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.
Calculus is sometimes described as the mathematics of change. The sequences and discrete dynamical systems discussed in this chapter involve one kind of change -- in situations like the population of temperate zone insects or the prices of farm products in which "generations" are distinct and the past and future history can be described by a sequence of numbers.
p(1), p(2), ... p(n), ...
We describe these situations with two pieces of information --
p(1) = ____ or
p(0) = ____
p(1995) = _____
p(1995) = _____
p(n) = f(p(n - 1))
or in the form
p(n + 1) = f(p(n))
One reason that mathematics is so powerful is that it allows us to express the same idea in many different ways. Different ways of expressing the same idea often highlight different aspects -- something that at first appears to be very mysterious may suddenly seem clear when looked at from another perspective.
This is particularly true for equations that describe change. In this section we look at three different ways of describing change.
p(n + 1) = f(p(n))
that focuses on the way each generation is determined by the preceding generation.
p(n + 1) - p(n) = g(p(n))
that focuses on the difference between each generation and the preceding generation.
that focuses on the ratio between each generation and the preceding generation.
For example, suppose that we are interested in a country whose 1995 population was 12,345,678 and whose population was rising by 0.2% per year. The snapshot condition can be expressed by
p(1995) = 12,345,678
and the change can be expressed by
p(n + 1) = 1.002 p(n)
or
p(n + 1) - p(n) = 0.002 p(n)
or
p(n + 1)/p(n) = 0.002
Notice that for this particular model the ratio equation
p(n + 1)/p(n) = 0.002
is the most illuminating since the right hand side of this equation is a constant and thus we see immediately that this is an exponential model.
If we were looking at a country whose primary source of population growth was immigration -- at the rate of 30,000 people per year -- then its population change might be expressed in one of the following three ways.
p(n + 1) = p(n) + 30,000
or
p(n + 1) - p(n) = 30,000
or
p(n + 1)/p(n) = 1 + 30,000/p(n)
Notice that in this example the difference equation
p(n + 1) - p(n) = 30,000
is the most illuminating because we see immediately that the difference from one year to the next is constant.
These different ways of modeling change suggest different ways of analyzing data. If you suspect, for example, that data might best be described by an exponential model then it is worthwhile to look at the ratios p(n + 1)/p(n) to see if these ratios are nearly constant. On the other hand, if you expect that the change from one generation to the next is nearly constant then it is worthwhile to look at the differences p(n +1) - p(n).
Express each of the following change equations as a difference equation and as a ratio equation.
Analyze each of the following data sets.
n p(n)
1 12000
2 13097
3 14115
4 15199
5 16202
6 17242
7 18292
8 19364
9 20431
10 21498
11 22500
12 23573
13 24614
14 25633
15 26671
16 27743
17 28805
18 29849
19 30878
20 31880
answer
n p(n)
1 12000
2 13455
3 15124
4 17134
5 19455
6 21839
7 24531
8 27786
9 31246
10 35140
11 39682
12 44805
13 50365
14 56934
15 64849
16 72636
17 81367
18 91684
19 103579
20 118072
answer
n p(n)
1 120.00
2 118.54
3 117.06
4 115.68
5 114.28
6 112.89
7 111.58
8 110.31
9 109.05
10 107.82
11 106.65
12 105.54
13 104.48
14 103.40
15 102.38
16 101.42
17 100.45
18 99.49
19 98.57
20 97.69
In this section we take our first look at a general application that will recur several times throughout this course -- the way in which an object, like a hot cup of coffee placed in a room at room temperature, changes temperature. Click on the button below.
