Unit - 3
Applications of Partial differentiation
- Find Maclaurin’s Expansion’s for log sec x.
Solution:
Let f(x) = log sec x
By Maclaurin’s Expansion’s,
(1)
By equation (1)
2. Prove that 
Solution:
Here f(x) = x cosec x
= 
Now we know that
3. Expand 
upto x6
Solution:
Here 
Now we know that
… (1)
… (2)
Adding (1) and (2) we get
4. Show that 
Solution:
Here 
Thus
5. Expand 
in power of (x – 3)
Solution:
Let 
Here a = 3
Now by Taylor’s series expansion,
… (1)
equation (1) becomes.
6. Using Taylors series method expand
in powers of (x + 2)
Solution:
Here 
a = -2
By Taylors series,
… (1)
Since
, 
, …..
Thus equation (1) becomes
7. Expand 
in ascending powers of x.
Solution:
Here
i.e. 
Here h = -2
By Taylors series,
… (1)
equation (1) becomes,
Thus
8. Decampere a positive number ‘a’ in to three parts, so their product is maximum
Solution:
Let x, y, z be the three parts of ‘a’ then we get.
… (1)
Here we have to maximize the product
i.e. 
By Lagrange’s undetermined multiplier, we get,
… (2)
… (3)
… (4)
i.e.
… (2)’
… (3)’
… (4)
And 
From (1)
Thus 
.
Hence their maximum product is 
.
9. Find the point on plane 
nearest to the point (1, 1, 1) using Lagrange’s method of multipliers.
Solution:
Let 
be the point on sphere 
which is nearest to the point 
. Then shortest distance.
Let 
Under the condition 
… (1)
By method of Lagrange’s undetermined multipliers we have
… (2)
… (3)
i.e. 
&
… (4)
From (2) we get 
From (3) we get 
From (4) we get 
Equation (1) becomes
i.e. 
y = 2
If 
where x + y + z = 1.
Prove that the stationary value of u is given by,
10. Expand 
in powers of x using Taylor’s theorem,
Solution:
Here
i.e. 
Here
h = 2
By Taylors series
… (1)
By equation (1)
