Let f be the function of x and y defined in a neighborhood of (3,2,1) by the sur
ID: 2872295 • Letter: L
Question
Let f be the function of x and y defined in a neighborhood of (3,2,1) by the surface (taking z = f(x,y)) xyz + xy2z3 = 18. Find first as a formula and then as a numerical value, inserting 3 for x, 2 for y, and 1 for z. Do the same for Use differentials to approximate f(3.2,2.1). The actual value of f(3.2,2.1) is different from what differentials predict. Using any approach you please, find f(3.2,2.1) to 3 places. Numerical methods and computers and calculators of any stripe may be employed. How does the predicted value compare to the better estimate? Now use the same surface and the same point, but consider y to be a function of x and z. So, instead of saying f(3,2) = 1, we now say g(3,1) = 2. FindExplanation / Answer
This question has more than 4 subparts, therefore I am going to answer only the first four. You have to post the remaining in another question.
So we have a function z = f(x,y) defined implicitly by the surface
x y z + x y^2 z^3 = 18.
1) We will find first df/dx = dz/dx. We derive with respect to x the equation for the surface, we obtain
y z + x y dz/dx + y^2 z^3 + 3 x y^2 z^2 dz/dx = 0.
We solve for dz/dx,
dz/dx ( xy + 3xy^2 z^2) = -yz - y^2 z^3,
dz/dx = -(yz + y^2 z^3)/(xy + 3xy^2 z^2).
Now we calculate this as a numerical value, replacing x=3, y=2, z=1, we get
dz/dx = -(2 + 4)/(6 + 36) = -1/7.
2) We calculate dz/dy:
x z + x y dz/dy + 2x y z^3 + 3 x y^2 z^2 dz/dy = 0,
dz/dy (xy + 3xy^2 z^2) = -(xz + 2xy z^3),
dz/dy = -(xz + 2xy z^3)/ (xy + 3xy^2 z^2) ,
and if we put numerical values,
dz/dy = -(3 + 12 )/ (6 + 36 ) = -15/42 = -5/14.
3) We have that
f(3.2, 2.1) = dz/dx * 0.2 + dz/dy * 0.1 = -0.02857 - 0.03571 = 0.06428.
4) I used an online numerical engine called wolframalpha in order to solve for z, it is obtained that
z = (81 x^2 y^4+sqrt(3) sqrt(x^6 y^9+2187 x^4 y^8))^(1/3)/(3^(2/3) x y^2)-(x y)/(3^(1/3) (81 x^2 y^4+sqrt(3) sqrt(x^6 y^9+2187 x^4 y^8))^(1/3))
when evaluating x=3.2, y=2.1, we obtain z=0.939.
The method of differentials gave a very different value from the original one.
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