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Linear functions, f(x) = m*x + b, where m and b are constants, have a constant growth rate (equal to the slope m). Simply put, each time x increases by 1, y or f(x) will increase by b. Equivalently, the change in y will always be equal to m times the change in x, so y grows mtimes as fast as x. The growth rates of many quantities are better measured using relative growth rates, as opposed to these absolute growth rates. For example, if the price of a new car in 1968 was $4000, and it increased to $4500 by 1969, the absolute change is 4500 - 4000 = 500 dollars, which could be expressed as the absolute rate 500 dollars per year. At the same time, suppose the average 3 bedroom house increased in price from $32000 in 1968 to $32500 in 1969. This gives the same absolute growth rate as for the cars, namely 500 dollars per year, but that 500 dollars is a much more significant increase when compared to the price of a car than it is when compared to the price of a home. For the car, we're talking about = 0.125, or an increase of 12.5%, while in relation to the cost of a home, we have = 0.015625, or roughly 1.56%. A percentage increaseis a relative growth rate. These can be computed by calculating ratios, usually one uses

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For the cars, we have = 1.125, which represents a 12.5% increase, and for the homes we have = 1.015625, which represents a 1.5625% increase. For the cars, this calculation can be rearranged as

i.e., the new price is the old price plus 12.5% of the old price. So multiplying by 1.125 is the same as increasing the price by 12.5%.

1. If the new price is obtained from the old price by multiplying by 1.05, what percentage growth, or relative growth does this represent?

2. If the population of Mexico was 74.66 million people in 1984 and 76.60 million people in 1985, what was the yearly percentage or relative growth?

3. If Mexico continued to grow at this same relative rate, what was the population (in millions) in 1986?

4. If a new population is obtained from an old population by multiplying the old one by 0.97, should we consider this a 97% growth rate, or a 3% decay (or decline) rate?

Here's a handy rule: If you obtain a new value from an old one by multiplying the old value by b, this can be viewed as a 100*(b-1) percent growth. We usually call it decay when b - 1 is negative. For example, here is a table with several possible "multipliers" and the corresponding growth rates.

A function which grows at a constant relative rate will satisfy an exponential formula of the form f(x) = k * ( b^x ), that is k times b to the x power. The variable x is in the exponent and both k and b will be fixed numbers (which change from problem to problem, but stay fixed throughout a given problem). The purpose of this activity is for you to discover a method for recovering a good estimate of both k and b, from a small number of data points of the form ( x, f(x) ).

5. Take k = 23 and b = 1.67, this gives f(x) = 23 * (1.67 ^x).
a) What are f(0), and k? Will it always be true (when f(x) = k*b^x) that f(0) = k?
 
 

b) Calculate f(3), f(6) and f(9)?
 
 
 

c) Now calculate f(3) / f(0), f(6) / f(3), and f(9) / f(6).
 
 
 

d) What is b^3?
 
 

e) Try f(7.2) / f(4.2) and f(-1.34) / f(-4.34).

6. Now try to show algebraically that f(x+3) / f(x) = b^3, regardless of the value of x.

7. This time, let g(x) be the function obtained by letting k = 3.2 and b = 0.88, so g(x) = 3.2 * (0.88^x). What are g(0) and k? Calculate g(5) / g(1), g(2) / g(-2), g(5.678) / g(1.678), and compare these ratios to b^4.

8. If you knew that h(x) = k*b^x, for some constants k and b, and you also knew h(7), h(8.5), h(10) and h(11.5), what power of b do you think the ratios h(11.5) / h(10), h(10) / h(8.5) and h(8.5) / h(7) would be equal to? Why?

9. Suppose that in the last question, all the ratios were approximately equal to 0.752. What equation of the form b^p = 0.752 would you solve in order to estimate the value of b (what number should p be)?

Recall a rule of exponents which says that . If we pick q = 1/p, this gives a way to solve equations like b^3.2 = 4.56 for b, just raise both sides to the 1/3.2 power. The left side becomes [b^3.2]^(1/3.2) = b^1 = b, and the right side is easy with the calculator, 4.56^(1/3.2) = 1.60667 (roughly).

10. Solve b^1.5 = 0.752 for b.

11. The grand finale! You know that f(x) = k*b^x for some constants k and b. You also know that f(0.2) = 1.864, f(0.5) = 1.572 and that f(0.8) = 1.326. Using ratios, derive an equation like b^(some number) = some ratio, that b should satisfy. Solve this equation for b. Now substitute your numeric value for b into the formula f(x) = k*b^x. Use any of the data points (like when x = 0.2, f(x) = 1.864) in your formula for f(x) to solve for k. Check your answer by calculating your values for f(0.2), f(0.5) and f(0.8).

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