Module 2

Limits, homogenous, Maxima Minima

Q1.   If then verify Euler’s theorem for u.

Sol 1.               

By symmetry,

b)    Put x=xt, y=yt we get

Thus u is homogenous function of

Hence, by Euler’s Theorem

Q2.   If prove that

Sol 2.       Putting x=xt, y=yt we get

Q3.   If find value of

Sol 3.       Let

Let u=v+w

Putting x=xt, y=yt, z=zt

Thus v is homogenous function of degree 6.

Q4. Find all stationary values of

Sol 4: We have

Step I:

Step II: We now solve

  i.e.

And     or

(i)        When

are stationary points.

(ii)      When

or .

and are stationary points.

Step III: (i) When

and

and

is minimum at.

The minimum value.

(ii)      When

and

is maximum at.

The maximum value

(iii)   When 

is neither maximum nor minimum.

(iv)    When  

is neither maximum nor minimum

Q5. Find the minimum distance from the point to the cone

Sol 5: If   is any point on the cone, its distance from the point is

When the distance is minimum, its square is also minimum.

Step I : We have to find the stationary value of

                             ………………(1)

With the condition that                            ………………….(2)

Consider the Lagrange’s function

  gives

     ………………….(3)

Step II : We have to eliminate x, y, z from (1), (2) and (3).

From (3),

Hence,     

Hence, the required point is .

The distance between and is

Q6. If , find the values of x, y, z for which is maximum.

Sol6 : We have to maximize with the condition that .

This means we have to maximize

i.e. we have to minimizei.e.

Step I : We have to minimize  ………………(1)

With the condition that    ……………….(2)

Consider the Lagrange’s function,

gives

But

and

Q7. Prove that

SOL7:   

Diff wrt

Hence by Maclaurin’s Series

we get

Q8. Expand upto

Sol 8:We have

Q9. Prove that

Sol 9. We have

Q10. Expand in Power of

Hence Prove that

Sol 10.