Estimation and Limits -- Multiplying Functions

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

This is the third in a series of modules that deal with the ways in which functions can be changed or combined. In this module we look at what happens when two functions are multiplied.

The product of two functions is useful in many different contexts.

If the function f(t) describes the per capita energy consumption in a particular country at time t and the function p(t) describes the population of the country then the product E(t) = f(t) p(t) describes the total energy consumption.
If a rectangular region is changing size and its length is described by the function f(t) and its width by the function g(t) then its area is described by the product A(t) = f(t) g(t).
The bouncing of a frictionless spring might be described by a function like f(t) = cos 2 pi t. Usually, however, there is some friction, so the bounces decrease in amplitude and are described by a function like h(t) = 0.85^t cos 2 pi t. This function is obtained by multiplying the function f(t) by the function g(t) = 0.85^t. The figure below shows these three functions.

Use your CAS window to compare the following functions

f(t) = sin 2 pi t
g(t) = t^2
h(t) = f(t) g(t)

Use your CAS window to compare the following functions

f(t) = sin 2 pi t
g(t) = 1/(1 + t^2)
h(t) = f(t) g(t)

Use your CAS window to compare the following functions

f(t) = sin 2 pi t
g(t) = sin 20 pi t
h(t) = f(t) g(t)

The basic fact that we discuss here is that if

  Lim  f(x) = L
x --> a

and

  Lim  g(x) = M
x --> a

then

  Lim  f(x) g(x) = L M
x --> a

We can express the same fact by saying that if f(x) is close to L when x is close to a and if g(x) is close to M when x is close to a then f(x) g(x) is close to L M when x is close to a.

However we express it, this descriptive fact is only part of a more interesting story. The first hypothesis -- the fact that

  Lim  f(x) = L
x --> a

requires that we can make f(x) as close as we want to L by making x sufficiently close to a. The second hypothesis -- the fact that

  Lim  g(x) = M
x --> a

requires that we can make g(x) as close as we want to M by making x sufficiently close to a. The conclusion -- the fact that

  Lim  f(x) g(x) = L M
x --> a

tells us that we can make f(x) g(x) as close as we want to L M by making x sufficiently close to a. The proof of this fact relies on a close look at how errors in the length and width of a rectangle affect its area. See the figure below.

Missing graphic

Suppose that we want to build a rectangle whose length is L and whose height is M. Thus, its area should be L M. This rectangle is shown in yellow in the figure above. If we make an error measuring the length and another error measuring the height then the area of the rectangle will be off. The error is shown in magenta in the lefthand side of the figure above. We can break this error up into three pieces as shown on the righthand side of the figure above. The red piece is the product of the error in measuring L multiplied by the height M. The blue piece is the error in measuring M multiplied by the width L. The small magenta piece is the product of the error in measuring L and the error in measuring M. Because we can control the size of these three pieces by controlling the size of the errors in measuring L and M we can control the total error.

Check Your Understanding

In manufacturing there is often a trade-off between precision and cost. Suppose that you want to make a device that will fill a bottle with one liter of water with a tolerance of 1 centimeter (that is, 0.01 liter). The device has two parts -- a pump and a timer and the precision of the whole device will depend on the precision of the two parts. The pump is supposed to pump 0.25 liters per minute. The cost for manufacturing the pump to several different tolerances is given below.
```
 Tolerance     Cost

0.010 lpm     $2.00
0.005 lpm     $5.00
0.002 lpm     $9.00
0.001 lpm    $20.00
```
The timer is supposed to turn the pump on for a period of 4 minutes. The cost for manufacturing the timer to several differrent tolerances is given below.
```
 Tolerance     Cost

0.10 min      $4.00
0.05 min     $10.00
0.02 min     $30.00
```
How much will it cost to manufacture this device?
answer
We can build many complicated functions starting with two simple functions, or building blocks,
```
A(x) = x
B(x) = 1
```
and the operations -- addition and multiplication -- for example, the function
```
h(x) = x^2
```
is just the product of A(x) and A(x).
It is easy to see that for the two basic building blocks
```
  Lim   A(x) = A(a)
x --> a

  Lim   B(x) = B(a)
x --> a
```
In the this module and the two preceding modules --
- Estimation and Limits -- Multiplying by a Constant and
- Estimation and Limits -- Adding Functions
we have discussed the following three facts

Use the facts discussed here to show that for any polynomial

Hint: First show that for the function h(x) = x^2
```
  Lim   h(x) = h(a)
x --> a
```
and then that for a power function like q(x) = x^n
```
  Lim   q(x) = q(a)
x --> a
```
answer