Fourier Series

Population of the Earth

Consider a smooth function f(x), and its expansion near x = x

S

and call E

E

(a) Apply this formula to f(x) = cos(x), with x

(b) Find a condition on |x| which ensures that |E

(c) Use a calculator or Matlab to check your answer to the previous question.

[from Math 485/585, Mathematical Modeling, Spring 2005, University of Arizona]

SOLUTION

(a) f(x) = cos x, x

f(x

f'(x) = - sin x => f'(x

f"(x) = - cos x => f"(x

f

f

f

S

= f(0) + f'(0)x + f"(0)x

= f(0) + f"(0)x

= 1 - x

E

(b) The complete Taylor expansion of f(x) = cos x near x = 0 is

cos x = 1 - x

Thus,

E

or

|E

|E

(c) To check the result in (b), we first calculate

cos(1.817121) = - 0.24384

Then we calculate

1 - (1.817121)

This differs from the exact value of cos(1.817121) by - 0.19668 + 0.24384 = 0.047157 < 0.05, so as long as |x| < 1.817121, |E

Show that

p = S

[Hint: Find a Fourier series expansion for the function f(x) = x defined for - p

SOLUTION

A function f(x) defined in the region - p

f(x) = a

where

a

a

b

Now suppose we let f(x) = x. Then we have

x = a

where

a

a

b

Let u = x and dv = sin nx dx. Then du = dx and v = - (1/n) cos nx, and

∫ x sin nx = - (x/n) cos nx + (1/n) ∫ cos nx dx = - (x/n) cos nx + (1/n

=> ∫

=> b

=> x = S

Let x = p/2. Then

p/2 = S

= S

Define m = (n + 1) / 2. Then n = 2m - 1 and

p/2 = S

Show that

S

[Hint: Find a Fourier series expansion for the function f(x) = x

SOLUTION

A function f(x) defined in the region - p

f(x) = a

where

a

a

b

Now suppose we let f(x) = x

x

where

a

a

b

Let u = x

∫ x

Let u = x and dv = sin nx dx. Then du = dx, v = - (1/n) cos nx, and

∫ x sin nx = - (x/n) cos nx + (1/n) ∫ cos nx dx = - (x/n) cos nx + (1/n

=> ∫ x

=> ∫

=> a

=> x

Let x = p. Then

p

=> S

Show that S

[Hint: Find a Fourier series expansion for the function f(x) = x

SOLUTION

A function f(x) defined in the interval - p

f(x) = a

where

a

a

b

Now suppose we let f(x) = x

a

a

b

Let u = x

and

∫ x

Let u = x

∫ x

Let u = x

∫ x

Let u = x and dv = sin nx dx. Then du = dx, v = - (1/n) cos nx, and

∫ x sin nx dx = - (1/n) x cos nx + (1/n) ∫ cos nx dx = - (1/n) x cos nx + (1/n

=> ∫ x

=> ∫ x

=> ∫ x

(24/n

=> ∫

=> a

=> x

Let x = p. Then

p

= p

It can be shown that

S

Thus,

p

=> S

=> S

Using the data in the following table, which shows the estimated population of the Earth at various points in time (Haub 1995, Curtin 2007), estimate the total number of people who have ever lived.

Year | Population |
---|---|

47994 BC^{1} |
2 |

8000 BC |
5,000,000 |

1 |
300,000,000 |

1650 |
500,000,000 |

1850 |
1,265,000,000 |

2002 |
6,215,000,000 |

2007 |
6,500,000,000 |

Curtin, Ciara 2007, "Fact or Fiction? Do Living People Outnumber the Dead?"

Haub, Carl 1995, "How Many People Have Ever Lived on Earth?"

SOLUTION

Let t be the average lifespan of a human being and assume that all human beings live this long and that the population is always evenly distributed in age from 0 to t. Then the fraction of the population which dies each year is 1/t. For example, if t = 70 years, the fraction of the population which dies each year is 1/t = 1/70 = 1.429%.

The above table contains seven data points and represents six time intervals. Let Y

P

where R

R

The relationship between the annual population growth rate R

R

From above, if t = 70 years, D

B

Thus, we can calculate the annual birth rates that are listed in the following table. The annual (birth, death) rate is the number which, when multiplied by the current population at the beginning of a particular year, gives the total number of (births, deaths) that occur during that year.

Data Point (i) | Year (Y_{i}) |
Population (P_{i}) |
Elapsed Years (E_{i}) |
Annual Growth Rate (R_{i}) |
Annual Death Rate (D_{i}) |
Annual Birth Rate (B_{i}) |
---|---|---|---|---|---|---|

1 |
47994 BC |
2 |
39994 |
0.000368 |
0.014286 |
0.014654 |

2 |
8000 BC |
5,000,000 |
8000 |
0.000512 |
0.014286 |
0.014798 |

3 |
1 |
300,000,000 |
1649 |
0.000310 |
0.014286 |
0.014596 |

4 |
1650 |
500,000,000 |
200 |
0.004652 |
0.014286 |
0.018938 |

5 |
1850 |
1,265,000,000 |
152 |
0.010528 |
0.014286 |
0.024814 |

6 |
2002 |
6,215,000,000 |
5 |
0.009008 |
0.014286 |
0.023293 |

7 |
2007 |
6,500,000,000 |
- |
- |
- |
- |

Using the values of B

If this is done for all the years from 47994 BC to 2007, the total number of births is 34 billion. If we assume different values for the average lifespan t, we get the total number of births shown in the following table.

Average Lifespan t |
Total Births |
---|---|

10 |
197,369,926,000 |

20 |
101,964,238,000 |

30 |
70,162,341,800 |

40 |
54,261,393,700 |

50 |
44,720,824,900 |

60 |
38,360,445,600 |

70 |
33,817,317,600 |

If we assume t = 20 years, we get a result which is close to the estimate given by Curtin (2007). The results in the table were generated by the program SciAm070824.f. Back to Main Menu