
#1
Apr309, 04:51 AM

P: 65

1. The problem statement, all variables and given/known data
Find a unit vector in the direction in which f(x,y,z) = (x3y+4z)^{1/2} increases most rapidly at P(0,3,0), and find the rate of increase of f in that direction 2. Relevant equations 3. The attempt at a solution I've calculated the unit vector to be <0,1,0> and the gradient to be <1/6,1/2,2/3> To find the directional derivative we find the dot product of <0,1,0> . <1/6,1/2,2/3> = 1/2 Do I than multiply that by cos(0)? regards Brendan 



#2
Apr309, 05:11 AM

Sci Advisor
HW Helper
P: 4,301

How did you calculate that unit vector?
And what does it matter whether you multiply by cos(0)? 



#3
Apr309, 05:12 AM

P: 15

cos(0) is just 1...




#4
Apr309, 06:58 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,890

Directional derivativesTo find the directional derivative we find the dot product of <0,1,0> . <1/6,1/2,2/3> = 1/2 Do I than multiply that by cos(0)? regards Brendan[/QUOTE] Again, <0, 1, 0> is the wrong vector. The rate of increase in the direction of the gradient vector is just the dot product of the unit vector in that direction with the gradient vector and is just the length of the gradient vector. 



#5
Apr309, 06:23 PM

P: 65

So If I use the gradient <1/6 ,1/2 ,2/3> and find its unit vector which is
<sqrt(26)/936, sqrt(26)/312, sqrt(26)/234 > than find the dot product of them both <1/6 ,1/2 ,2/3> . <sqrt(26)/936, sqrt(26)/312, sqrt(26)/234 > which is sqrt(26)/6 the magnitude of the gradient vector? regards Brendan 



#6
Apr309, 07:04 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,890





#7
Apr309, 09:05 PM

P: 65

Sorry,
the unit vector would be < 1/6 , 1/2 , 2/3 > / < sqrt(26)/6 , sqrt(26)/6 , sqrt(26)/6 > regards Brendan 


Register to reply 
Related Discussions  
estimating partial derivatives/directional derivatives  Calculus & Beyond Homework  1  
Directional derivatives  Calculus & Beyond Homework  9  
Directional derivatives  Calculus & Beyond Homework  4  
Directional derivatives  Introductory Physics Homework  3  
Directional Derivatives  General Math  2 