Question

Use the gradient to find the directional derivative of the function at P in the direction of Q:

f(x,y) = sin(2x)cos(y), P(pi,0),Q(pi/2,pi)

Answer 1

first of all find PQ

Answer 2

I need help finding u, I'm not sure if I've done this right:

PQ = \[PQ=<-\pi/2,\pi>\]
\[u=-\pi/2,\pi/\sqrt{(\pi^{2}/4)+(\pi^2)}=5\pi^2/4\]

Answer 3

let us suppose PQ = r

Answer 4

df/dr = grad(f).r' --- where r' is unit vector along r

Answer 5

PQ = \[i \pi/2 - j \pi\]

Answer 6

gradF with P applied, dotted with Q right?

Answer 7

\[D_{u}(\pi,0)=2/\sqrt{5}\]

Answer 8

@JerJason, the u should be a vector which is not just represented by that only. the u is (1/sqrt(5),2/sqrt(5))

Answer 9

OP and OQ are two position vectors, PQ is a vector that you get from OQ - OP

and pi sqrt(5)/2 is the magnitude of PQ, yo can use this magnitude to find the unit vector ... and if you can find the gradient of the function then you can find the directional derivative by taking dot product of gradient and this newly found unit vecotr

Answer 10

@experimentX P is just a point on the function and Q is the vector which is parallel to the directional derivative

Answer 11

gradF at P = <-1,0>
unit Q-P = <-sqrt(5)pi^2/4, sqrt(5)pi^2/2>

dotted = sqrt(5)/4 pi^2

Answer 12

is Q a vector? or a point? since notation is not consistent between authors

Answer 13

@anonymoustwo44 I think .. P and Q are two points. ---> ' P(pi,0),Q(pi/2,pi) ' and
P in the direction of Q ---> along PQ

Answer 14

thats what i interped it as as aswell, the vector from P to Q

Answer 15

it is stated that we are to the the directional derivative of the function a POINT P in the DIRECTION OF Q. so the (x,y) here is P and the u here is vector Q divided by it's magnitude

Answer 16

@ experimentX at P in the direction of Q not just P in direction of Q

Answer 17

i would have to defer to the person who posted the question for clarification

Answer 18

u = PQ/(pi*sqrt(5)/2)

Answer 19

Yes P and Q are both two points.

Answer 20

then use the method as i described above