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QUESTION: 1

If then the value of is

Solution:

**u = sin ^{-1}(x + y/(√x + √y))**

**Put x = xt and y = yt **

**u = sin ^{-}^{1}[(xt-yt)/(√xt + √yt)]**

**sinu = t ^{1/2}[(x+y)/(√x + √y)]**

**= t ^{1/2} f(x,y)**

**This function sin u is homogeneous with degree 1/2.**

**By Euler's theorem**

**xu _{x} + yu_{y} = G(u) = nf(u)/f,(u) = 1/2 tan u**

**xu _{x} + yu_{y} = 1/2 tan u**

**x2u _{xx} + 2xyu_{yy} + y2u_{yy} = G(u)[G'(u) - 1]**

**= 1/2 tan u [(sec ^{2}u - 2)/2]**

**= 1/4 tan u [(tan ^{2}u - 1)/1]**

**= 1/4 * sin u/cos u [(sin ^{2}u - cos^{2}u)/cos^{2}u]**

**x ^{2}u_{xx} + 2xyu_{yy} + y2u_{yy} = -(sinu cos^{2}u)/4ucos^{3}u**

QUESTION: 2

then the value of is equal to

Solution:

We have

** v** is homogeneous function of degree

The correct answer is: n(n - 1)

QUESTION: 3

The accompanying figure shows the graph of an unspecified function of f(x, y) and partial derivatives f_{x}(x, y) and f_{y}(x, y). Determine which is which and explain.

Solution:

The correct answer is: II - *f*(*x*, *y*), I - f_{x}(x, y), III - f_{y}(x, y)

QUESTION: 4

Solution:

We are given

f(u) is homogeneous function degree 1, then

[By dividing with cos^{3} u]

The correct answer is:

QUESTION: 5

If u = f(t) and v = f(t) and t = φ (x, y) then

Solution:

The correct answer is:

QUESTION: 6

If w = f(u, v) where u = x + y and v = x – y then

Solution:

We have w = f(u, v)

The correct answer is:

QUESTION: 7

If f is differentiable and z = u + f(u^{2}v^{2}), then

Solution:

Let w = u^{2}v^{2}

then z = f(w) + u

So,

The correct answer is:

QUESTION: 8

If then the value of is

Solution:

f(u) is homogeneous function of degree 2

Now, let g(u) = 2tan u

The correct answer is: 2tan *u*(2sec^{2}*u* –1)

QUESTION: 9

If f(x, y, z) = 0 then the value of equal to

Solution:

**(1) Differentiate with respect to y, I get:**

**0+F2+F3∂z/∂y=0**

**So **

**F3 ∂z/∂y = −F2**

**(2) Differentiate with respect to x, I get:**

**F1 + F2 ∂y/∂x + 0 = 0**

**So F2 ∂y/∂x = −F1**

**(3) Differentiate with respect to z, I get:**

**F1 ∂x/∂z +0 + F3 = 0**

**4) After some manipulations with the Fi, I get to the conclusion that **

**∂z/∂y∗∂y/∂x∗∂x/∂z=−1, so when evaluated with x, z, y respectively, conclusion is still true**

QUESTION: 10

Use the information the figure to find the first order partial derivatives of ** f** at the point (1, 2)

Solution:

The correct answer is:

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