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TranscendArcu
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Homework Statement
Suppose a mountain is described by the function z = 10x^2 * y − 5x^2 − 4y^2 − x^4 − 2y^4 and that you are standing at the point (1,1,−2). The positive x-axis points east and the positive y- axis points north. If you walk in the northeast direction what angle above the horizontal does your path make?
Homework Equations
cos(θ) = (v • w)/(|v||w|) (But I'm not sure this is even relevant.)
z - z0 = fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0)
The Attempt at a Solution
I am not convinced that this will work, but I intend to find a tangent plane at (1,1,-2). Then, using the normal vectors I have between the tangent and the xy-plane, I will calculate the angle.
fx = 20xy - 10x - 4x^3
fy = 10x^2 - 8y - 8y^3
Evaluated at (1,1) gives,
fx = 6
fy = -6
so,
z + 2 = 6(x-1) - 6(y-1), which I rearrange to give,
0 = 6(x-1) - 6(y-1) - 1(z + 2)
So, I think I have normal vector <6,-6,-1>, and a normal vector on the xy-plane, <0,0,1>.
So,
cos(θ) = (-1)/sqrt(73). So θ = arccos((-1)/sqrt(73)), which is obtuse, so, replacing <0,0,1> with its negative gives θ = arccos((1)/sqrt(73))
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