Homework SourseCompute the rate of change of the function fxyz zsinx1 144x
Compute the rate of change of the function
f(x,y,z)= z*sin(x-1) + (144*x*e^(y-12))/y + z
along the line
r = r(t) = <cos(t^2 -9), t^2 - t, t-3>
with respect to the arc length parameter of this line at the point P0 (1, 12, -6)
f(x,y,z)= z*sin(x-1) + (144*x*e^(y-12))/y + z
along the line
r = r(t) = <cos(t^2 -9), t^2 - t, t-3>
with respect to the arc length parameter of this line at the point P0 (1, 12, -6)
Solution
First, find the three partial derivatives of f:
df/dx = (144 e^(-12 + y))/y + z Cos[1 - x]
df/dy = -((144 e^(-12 + y) x)/y^2) + (144 e^(-12 + y) x)/y
df/dz = 1 - Sin[1 - x]
Substituting in the point (1,12,-6), we get:
df/dx = 6
df/dy = 11
df/dz = 1
We now have the gradient of f(x):
f(x) = <6, 11, 1>.
We then have the directional derivative as:
f(x)·r = <6, 11, 1> · <cos(t^2 -9), t^2 - t, t-3>
= -3 + t + 11 (-t + t^2) + 6 Cos[9 - t^2]
