Estimation and Limits -- Multiplication by a Constant

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 first of a series of modules that deal with various ways in which functions can be changed. In this module we look at what happens when a function is multiplied by a constant as shown in the figure below.

Missing graphic

Multiplying a function by a constant has many practical uses. For example, if a function f(t) represents a sound recording then the function 2 f(t) effectively amplifies the sound by a factor of 2. This is one of the principles behind digital processing of audio and video signals.

Use your CAS window to compare the following functions:

f(x) = x^2
g(x) = 2 x^2
h(x) = 0.5 x^2

Use your CAS window to compare the following functions:

f(x) = cos x
g(x) = -2 cos x
h(x) = -0.5 cos

Use your CAS window to compare the following functions:

f(x) = x^3 - x + 1
g(x) = 2 f(x)
h(x) = -2 f(x)

The basic fact we discuss here is, the fairly obvious fact, that if

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

and if k is any constant then

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

We can also express this fact by saying that if f(x) is close to L when x is close to a then k f(x) is close to k L when x is close to a.

However we express it, this descriptive fact is only part of a more interesting story. The 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 sufficently close to a. The conclusion -- the fact that

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

tells us that we can make k f(x) as close as we want to k L by making x sufficiently close to a. The proof of this fact is based on the observation that multiplying f(x) by the constant k not only multiplies the value of f(a) by k but it also multiplies any error by |k| . See the figure below.

Missing graphic

The blue markss the difference between f(x) and L and the magenta marks the difference between k f(x) and k L.

Suppose we are given a permissible error epsilon for k f(x). That is, suppose that we want to be sure that

|k f(x) - k L| < epsilon.

Because any error in f(x) will be multiplied by |k| we need to be sure that

|f(x) - L| < epsilon/|k|

This is easy to do. Since epsilon/|k| is positive and since

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

there is a positive delta such that if

|x - a| < delta

then

|f(x) - L| < epsilon/|k|

and, therefore,

|k f(x) - k L| = |k| |f(x) - L|
               < |k| epsilon/|k| = epsilon

Check Your Understanding

For each of the following problems you are given a function f(x), a point a and a permissible error, epsilon. Find a tolerance that will work for the given permissible error with the function f(x) and then find a tolerance that will work for the same permissible error but for the function 2 f(x).

```
   f(x) = x
      a = 2
epsilon = 0.01
```
answer

   f(x) = x^2
      a = 2
epsilon = 0.01

answer

   f(x) = sin x
      a = 0
epsilon = 0.01

answer

In manufacturing there is often a trade-off between precision and cost. The table below, for example, gives the cost of manufacturing a particular automobile part to a given tolerance -- that is, with a given permissible error.
```
 Tolerance     Cost

0.10 inches   $2.00
0.05 inches   $6.00
0.02 inches  $25.00
0.01 inches  $85.00
```
The ABC Automobile company has two lines of cars -- one line is aimed at the "price conscious" market and the other line is aimed at the "luxury" market. When the company manufactures cars for the luxury market this part must be manufactured to a tolerance of 0.02 inches. When it manufactures cars for the price conscious market this part is only manufactured to a tolerance of 0.10 inches. Discuss the implications of this for the consumer.
answer