Homework SourseCompute the rate of change of the function fxyz yx3 zx2e4z
Compute the rate of change of the function
f(x,y,z)= (y/x^3) + z*(x^2)*(e^4-z)
along the line
r = r(t) = < e^[(t^2)-4] , (t^3)-1, 2-t >
with respect to the arc length parameter of this line at the point P0 (1, -9, 4)
f(x,y,z)= (y/x^3) + z*(x^2)*(e^4-z)
along the line
r = r(t) = < e^[(t^2)-4] , (t^3)-1, 2-t >
with respect to the arc length parameter of this line at the point P0 (1, -9, 4)
Solution
First, find the three partial derivatives of f: df/dx = -((3 y)/x^4) + 2 x (e^4 - z) z df/dy = 1/x^3 df/dz = x^2 (e^4 - z) - x^2 z Substituting in the point (1,-9,4), we get: df/dx = -5 + 8 e^4 df/dy = 1 df/dz = -8 + e^4 We now have the gradient of f(x): ?f(x) = <-5 + 8 e^4, 1, -8 + e^4>. We then have the directional derivative as: ?f(x)·r = <-5 + 8 e^4, 1, -8 + e^4> · < e^[(t^2)-4] , (t^3)-1, 2-t > = -1 + e^(-4 + t^2) (-5 + 8 e^4) + (-8 + e^4) (2 - t) + t^3![Compute the rate of change of the function f(x,y,z)= (y/x^3) + z*(x^2)*(e^4-z) along the line r = r(t) = < e^[(t^2)-4] , (t^3)-1, 2-t > with respect to th](/WebImages/27/compute-the-rate-of-change-of-the-function-fxyz-yx3-zx2e4z-1071440-1761561297-0.webp "Compute the rate of change of the function f(x,y,z)= (y/x^3) + z*(x^2)*(e^4-z) along the line r = r(t) = < e^[(t^2)-4] , (t^3)-1, 2-t > with respect to th")
