Question

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) ≈

Answer #1

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)

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

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

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)?

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.

Find the minimum of f(x,y,z) = x2 + y2 +
z2 subject to the two constraints x + 2y + z = 3 and x -
y = 4 by answering following questions
a) write out the lagrange equation involving lagrange
multipliers λ(lamba) and μ(mu)
b) solve for lamba in terms of x and y
c) solve for x,y,z using the constraints
d) determine the minimum value

Let H be the hemisphere x2 + y2 + z2 = 54, z ≥ 0, and suppose f
is a continuous function with f(2, 5, 5) = 9, f(2, −5, 5) = 11,
f(−2, 5, 5) = 12, and f(−2, −5, 5) = 13. By dividing H into four
patches, estimate the value below. (Round your answer to the
nearest whole number.)
H
f(x, y, z) dS

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