anonymous
  • anonymous
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)
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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experimentX
  • experimentX
first of all find PQ
anonymous
  • anonymous
I need help finding u, I'm not sure if I've done this right: PQ = \[PQ=<-\pi/2,\pi>\] \[u=-\pi/2,\pi/\sqrt{(\pi^{2}/4)+(\pi^2)}=5\pi^2/4\]
experimentX
  • experimentX
let us suppose PQ = r

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experimentX
  • experimentX
df/dr = grad(f).r' --- where r' is unit vector along r
experimentX
  • experimentX
PQ = \[i \pi/2 - j \pi\]
amistre64
  • amistre64
gradF with P applied, dotted with Q right?
anonymous
  • anonymous
\[D_{u}(\pi,0)=2/\sqrt{5}\]
anonymous
  • anonymous
@JerJason, the u should be a vector which is not just represented by that only. the u is (1/sqrt(5),2/sqrt(5))
experimentX
  • experimentX
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
anonymous
  • anonymous
@experimentX P is just a point on the function and Q is the vector which is parallel to the directional derivative
amistre64
  • amistre64
gradF at P = <-1,0> unit Q-P = <-sqrt(5)pi^2/4, sqrt(5)pi^2/2> dotted = sqrt(5)/4 pi^2
amistre64
  • amistre64
is Q a vector? or a point? since notation is not consistent between authors
experimentX
  • experimentX
@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
amistre64
  • amistre64
thats what i interped it as as aswell, the vector from P to Q
anonymous
  • anonymous
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
anonymous
  • anonymous
@ experimentX at P in the direction of Q not just P in direction of Q
amistre64
  • amistre64
i would have to defer to the person who posted the question for clarification
experimentX
  • experimentX
u = PQ/(pi*sqrt(5)/2)
anonymous
  • anonymous
Yes P and Q are both two points.
experimentX
  • experimentX
then use the method as i described above

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