CHM2132 tutorial notes for Sept. 11
th
, 15
th
.
1. What is the slope of the curve
f
(
x
)
=
3ln
x
+
2
x
!
4
x
3
at the point
x
=3?
Since the question is asking for the slope of a curve at a specific point we need to know
the slope of the tangent to the curve at that point – the derivative of
f
(
x
,
y
,
z
)
.
d
f
(
x
)
[ ]
dx
=
d
3ln
x
+
2
x
!
4
x
3
#
%
dx
Some handy general formulas for derivatives that we will need to solve this are:
dx
n
dx
=
nx
n
!
1
and
d
ln
x
dx
=
1
x
(These are also listed in the inside cover of the text.)
Now the derivative is:
d
f
(
x
)
[ ]
dx
=
3
x
!
2
x
2
!
12
x
2
Substitute in
x
=3:
d
f
(
x
)
[ ]
dx
=
3
3
!
2
3
2
!
12 3
( )
2
=
!
107
Therefore the slope of the curve is 107 at
x
=3.
2. Determine the expression for an ideal gas that describes how the temperature of a gas
changes with the pressure for a fixed amount of gas at a constant volume.
We know how the temperature of an ideal gas is related to its other variables by the ideal
gas equation:
T
=
pV
nR
The question is, what happens when we change the pressure and keep everything else
constant? Mathematically this corresponds to a derivative – we need to know the
derivative of
T
with respect to
p
.
However, there are more variables that define this
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 Fall '10
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