Find an equation of the tangent plane to the surface
at the point (-3, 1, -13).
z =

z = 12x + 10y + 13

The work

2.

Find the equation of the tangent plane to the surface
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
at the point (1, 4, -5).
z =

z = -x - 5y + 16

The work

5.

Find the equation of the tangent plane to the surface at the point (1, 4, 1.654).

z =

z = 0.292x + 0.298y + 0.169

The work

6.

Find the linearization of the function
at the point (-3, 9).

3x - 0.5y + 4.5

The work

7.

Find the linearization of the function
at the point (1, 4).

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, and . 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
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.