"The only useful function of a statistician is to make predictions, and thus to provide a basis for action" -- (W. Edwards Deming, 1980)
Joseph Louis Lagrange (1736-1813): Born in Italy, Lagrange lived mostly in France and made mathematical contributions in many areas. Among other things, he took a critical look at the mathematical works of Newton, Euler, and Leibniz, who had created the foundations for the amazing branch of mathematics we know as calculus. Lagrange refined the works of these great mathematicians. In doing so, he created the calculus notation f'(x) and f''(x), which we still use today. He also made contributions in the field of astronomy and to the development of the theory of finite groups. Lagrange and other French mathematicians flourished under the influence of Napoleon Bonaparte (1769-1821), who was a strong advocate of mathematics education. One of the interesting theorems proved by Lagrange states that every natural number can be written as the sum of the squares of four integers. For instance,
23 = 32 + 32 + 22 + 12, and
59 = 72 + 32 + 12 + 02.
What excuse did Herkimer give for the difficulty he had when sailing from California to Hawaii?
Answer: He said "I left my chart in San Francisco."
DOUG GRAVES...he had a successful career as an undertaker.
FILLMORE WELLES...he supplied water during periods of drought.
Reading: Section 8.1, pages 465-468.
Written: None, but read items Items for reflection (below).
Note that in the reading, you will find financial problems. Many financial problems involve exponential functions. Look at problem numbers 52 and higher beginning on page 470. Most of these involve finance.
Mathematical word analysis:
TRUNCATE: When you "truncate" a number, such as 72.768, you simply drop the decimal portion. The roots of this word come from the Early French truncus, which referred to a cutting from a tree. One could make a club or staff from a tree cutting. When you truncate a number, you simply "cut off" the decimal portion.
Here is an exponential function that relates to finance:
y = f(x) = $1000(1.09)x
Note that the form of the function is y = abx, where a = $1000 and b = 1.09. This is a function that will produce the accumulation of a deposit of $1,000 at an interest rate of 9% if you input a number of years. For instance
f(10) = $1000(1.09)10 = $2,367.36
That is, a deposit of $1,000 will grow to $2,367.36 after 10 years if the annual interest rate is 9%.
Note that your TI-83 can fit exponential functions to models. Check this out. Here is the path:
STAT -> CALC -> option 10, which is labeled 0.
MATH IS POWER!
Problem: If $1,000 is invested at an annual rate of 7.6% compounded daily, to what amount will this accumulate to in 10 years?
Solution (with communication):
$1,000(1 + .076/365)365(10)
Problem: What is the domain and range of the function
y = 22x-1+ 45
Solution (with communication): The function is defined for any real number one could substitute for x. Hence, the domain is the set of real numbers. Now 22x-1 cannot be 0 or negative for any domain value, so the range is y > 45.
Problem: I have $12,945 now as a result of a single investment I made 13 years ago. If the investment earned an annual rate of 8.2% compounded monthly, what was the amount of the original investment?
Solution (with communication): If x is the original amount invested, then
x(1+.082/12)13(12) = $12,945
==> x = ($12,945)(1 + .082/12)-156
==> x = $4,474 (to the nearest dollar).
Problem: If y= 2(3x) + 4, what is the value of x when y = 22?
Solution (with communication):
y = 22 ==> 22 = 2(3x) + 4
==> 2(3x) = 18
==> 3x = 9
==> x = 2.