Question

Find the linear approximation of the function f(x, y, z) = sqrt x2 + y2 + z2 at (3, 6, 6) and use it to approximate the number sqrt3.01^2 + 5.97^2 + 5.98^2 . (Round your answer to five decimal places.) f(3.01, 5.97, 5.98)

Answer #1

Find the linear approximation of the function f(x, y, z) = x2 +
y2 + z2 at (6, 2, 9) and use it to approximate the number 6.012 +
1.972 + 8.982 . (Round your answer to five decimal places.) f(6.01,
1.97, 8.98) ≈

Calculate ∫ ∫S f(x,y,z)dS for the given surface and function.
x2+y2+z2=144, 6≤z≤12; f(x,y,z)=z2(x2+y2+z2)−1.

Use the method of Lagrange multipliers to find the minimum value
of the function
f(x,y,z)=x2+y2+z2
subject to the constraints x+y=10 and 2y−z=3.

Find the minimum of f(x, y, z) = x2 + y2 +
z2 subject to the two constraints x + y + z = 1 and 4x +
5y + 6z = 10

Evaluate ∫∫Sf(x,y,z)dS , where f(x,y,z)=0.4sqrt(x2+y2+z2)) and S
is the hemisphere x2+y2+z2=36,z≥0

Given the function f(x, y, z) = (x2 + y2 +
z2 )−1/2
a) what is the gradient at the point (12,0,16)?
b) what is the directional derivative of f in the direction of
the vector u = (1,1,1) at the point (12,0,16)?

-find the differential and linear approximation of f(x,y) =
sqrt(x^2+y^3) at the point (1,2)
-use tge differential to estimate f(1.04,1.98)

Use Lagrange multipliers to find the extremal values of
f(x,y,z)=2x+2y+z subject to the
constraint
x2+y2+z2=9.

Show that the origin is a spiral point of the system
x' = -y - x(sqrt(x2 + y2))
y' = x - y(sqrt(x2 +
y2)) but a center for its linear approximation

show that the function f(x,y,z) =
1/√(x2+y2+z2) provides the
equation fxx + fyy + fzz = 0, called the 3−D Laplace equation.

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