


Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
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
1 / 4
This page cannot be seen from the preview
Don't miss anything!
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 i j 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 b i 3 j 4 k c 3 i 2 j 4 k
Solution: