Chapter 7
The Chain Rule, Related Rates, and
Implicit Differentiation
7.1
For each of the following, find the derivative of
y
with respect to
x
.
(a)
y
6
+ 3
y

2
x

7
x
3
= 0
(b)
e
y
+ 2
xy
=
√
3
(c)
y
=
x
cos
x
7.2
Consider the growth of a cell, assumed spherical in shape.
Suppose that the radius of the cell
increases at a constant rate per unit time. (Call the constant
k
, and assume that
k >
0.)
(a) At what rate would the volume,
V
, increase ?
(b) At what rate would the surface area,
S
, increase ?
(c) At what rate would the ratio of surface area to volume
S/V
change? Would this ratio increase
or decrease as the cell grows? [Remark: note that the answers you give will be expressed in
terms of the radius of the cell.]
7.3
Growth of a circular fungal colony
A fungal colony grows on a flat surface starting with a single spore. The shape of the colony edge
is circular (with the initial site of the spore at the center of the circle.) Suppose the radius of the
colony increases at a constant rate per unit time. (Call this constant
C
.)
(a) At what rate does the area covered by the colony change ?
(b) The biomass of the colony is proportional to the area it occupies (factor of proportionality
α
). At what rate does the biomass increase?
v.2005.1  September 4, 2009
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Math 102 Problems
Chapter 7
7.4
Limb development
During early development, the limb of a fetus increases in size, but has a constant proportion.
Suppose that the limb is roughly a circular cylinder with radius
r
and length
l
in proportion
l/r
=
C
where
C
is a positive constant.
It is noted that during the initial phase of growth, the radius
increases at an approximately constant rate, i.e. that
dr/dt
=
a.
At what rate does the mass of the limb change during this time? [Note: assume that the density of
the limb is 1 gm/cm
3
and recall that the volume of a cylinder is
V
=
Al
where
A
is the base area (in this case of a circle) and
l
is length.]
7.5
A rectangular trough is 2 meter long, 0
.
5 meter across the top and 1 meter deep.
At what rate
must water be poured into the trough such that the depth of the water is increasing at 1 m
/
min
when the depth of the water is 0
.
7 m?
7.6
Gas is being pumped into a spherical balloon at the rate of 3
cm
3
/s
.
(a) How fast is the radius increasing when the radius is 15
cm
?
(b) Without using the result from (a), find the rate at which the surface area of the balloon is
increasing when the radius is 15
cm
.
7.7
A point moves along the parabola
y
=
1
4
x
2
in such a way that at
x
= 2 the
x
coordinate is increasing
at the rate of 5 cm/s. Find the rate of change of
y
at this instant.
7.8
Boyle’s Law
In chemistry, Boyle’s law describes the behaviour of an ideal gas:
This law relates the volume
occupied by the gas to the temperature and the pressure as follows:
PV
=
nRT
where
n, R
are positive constants.
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 Winter '09
 Allard
 Derivative

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