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MTH301 Assignment #2 Solutions - Derivatives, Vectors, Projections, Volumes (Spring 2012), Exercises of Mathematics

The solutions to assignment #2 in the mth301 course taken during spring 2012. The assignment covers topics such as directional derivatives, unit vectors, vector projections, and volumes. The solutions include step-by-step calculations and explanations for each question.

Typology: Exercises

2011/2012

Uploaded on 08/03/2012

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Solution of Assignment # 2
MTH301 (Spring 2012)
Total marks: 10
Lecture # 7 to 15
Question # 1
Find the directional derivative of 2
(, ) ln
f
xy y x at P(1,4) in the direction of
33aij .
Solution:
(, ) (, )
xy
f
fxyifxyj 
2ln
y
f
iyxj
x
 
16
(1, 4) ln 1
1
f
ij
i
i

  

ˆ()/2uij 
ˆ
./2
u
Df fu  
Question # 2
Find a unit vector in the direction in which function 22
(, )
f
xy x y
 increases most
rapidly at P(4, -3}. .
Solution:
A function increases most rapidly in the direction of the gradient vector
f
at P.
22
(, )
f
xy x y
docsity.com
pf3
pf4

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Download MTH301 Assignment #2 Solutions - Derivatives, Vectors, Projections, Volumes (Spring 2012) and more Exercises Mathematics in PDF only on Docsity!

Solution of Assignment # 2

MTH301 (Spring 2012) Total marks: 10 Lecture # 7 to 15

Question # 1

Find the directional derivative of f ( , x y  )  y^2 ln x at P(1,4) in the direction of a  3 i  3 j.

Solution:

f  f (^) x ( , x y i  )   f (^) y ( , x y  ) j 2 f y i y ln x j x

(1, 4) 16 ln 1 f i j i i

u ˆ   ( i j ) / 2

D fu  f u. ˆ / 2

Question # 2

Find a unit vector in the direction in which function f ( , x y  ) x^2  y^2 increases most rapidly at P(4, -3}..

Solution:

A function increases most rapidly in the direction of the gradient vector  f at P.

f ( , x y  ) x^2  y^2

f  f (^) x ( , x y i  )   f (^) y ( , x y  ) j

2 2 2 2 f x^ i y j x y x y

f i j

i j

Hence u 4 / 5     i j is the required unit vector.

Question # 3

Find the vector projection of vector b  2 i  6 j  4 k on vector a  3 ij  5 k

Solution:

vector rojection of b on a. . (^6 6 20) (3 5 ) 9 1 25 (^32) (3 5 ) 35 96 32 32 35 35 7

p b a a a i j k

i j k

i j k

 ^   

Question # 4

Find the volume of the parallelepiped

Where a  2 i  3 j  5 k bi  3 j  4 k c  3 i  2 j  4 k

Solution:

 x^1 ^  y^4   z^3 ^0