ∂ u i s e q u a l t o

x y log ⁡ x x^y\ \log x x y lo g x

y x ( y − 1 ) y\ x^{\left(y\ -\ 1\right)} y x ( y − 1 )

What is the saddle point?

Point where function neither have maximum value nor minimum value

Point where function has maximum value

Point where function has minimum value

Point where function has zero value

The drawback of Lagrange’s Method of Maxima and minima is?

Nature of stationary point is can not be known

Nature of stationary point is known but can not give maxima or minima

I f u = x log ⁡ ( x y ) , t h e n d u d x i s If\ \ u=\ x\ \log\left(xy\right)\ ,\ then\ \frac{du}{dx\ }is I f u = x lo g ( x y ) , t h e n d x d u i s

1 + ( log ⁡ x + log ⁡ y ) + x y d y d x 1\ +\ \left(\log x\ +\ \log\ y\right)+\ \frac{x}{y\ }\ \ \frac{dy}{dx} 1 + ( lo g x + lo g y ) + y x d x d y

1 + ( log ⁡ x − log ⁡ y ) + x y d y d x 1\ +\ \left(\log x\ -\ \log\ y\right)+\ \frac{x}{y\ }\ \ \frac{dy}{dx} 1 + ( lo g x − lo g y ) + y x d x d y

1 + ( log ⁡ x + log ⁡ y ) + x y d y d x 1\ +\ \left(\log x\ +\ \log\ y\right)+\ \frac{x}{y\ }\ \ \frac{dy}{dx} 1 + ( lo g x + lo g y ) + y x d x d y

1 + ( log ⁡ x + log ⁡ y ) 1\ +\ \left(\log x\ +\ \log\ y\right) 1 + ( lo g x + lo g y )

I f u = f ( x z , y z ) t h e n t h e v a l u e o f x ∂ u ∂ x + y ∂ u ∂ y + z ∂ u ∂ z i s If\ u\ =f\left(\frac{x}{z\ },\frac{y}{z}\right)\ then\ the\ value\ of\ x\ \frac{\partial u}{\partial x}+y\frac{\partial u}{\ \partial y}+z\ \frac{\partial u}{\partial z}\ is I f u = f ( z x , z y ) t h e n t h e v a l u e o f x ∂ x ∂ u + y ∂ y ∂ u + z ∂ z ∂ u i s

x + y + z x\ +\ y\ +z x + y + z

Necessary conditions for Maximum or Minimum

∂ f ∂ x = 0 & ∂ f ∂ y = 0 \frac{\partial f}{\partial x}\ =\ 0\ \&\ \ \frac{\partial f}{\partial y}\ =\ 0\ ∂ x ∂ f = 0 & ∂ y ∂ f = 0

∂ f ∂ x = 0 \frac{\partial f}{\partial x}\ =\ 0 ∂ x ∂ f = 0

∂ f ∂ y = 0 \frac{\partial f}{\partial y}\ =\ 0 ∂ y ∂ f = 0

∂ f ∂ x < = 0 & ∂ f ∂ y > = 0 \frac{\partial f}{\partial x}\ <=\ 0\ \&\ \ \frac{\partial f}{\partial y}\ >=\ 0\ ∂ x ∂ f < = 0 & ∂ y ∂ f > = 0

Write the condition of the rectangular box open at the top of maximum capacity whose surface area is 432 sq.m

F = x y z + λ ( 2 x y + 2 x z + y z − 432 ) F\ =\ xyz\ +\lambda\left(2xy\ +2xz+yz-432\right) F = x y z + λ ( 2 x y + 2 x z + y z − 4 3 2 )

F = 8 x y z + λ ( x y + x z + y z − 432 ) F\ =\ 8xyz\ +\lambda\left(xy\ +xz+yz-432\right) F = 8 x y z + λ ( x y + x z + y z − 4 3 2 )

F = x y z + λ ( x y − x z + y z − 432 ) F\ =\ xyz\ +\lambda\left(xy\ -xz+yz-432\right) F = x y z + λ ( x y − x z + y z − 4 3 2 )

F = x y z + λ ( 2 x y + 2 x z + 2 y z − 432 ) F\ =\ xyz\ +\lambda\left(2xy\ +2xz+2yz-432\right) F = x y z + λ ( 2 x y + 2 x z + 2 y z − 4 3 2 )

I f u = f ( x , y ) , x = ϕ ( t ) , y = ψ ( t ) , t h e n d u d t = . . . If\ u\ =\ f\left(x,y\right),x\ =\ \phi\left(t\right),\ y\ =\ \psi\left(t\right),then\ \frac{du}{dt}\ =... I f u = f ( x , y ) , x = ϕ ( t ) , y = ψ ( t ) , t h e n d t d u = . . .

