Estimation and Limits -- A More Precise Look

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.

• The way in which the flow of fresh water cleans up a polluted lake.

• The way in which the area of a region like the yellow region in the figure below can be estimated using rectangles.

  

• The way in which the slope of a tangent to a curve can be estimated by looking at secant lines.

  

• The way in which velocity can be estimated from position measurements.

In each case notation like

  Lim     p(n) = L
n --> oo

or

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

can be thought of in several different ways.

• Descriptively. For example, when we look at a polluted lake and write

    Lim     p(n) = 0
  n --> oo
  

  we are saying that after many years the level of pollution will approach zero.

• As describing a mechanism for making estimates. For example, when we are looking at the area of a region and write

  

  we are describing a mechanism for estimating the area of a region.

• More precisely. In any practical situation a general description like

  "After many years the level of pollution will be close to zero."

  or

  "With many rectangles the estimate will be very close to the actual area."

  may be close to useless. If a polluted lake won't be cleaned up enough for swimming for 12 million years then we will hardly be able to appreciate its eventual unpolluted state. Similarly, if an estimate that is within 0.001 of the exact area of a region requires 30 billion rectangles then this mechanism for estimating area is of little practical use.

• Suppose that a lake currently has 10 ppm of a toxic chemical and that each year the level of pollution drops by 20%. This situation can be described by the dynamical system

  p(1) = 100
  p(n + 1) = 0.80 p(n)
  

  If the maximum permissible level of pollution in a lake that is used for swimming is 2 ppm when will the lake be safe for swimming?

  If the maximum permissible level of pollution in a lake that is used as a source of drinking water is 0.05 ppm when will it be safe to use this lake as a source of drinking water?

• Consider the function

  f(x) = x^2

  shown in the graph below.

  

  We are interested in the slope of the tangent to this curve at the point (0.5, 0.25). This tangent is shown in black in the figure. We can estimate the slope of this tangent by choosing another point on the curve -- for example, the point (0.7, 0.49) marked by a red dot on the figure and computing the slope of the line through the two points (0.5, 0.25) and (0.7, 0.49).

  

  This line is shown in red in the figure. Notice that this line is steeper than the tangent, so this is an overestimate for the slope of the tangent. You can see from the figure that if we use any point to the right of the point (0.5, 0.25) as our second point then the resulting estimate will be an overestimate. If we use a point to the left of the point (0.5, 0.25) as our second point -- for example, the point (0.3, 0.09) -- then we will get an underestimate.

  

  The line through the two points (0.5, 0.25) and (0.3, 0.09) is shown in blue in the figure.

  Putting this all together we know that the slope of the tangent is between 0.8 and 1.2.

  We have a general procedure for estimating the slope of the tangent to the curve f(x) = x^2 at the point (0.5, 0.25) by computing the slope of the line through the two points (0.5, 0.25) and (x, f(x)). We can write this

  

  where F(x) denotes the estimate obtained using the two points (0.5, 0.25) and (x, f(x)) and f(x) denotes the function f(x) = x^2.

  Thus,

  Slope of tangent =  Lim     F(x).
                    x --> 0.25
  

  Find an estimate for the slope of the tangent that is within 0.1 of the exact answer.

  Find an estimate for the slope of the tangent that is within 0.05 of the exact answer.

  Find an estimate for the slope of the tangent that is within 0.01 of the exact answer.

  Find an estimate for the slope of the tangent that is within 0.001 of the exact answer.

The point of all this is that whenever we have a limit we have a way of getting estimates that enables us to obtain estimates that are as close to the exact answer as might be required for any practical purpose. A more precise definition of the idea of a limit, written

emphasizes this ability to obtain arbitrarily good estimates.

We want to consider the following elements of this whole process.

