-
Use the chain rule to find
if ,
,
,
and .
-
Use the chain rule to find and
if ,
,
and .
,
.
-
Use the chain rule to find and
if ,
,
and .
-
If where is differentiable,
,
,
,
,
,
,
,
and ,
find when .
-
Let , where
,
,
and
are differentiable,
,
,
,
,
,
,
,
.
Find
and .
-
-
Suppose that is a differentiable function of and
and
.
Use the table of values to calculate
and .
-
Suppose that is a differentiable function of and
and
.
Use the table of values to calculate
and .
| | | | |
| | | | |
| | | | |
-
Use the tree diagram to write out the chain rule for differentiating
,
where
and .
,
,
.
-
Let ,
,
and .
Use the chain rule to find ,
,
and
at .
, , and .
-
The temperature at a point
is
,
measured in kelvin. A
Mars
Exploration Rover
crawls so that its position after
seconds is given by
and
,
where
and
are in centimeters. The temperature function
satisfies
and
.
How fast is the temperature rising on the rover's path after
seconds?
[K/s].
-
Soylent Green production in a given year, , depends on the average
temperature and the annual rainfall . It is estimated that
the average temperature is rising at a rate of K/year and
rainfall is decreasing at a rate of cm/year. It is also known
that at current production levels, and
. What is the significance of the signs of these
partial derivatives? Estimate the current rate of change of Soylent
Green production, .
-
The radius of a right circular cone is increasing at a rate of
cm/s and its height is decreasing at the rate of
cm/s. At what rate is the volume of the cone changing when the
radius is cm and the height is cm?
Approximately cubic cm per second.
-
The length
, width
, and height
of a rectangular box
change with respect to time
. At a certain time the
dimensions are
. At the same
time,
and
are increasing at a rate of
m/s, while
is decreasing at a rate of
m/s. Find the rates at
which the following are changing:
- The volume of the box.
- The surface area of the box.
- The length of the diagonal of the box.
- cubic meters per second.
- square meters per second.
- m/s.
-
Assuming that is differentiable and ,
show that .