∂ u ∂ x d x d t + ∂ u ∂ y d y d t \frac{\partial u}{\partial x}\frac{dx}{dt}\ +\ \frac{\partial u}{\partial y}\frac{dy}{dt} ∂ x ∂ u d t d x + ∂ y ∂ u d t d y

∂ u ∂ x ∂ x ∂ t + ∂ u ∂ y ∂ y ∂ t \frac{\partial u}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial t} ∂ x ∂ u ∂ t ∂ x + ∂ y ∂ u ∂ t ∂ y

∂ u ∂ t d x d t + ∂ u ∂ t d y d t \frac{\partial u}{\partial t}\frac{dx}{dt}\ +\ \frac{\partial u}{\partial t}\frac{dy}{dt} ∂ t ∂ u d t d x + ∂ t ∂ u d t d y

I f z = f ( x , y ) , x = ϕ ( u , v ) , y = ψ ( u , v ) , t h e n ∂ z ∂ u = . . . If\ z\ =f\left(x,y\right),\ x\ =\ \phi\left(u,v\right)\ ,\ y\ =\ \psi\left(u,v\right),then\ \frac{\partial z}{\partial u}\ =\ ... I f z = f ( x , y ) , x = ϕ ( u , v ) , y = ψ ( u , v ) , t h e n ∂ u ∂ z = . . .

∂ z ∂ x ∂ x ∂ u + ∂ z ∂ y ∂ y ∂ u \frac{\partial z}{\partial x}\frac{\partial x}{\partial u}\ +\ \frac{\partial z}{\partial y}\frac{\partial y}{\partial u} ∂ x ∂ z ∂ u ∂ x + ∂ y ∂ z ∂ u ∂ y

∂ z ∂ x d x d u + ∂ z ∂ y d y d u \frac{\partial z}{\partial x}\frac{dx}{du}\ +\ \frac{\partial z}{\partial y}\frac{dy}{du} ∂ x ∂ z d u d x + ∂ y ∂ z d u d y

∂ z ∂ x d x d v + ∂ z ∂ y d y d v \frac{\partial z}{\partial x}\frac{dx}{dv}\ +\ \frac{\partial z}{\partial y}\frac{dy}{dv} ∂ x ∂ z d v d x + ∂ y ∂ z d v d y

A f u n c t i o n f ( x , y ) Afunction\ f\left(x,y\right) A f u n c t i o n f ( x , y ) is said to be homogeneous function of degree n,if f ( x , y ) f\left(x,y\right) f ( x , y )

x n ϕ ( y x ) x^n\phi\left(\frac{y}{x}\right) x n ϕ ( x y )

x ϕ ( y x ) x^{ }\phi\left(\frac{y}{x}\right) x ϕ ( x y )

y ϕ ( y x ) y\phi\left(\frac{y}{x}\right) y ϕ ( x y )

Maclaurin's theorem for a function of two variable obtained by Taylor's theorem by putting

a = 0 , b = 0 , h = x , k = y a\ =\ 0,\ b=\ 0,\ h\ =\ x,\ k\ =\ y a = 0 , b = 0 , h = x , k = y

a = x , b = y , h = 0 , k = 0 a\ =\ x,\ b=\ y,\ h\ =\ 0,\ k\ =0 a = x , b = y , h = 0 , k = 0

a = 0 , b = x , h = y , k = 0 a\ =\ 0,\ b=\ x,\ h\ =\ y,\ k\ =\ 0 a = 0 , b = x , h = y , k = 0

UNIT II- FUNCTIONS OF SEVERAL VARIABLES

Authored by Shyam Kannan

Mathematics

12th Grade - University

Used 6+ times

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35 questions

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MULTIPLE CHOICE QUESTION

5 mins • 1 pt

If u and v are functions of x and y then u and v are said to be functionally related if ,

