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Calc 3 11.4

 

<== Calculus 3
Number Question Answer
See the work
1. Find an equation of the tangent plane to the surface z = 5 y^2 - 2 x^2 at the point (-3, 1, -13).
z =
z = 12x + 10y + 13
The work
2. Find the equation of the tangent plane to the surface
z = 3x^3 + y^3 + 4 xy at the point (0, -3, -27).
z =
z = -12x + 27y + 54
The work
3. Find the equation of the tangent plane to the surface z = cos(6x)*cos(7y) at the point ( π , π /2, 0).

z =
z = 7(y-π/2)
The work
4. Find an equation of the tangent plane to the surface z = 1 x^2  - 1 y^2  - 3 x + 3 y + 1
at the point (1, 4, -5).
z =
z = -x - 5y + 16
The work
5. Find the equation of the tangent plane to the surface z = e^{3 x/17} \ln \left(
1 y 
ight) at the point (1, 4, 1.654).

z =

z = 0.292x + 0.298y + 0.169
The work
6. Find the linearization of the function z = x \sqrt{y} at the point (-3, 9).
L(x,y) =
3x - 0.5y + 4.5
The work
7. Find the linearization of the function f(x,y) = \sqrt{45 - 4 x^2 - 1 y^2} at the point (1, 4).
L(x,y) =
Use the linear approximation to estimate the value of
f (0.9, 4.1) =
L(x,y) = -0.8x - 0.8y + 9

f (0.9, 4.1) = 5
The work
8. Suppose that f(x,y) is a smooth function and that its partial derivatives have the values, f_x(3, -7) = 2 and f_y(3, -7) =
-3. Given that f (3, -7) = 9, use this information to estimate the following values:
Estimate of f (3, -6)
Estimate of f (4, -7)
Estimate of f (4, -6)
f (3, -6) = 6
f (4, -7) = 11
f (4, -7) = 8
The work
9. Find the differential of the function w = x^{5} \sin(y^{2} z^{5})
dw = dx + dy + dz
dw =
dx +
dy +
dz
The work
10. Use differentials to estimate the amount of material in a closed cylindrical can that is 50 cm high and 20 cm in diameter if the metal in the top and bottom is 0.2 cm thick, and the metal in the sides is 0.05 cm thick. Note, you are approximating the volume of metal which makes up the can (i.e. melt the can into a blob and measure its volume), not the volume it encloses.
The differential for the volume is
dV = dr + dh
dr = and dh =
The approximate volume of material is cm^3.
dV = (2πrh)dr + (πr^2)dh
dr = (0.05)
dh = (0.4)
Volume = 282.743
The work
11. The dimensions of a closed rectangular box are measured as 50 centimeters, 70 centimeters, and 70 centimeters, respectively, with the error in each measurement at most .2 centimeters. Use differentials to estimate the maximum error in calculating the surface area of the box.

square centimeters

152
The work