Use the chain rule to find , where
First the pieces:

2xy + y^2

x^2 + 2xy

-4t^3

4t^3

36t^3+72t^7

The work

2.

Suppose , where
, and
.

A ) Use the chain rule to find as a
function of x, y, z, and t.
=
B ) Use part A to evaluate when t = 0.

A)

B) 35/12

The work

3.

Suppose ,

A. Use the chain rule to find and
as functions of x, y, s and t.
=
=

B. Find the numerical values of
and when (s , t) = ( 4 , 4 ).

A) =
=

B)
-17070.39
-14952.91

The work

4.

Let
Compute
=
=

= 375
= -900

The work

5.

Let where

3
-13

The work

6.

Consider the curve
The equation of the tangent line to the curve at the point (1,1) has the form y = mx + b where
m = and
b =

m = -8/11
b = 19/11

The work

7.

Find the slope of the tangent line to the curve
at the point ( 3,4 ).
The slope is .

m = -0.991

The work

8.

Consider the surface
Find the following partial derivatives
∂z/∂x =
∂z/∂y =

∂z/∂x =
∂z/∂y =

The work

9.

Consider the surface
Describe the set of points on the surface for which it is not possible to define the surface as the graph of a differentiable function z = f(x,y).
Your answer should be in the form g(x,y,z) = 0
= 0.

= 0

The work

10.

Suppose that z = f(x,y) is defined implicitly by an equation of the form
F (x,y,z) = 0. Find formulas for the partial derivatives
∂f/∂x and ∂f/∂y
in terms of F1, F2, F3.

∂f/∂x =
∂f/∂y =

∂f/∂x = -F1 / F3
∂f/∂y = -F2 / F3

The work

11.

The radius of a right circular cone is increasing at a rate of 3
inches per second and its height is decreasing at a rate of 5 inches
per second. At what rate is the volume of the cone changing when the
radius is 30 inches and the height is 50 inches?

cubic inches per second

4712.39

The work

12.

In a simple electric circuit, Ohm's law states that V = IR, where V is the
voltage in volts, I is the current in amperes, and R is the resistance in ohms.
Assume that, as the battery wears out, the voltage decreases at 0.03 volts
per second and, as the resistor heats up, the resistance is increasing at
0.02 ohms per second. When the resistance is 300 ohms and the current
is 0.02 amperes, at what rate is the current changing?