$J=\frac{\partial\left(u\ ,v\right)}{\partial\left(x\ ,\ y\right)}\ =\ 1$

$J\ =\ \frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}\ =\ 0$

$J\ =\ \frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}\ =\ \infty$

None of these

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

If u is a homogeneous function of degree ' n' in x and y , then

$x\frac{\partial u}{\partial x}\ +\ y\ \ \frac{\partial u}{\partial y}\ =\ n$

$x\frac{\partial u}{\partial x}\ +\ y\ \ \frac{\partial u}{\partial x}\ =\ nu$

$x\frac{\partial u}{\partial x}\ +\ y\ \ \frac{\partial u}{\partial y}\ =\ nu$

$x\frac{\partial u}{\partial x}\ +\ y\ \ \frac{\partial u}{\partial y}\ =\ u$

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

If u , v are functions of r, s and r , s are functions of x , y then

$\frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}\ =\ \frac{\partial\left(u\ ,\ v\right)}{\partial\left(r\ ,\ s\right)}\ \times\frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}$

$\frac{\partial\left(u\ ,\ v\right)}{\partial\left(x\ ,\ y\right)}\ =\ \frac{\partial\left(x\ ,\ v\right)}{\partial\left(r\ ,\ s\right)}\ \times\frac{\partial\left(r\ ,\ s\right)}{\partial\left(x\ ,\ y\right)}$

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

For function f(x,y) of two variables condition for minimum at certain point is

$rt\ -\ s^2\ >\ 0\ and\ r\ >\ 0$

$rt\ -\ s^2\ <\ 0\ and\ r\ <\ 0$

$rt\ -\ s^2\ >\ 0\ and\ r\ <\ 0$

$rt\ -\ s^2\ =\ 0\ and\ r\ >\ 0$

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$x\ =\ r\cos\theta\ ,\ y\ =\ r\sin\theta\ then\ \frac{\partial\left(r,\ \theta\right)}{\partial\left(x\ ,\ y\right)}\ =\ ?$
If

$\frac{1}{r}$

$r$

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

If $f\left(x\ ,\ y\ ,z\right)$ be a function of $x\ ,\ y\ ,\ z$ which is to be examined for maximum or minimum value.Let the variables $x\ ,\ y\ ,\ z$ be connected by the relation $\phi\left(x\ ,\ y\ ,z\right)\ =\ 0\$ then by Lagrange's method of multiplier....

$\frac{\partial f}{\partial x}\ +\lambda\ \frac{\partial\phi}{\partial x}=1\ ,\frac{\partial f}{\partial y}\ +\lambda\ \frac{\partial\phi}{\partial y}\ =1\ ,\frac{\partial f}{\partial z}\ +\lambda\ \frac{\partial\phi}{\partial z}\ =\ 1$

$\frac{\partial f}{\partial x}\ +\lambda\ \frac{\partial\phi}{\partial x}\ =0\ ,\frac{\partial f}{\partial y}+\lambda\ \frac{\partial\phi}{\partial y}=0\ ,\frac{\partial f}{\partial z}\ +\lambda\ \frac{\partial\phi}{\partial z}\ =0$

$\frac{\partial f}{\partial x}+\lambda\ \frac{\partial\phi}{\partial x}\ =1\ ,\frac{\partial f}{\partial y}+\lambda\ \frac{\partial\phi}{\partial y}\ =0\ ,\frac{\partial f}{\partial z}\ +\lambda\ \frac{\partial\phi}{\partial z}\ =1$

None of these

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$If\ u\ =\ \frac{x\ +\ y}{x\ -\ y}\ and\ v\ =\ \frac{x\ -\ y}{\left(x\ -\ y\right)^2}\$ then the relation is ....

