Download Review problems: Linearization, Differential, Chain Rule

Review problems: Linearization, Differential, Chain Rule
These problems are designed to deepen your understanding about when to use linearization, differentials, or the
chain rule. In deciding which method to use, it’s helpful to see what units the answer should be in. For linearization
and differential problems the units are the same units as the values of the function being studied, for example if f is
area then for linearization or differential problems the units would be square feet or meters etc. For chain rule problems
the answer is always a rate, usually involving time, so the units are, for example ft/sec. If the answer is a percent, then
we use df /f to calculate percent error or percent change.
1) A metal plate is situated in the xy-plane and the temperature T on the plane is given by the following table in
degrees Centigrade, x and y measured in centimeters.
x = 1.5
x = 2
x = 2.5
y = 2.5
32.5
45.6
62.5
y=3
47.3
63
83.3
y = 3.5
66.5
84.9
108.5
a) Use the table to find approximate values for
y=4
91
112
139
∂T
∂T
∂x (2, 3), ∂y
(2, 3).
b) Using the table and your answers to a) give the local linearization for T at (2, 3).
c) What is the approximate value of T at (2.2, 2.9)?
d) Suppose a metal tip is moving along the plate. When the tip is at (2, 3), it has velocity < .2, −.1 > centimeters/sec.
How fast is the temperature changing at the tip when the tip is at (2, 3)?
e) Using your answers to a), approximate the change in T as you move from (2, 3) to (2.2, 2.9).
2 ) Suppose that the current I (amperes) in an electric circuit is related to the voltage V (volts) and the resistance R
(ohms) by the equation I = V /R.
What is the rate of change of I if the voltage V = 25 volt and drops with the rate 2 volt/sec and the resistance
R = 100 ohm and increases with the rate 10 ohm/sec?
3a) The current I, voltage V and resistance R of a resistor are related by
R=
V
.
I
When you apply 100 volts to the resistor, the amp meter registers 10 amps. If the error in the amp. meter is ±.3
amps and the error in the volt meter is ±2 volts, approximate the error in the measurement of the resistance of the
resistor. (Hint: To improve your understanding try to do this in two different ways, by using the definition of the
differential directly, and by going through percent error. Check that both methods give the same answer.)
3b) Suppose you don’t know the error in the amp. meter or the volt meter. Instead, you know the percent error of
the voltmeter is ±1% and the percent error of the amp meter is ±2%; what is the percent error in the measurement of
resistance?