• We have a process, often written F(x), that gives us a family of estumates. Think of F(x), for example, as the estimate for the slope of the tangent

  

• There is an "ideal" value a that we would like x to have. In our example, the tangent is the line through one point which means that the second point should be right on top of the first point -- that is, we would like x to have the "ideal" value 0.5. Usually the ideal value is really impossible. In our example, if x = 0.5 then the procedure tells us to divide zero by zero which is, of course, impossible.

• In most practical situations we do not need an exact answer but we do need an answer that is within some allowable error of the exact answer. The Greek letter epsilon is traditionally used to denote an allowable error. Depending on the practical problem epsilon might be fairly large or it might be very, very tiny.

• Whatever the allowable error there is some tolerance such that as long as the value of x is within the tolerance of the ideal value a then the estimate is within the allowable error. The Greek letter delta is traditionally used to denote a tolerance. The value of F(a) has nothing at all to do with the limit. In fact, usually we cannot even compute F(a).

Putting all this together we write the more precise definition

means

The module Estimation and Limits -- A Graphic Approach describes a nice way of visualizing this definition.

The same ideas are involved when we estimate the area of a region like the yellow region in the figure below using rectangles.

In this case we can obtain an estimate that is within some allowable error by using enough rectangles. If we denote the estimate obtained using n rectangles by E(n) then we say

  Lim   E(n)   = Area
n --> oo

In a situation like this our more precise definition of a limit becomes

means

The same mathematics captures the essential points in our polluted lake. Recall that the initial level of pollution was 10 ppm and the level of pollution dropped by 20% each year. We were interested in when the level of pollution would drop below 2 ppm so that the lake would be safe for swimming. Making some computations, we see that

p(1) = 10.00
p(2) =  8.00
p(3) =  6.40
p(4) =  5.12
p(5) =  4.10

p(6) =  3.28
p(7) =  2.62
p(8) =  2.10
p(9) =  1.68

and since the level continues to decrease -- for all n >= 9, |p(n) - 0| <= 2.0 -- thus, if the lake will be safe for swimmming when the level of pollution is below 2 ppm then it will be safe to swim in nine years. For example, if p(1) is the level of pollution in 1995 then in the year 2003 it will be safe to swim and it will remain safe after that barring some unforeseen problems like another toxic waste dumping incident or a prolonged drought. A prolonged drought can raise the level of pollution since only the water evaporates leaving a higher concentration of the pollutant.

Check Your Understanding

1. Explain how to estimate the slope of the tangent to the curve

   y = x^2

   at the point (0.8, 0.64). Using your method find an estimate that is within 0.01 of the exact answer.

   answer

2. Explain how to estimate the slope of the tangent to the curve

   y = x(1 - x)

   at the point (0.8, 0.16). Using your method find an estimate that is within 0.01 of the exact answer.

   answer

3. Explain how to estimate the slope of the tangent to the curve

   y = x^2

   at the point (a, a^2). Find the exact answer by looking at the algebra.

   answer

4. Explain how to estimate the area under the curve

   y = cos x

   between x = 0 and x = pi/2. Using your method find an estimate that is within 0.1 of the exact answer. Find an estimate that is within 0.01 of the exact answer.

   answer

5. The Fly-by-Night Toxic Waste Disposal Company has been caught dumping toxic waste in Long Lake. Scientists working at the state university have built the following model for the level of pollution in Long Lake.

   p(1) = 20 ppm
   p(n) = 0.9 p(n - 1)
   

   The state, along with one nearby city that draws its drinking water from Long Lake and several environmental groups have sued the Fly-by-Night Toxic Waste Disposal Company and the court has agreed to fine the company $100,000 for each year that the water in the lake is unfit to be used a source of drinking water. Several different "experts" have testified the the following:

   Prof. Forhire     "The safe level is 12 ppm."
   Prof. Smith       "The safe level is  3 ppm."
   Prof. Jones       "The safe level is  2 ppm."
   Dr.   Babydoctor  "The safe level is  1 ppm."
   

   Make a table showing when the lake can safely be used as a source of drinking water and the fine the company should pay based on the different experts' testimony.

   answer

Stretch Your Understanding