$u^2\ =\ 1+4v$

$u\ =\ 1+\ 4\ v^2$

$u^2\ =\ 4v$

$u^2\ =\ u+4v$

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$If\ x\ =\ \sqrt{vw}\ ,\ y\ =\ \sqrt{wu}\ ,\ z\ =\sqrt{uv}\ then\frac{\partial\left(x\ ,\ y\ ,\ z\right)}{\partial\left(u\ ,\ v\ ,\ w\right)}\ =\ ?$

$\frac{1}{4}$

$\frac{1}{8}$

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$If\ u\ =\ x^y\ then\ \frac{\partial u}{\partial y}\ is\ equal\ to\$

$y\ x^{\left(y\ -\ 1\right)}$

$x^y\ \log x$

None

10.

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

$If\ u\ =\ \tan^{-1}\left(\frac{x^{3\ }+y^{3\ }\ }{x\ +\ y}\right)\ then\ ...\ x\frac{\partial u}{\partial x}\ +y\ \frac{\partial u}{\partial y}\ =\ ?$

$\sin\ 2u$

$\cos\ 2u$

$2\ \tan\ u$

$2\ \sin\ u$

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UNIT II- FUNCTIONS OF SEVERAL VARIABLES

If u and v are functions of x and y then u and v are said to be functionally related if ,

If u is a homogeneous function of degree ' n' in x and y , then

If u , v are functions of r, s and r , s are functions of x , y then

For function f(x,y) of two variables condition for minimum at certain point is

x = rcos⁡θ , y = rsin⁡θ then ∂(r, θ)∂(x , y) = ?x\ =\ r\cos\theta\ ,\ y\ =\ r\sin\theta\ then\ \frac{\partial\left(r,\ \theta\right)}{\partial\left(x\ ,\ y\right)}\ =\ ?x = rcosθ , y = rsinθ then ∂(x , y)∂(r, θ)​ = ? If

If u = x + yx − y and v = x − y(x − y)2 If\ u\ =\ \frac{x\ +\ y}{x\ -\ y}\ and\ v\ =\ \frac{x\ -\ y}{\left(x\ -\ y\right)^2}\ If u = x − yx + y​ and v = (x − y)2x − y​ then the relation is ....

If x = vw , y = wu , z =uv then∂(x , y , z)∂(u , v , w) = ?If\ x\ =\ \sqrt{vw}\ ,\ y\ =\ \sqrt{wu}\ ,\ z\ =\sqrt{uv}\ then\frac{\partial\left(x\ ,\ y\ ,\ z\right)}{\partial\left(u\ ,\ v\ ,\ w\right)}\ =\ ?If x = vw​ , y = wu​ , z =uv​ then∂(u , v , w)∂(x , y , z)​ = ?

If u = xy then ∂u∂y is equal to If\ u\ =\ x^y\ then\ \frac{\partial u}{\partial y}\ is\ equal\ to\ If u = xy then ∂y∂u​ is equal to

If u = tan⁡−1(x3 +y3 x + y) then ... x∂u∂x +y ∂u∂y = ?If\ u\ =\ \tan^{-1}\left(\frac{x^{3\ }+y^{3\ }\ }{x\ +\ y}\right)\ then\ ...\ x\frac{\partial u}{\partial x}\ +y\ \frac{\partial u}{\partial y}\ =\ ?If u = tan−1(x + yx3 +y3 ​) then ... x∂x∂u​ +y ∂y∂u​ = ?

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$x\ =\ r\cos\theta\ ,\ y\ =\ r\sin\theta\ then\ \frac{\partial\left(r,\ \theta\right)}{\partial\left(x\ ,\ y\right)}\ =\ ?$
If

$If\ u\ =\ \frac{x\ +\ y}{x\ -\ y}\ and\ v\ =\ \frac{x\ -\ y}{\left(x\ -\ y\right)^2}\$ then the relation is ....

$If\ x\ =\ \sqrt{vw}\ ,\ y\ =\ \sqrt{wu}\ ,\ z\ =\sqrt{uv}\ then\frac{\partial\left(x\ ,\ y\ ,\ z\right)}{\partial\left(u\ ,\ v\ ,\ w\right)}\ =\ ?$

$If\ u\ =\ x^y\ then\ \frac{\partial u}{\partial y}\ is\ equal\ to\$

$If\ u\ =\ \tan^{-1}\left(\frac{x^{3\ }+y^{3\ }\ }{x\ +\ y}\right)\ then\ ...\ x\frac{\partial u}{\partial x}\ +y\ \frac{\partial u}{\partial y}\ =\ ?$