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HILL: PARtial
derivative APPLICATIONS
Applications with Rates of Change
Real-World Applications: Rates of Change
In the Java applet we saw how the concept of partial
derivative could be applied geometrically to find the slope of the surface in the x and y directions. In the following two examples we present� partial derivatives as
rates of change. Specifically we explore an application to a temperature function ( this example does have a geometric aspect in terms of
the physical model itself) and a
second application to electrical circuits, where no geometry is involved.
I.
Temperature on a Metal Plate
The screen
capture below shows a current website illustrating thermal flow for chemical
engineering. Our first application will deal with a simialr flate plate where
temperature varies with position. * The example following the picture below
is taken from the current text in SM221,223: Multivariable Calculus
by James Stewart.
����� Suppose we have a flat metal plate where
the temperature at a point (x,y) varies according to position. In
particular,� let the temperature at a
point (x,y) be given by  , where T is measured in oC and x and y in
meters. Question: what is
the rate of change of temperature with respect to distance at the point (2,1)
in (a) the
x-direction? and (b) in the
y-direction ?
Let's take (a) first.
What is
the rate of change of
temperature with respect to distance at
the point (2,1) in (a) the
x-direction?
What
observations and translations can we make here?
Rate of change of temperature indicates that we will be
computing a type of derivative. Since the temperature function is defined on
two variables we will be computing a partial derivative. Since the
question asks for the rate of change in the x-direction, we will be holding y
constant. Thus, our question now becomes:
What is  �at the point (2,1)?
Since  �
and �� �
Our final
verbal conclusion is:
������������� The rate of change of temperature in the
x-direction at (2,1) is - degrees per meter;
�note this means that the temperature is decreasing !
Part
(b): The rate of change of temperature in the
y-direction at (2,1) is computed in a similar manner.
 �and
and �� �
Our verbal
conclusion is: the
temperature is decreasing at a rate of  oC/m in the y-direction.
II.
Electrical Circuits:
Changes in Current
The following is adapted from an
example in a former text for SM221,223 Multivariable Calculus
by Bradley and Smith.
* �� In an electrical circuit with electromotive
force (EMF) of E volts and resistance R ohms, the current, I,� is
I=E/R
amperes.
Question: (a) At the instant when E=120 and R=15 ,
what is the rate of change of current with respect to voltage ) (b)What is
the rate of change of current with respect to resistance?
(a) Even though no geometry is involved in this example,
the rate of change questions can be answered with partial derivatives.; we first note that I is a function of E and R, namely,
I(E,R) = ER-1
The rate of change of current with respect to voltage =
�the partial derivative of I with respect to
voltage, holding resistance constant =
 ��
when E=120 and R=15 , we have
 .
Our verbal conclusion becomes: If the resistance
is fixed at 15 ohms, the current is increasing with
respect to voltage at the rate of 0.0667 amperes per volt when the EMF
is 120 volts.
Part
(b): What is the rate of change of current with respect to resistance?
Using
similar observations to part (a) we conclude:
the partial derivative of I with respect to
resistance, holding voltage constant =
 ��
when E=120 and R=15 , we have
Our verbal conclusion becomes: If the EMF is
fixed at 120 volts, the current is decreasing with respect to resistance at the rate of 0.5333 amperes
per ohm when the resistance is 15 ohms.
 USNA Mathematics
Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil
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