Estimation and Limits -- A Descriptive 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.
There are many situations where it is difficult or even impossible to find an exact
answer but it is possible to make estimates. In fact, it is often possible to make
very good estimates, estimates that are good enough for any purpose, no matter how
good the estimate must be. If we can make arbitrarily good estimates then that is
sufficient for any practical purpose.
Limits and estimation are two sides of the same coin. Suppose, for
example, that we want to determine the length of a curve described by a function
y = f(x)
on an interval [a, b] as shown in the figure below.
We can estimate the length of this curve by approximating it by another curve made up
of straight line segments as shown in the figure below.
For this particular estimate we used six straight line segments but we can use any
number n of segments. The more segments we use the better our estimate will
be.
To compute an estimate based on n segments we proceed as follows.
- First divide the interval [a, b] into n pieces of equal
length
h = (b - a)/n
Let x(i) denote the point
x(i) = a + i h
as shown in the figure below
- The i-th piece stretches from x(i - 1) to x(i) and
the i-th straight line segment stretches from
(x(i - 1), f(x(i - 1)) to (x(i), f(x(i))). Its length is
See the figure below.
By adding up the lengths of the n segments we get an estimate
for the length of the curve.
Use your CAS window to find estimates of the length of each of the following curves.
Notice there is a trade-off -- time versus the quality of the estimate. Using a
larger number of line segments gives a better estimate but also requires more time
than a smaller number of line segments. Find the best estimates that you can in
a reasonable amount of time.
The curve described in the first problem above is one-fourth of a circle of radius 1.
Since the circumference of a circle is radius 1 is 2 pi the length
of this curve is
2 pi / 2 = pi/2
Notice by using larger values of n -- that is, by using more straight line
segments to approximate the curve -- we get better estimates for the true length
of this curve.
In fact, by using enough segments we can get as good an estimate as might be needed
for any particular purpose. Let E(n) denote the estimate we get using
n segments. Notice that the exact answer is the limit
of these estimates
Lim E(n) = Length of the curve
n --> oo
The modules
Area
and
Tangents
give other examples of similar situations in which the exact answer is the limit
of estimates.
Check Your Understanding
In this set of exercises we look at a simple model of tennis. Suppose that you
are playing tennis against a familiar opponent. The two of you have always
played very consistently. You always make 85% of your shots and your opponent
always makes 80% of her shots. For this simple model we will ignore the differences
between serving and ordinary shots. The drawing below shows what happens on the first
few shots when you serve first.
If you miss the first shot then you lose. If you make the first shot then there are
two possibilities -- either your opponent misses her first shot and then she loses
OR your opponent makes her first shot. If she makes her first shot then there are
two new possibilities -- either you miss your second shot and lose or you make
your second shot. If you make your second shot then there are two new possibilities.
...
The drawing below shows the probability of each of the various possibilities.
- Notice that the probability of you missing your first shot is 0.15 --
resulting in an immediate loss.
- The probability of you making your first shot is 0.85.
If this happens then there are two possibilities.
- Your opponent misses her first shot. This happens with probability
0.20, so the net probability of you making your first shot followed by your
opponent missing her first shot is (0.85) * (0.20) = 0.17. In this case you
win.
- Your opponent makes her first shot. This happens with probability
0.80, so the net probability of you making your first shot followed by
your opponent making her first shot is (0.85) * (0.80) = 0.68. In this
case the point continues.
Looking at the first two shots we see there are three possibilities.
- You lose on your first shot -- this happens with probability
0.15.
- You win on your opponent's first shot -- this happens with probability
0.17
- The point continues -- this happens with probability
0.68.
This gives us our first estimates for the outcome of this point
- You win -- probability 0.17.
- You lose -- probability 0.15.
- Nobody wins -- probability 0.68.
We use the following notation
- W(n) -- the probability that you win considering n shots
by each player. Notice W(1) = 0.17.
- L(n) -- the probability that you lose considering n shots
by each player. Notice L(1) = 0.15.
- T(n) -- the probability of neither side winning after n shots
by each player. Notice T(1) = 0.68.
Notice that
Lim W(n) = Probability that you win the point.
n --> oo
Lim L(n) = Probability that you lose the point.
n --> oo
Lim T(n) = Probability that the point goes on forever.
n --> oo
- Find W(2), L(2), and T(2).
answer
- Find W(3), L(3), and T(3).
answer
- Find good estimates for the probability that you win, the probability that you
lose, and the probability that the point goes on forever.
answer
- Answer the questions above for a point in which your opponent makes the first
shot.
answer
- If you flip a coin to determine who chooses whether to make the first shot and
you win the coin toss should you choose to make the first shot?
answer
- You have been debating whether to play a more aggressive game. If you play
a more aggressive game then your probability of making each shot goes down to
0.70 but your opponent's probability of returning each shot goes down to 0.60.
Discuss the pros and cons of this new strategy. Notice that your new strategy does
not affect the probability of your opponent making her first shot if she makes
the first shot -- it only affects the probability of her making each return.
answer
Stretch Your Understanding
There are two Stretch Your Understanding Modules for this topic.
Copyright c 1995 by
Frank Wattenberg, Department of Mathematics, Carroll College,
Helena, MT 59625