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MAT122 :: Lecture Note :: Week 08
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Quote of the Week
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Leave the beaten track occasionally and dive into the woods.
You will be certain to find something you have never seen before.
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Next two BABs were covered last week; re-visit comparing percent changes.
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An exponential function is a function that has the input variable used as an exponent.
exp(x) = e^x where 'e' is the mathematical constant 2.718281828459045... * http://antwrp.gsfc.nasa.gov/htmltest/gifcity/e.2milExponential functions are also defined as follows.
f(x) = ab^x where 'b' is the base and 'a' is a constant * 'a' is the initial value (value of f(x) when x = 0) * 'b' is the growth (b > 1) or decay factor (0 < b < 1)Exponential functions provide mathematical models for exponential growth and exponential decay.
Exponential functions have outputs that have a constant percent change (either growth or decay) given equally spaced input values.
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Growth Factors
A growth factor is a number that is multiplied with an "original" number to give a "new" number that is some percentage more than the "original" number.
Example: If the price of a
$5item is increased by a factor of1.2, then the new price is$6.$5 * 1.2 = $6Growth factors are used in lieu of percent increases.
Example: If the price of a
$5item is increased by20%, then the new price is$6.$5 is the "original" number (amount or value) $5 + (20% of $5) $5 + (0.2 * $5) $5 + $1 = $6 $6 is the "new" number (amount or value)In general, the growth factor for a
n%increase is calculated as follows.1) add n% to 100% ... n% is the percent increase ... 100% represents the original number 2) convert (100% + n%) to a decimal number ... (100% + n%) / 100Example: The growth factor for a
55%increase is calculated as follows.In this example, n% is 55% 1) 100% + 55% = 155% 2) 155% = 155 / 100 = 1.55 the growth factor for a 55% increase is 1.55 specific cases... 10 increased 55% is 10 * 1.55 = 15.5 ...or... 10 increased 55% is 10 + (55% of 10) (0.55 * 10) 10 + 5.5 = 15.5 33 increased 55% is 33 * 1.55 = 51.15 ...or... 33 increased 55% is 33 + (55% of 33) (0.55 * 33) 33 + 18.15 = 51.15 1000 increased 55% is 1000 * 1.55 = 1550.0 ...or... 1000 increased 55% is 1000 + (55% of 1000) (0.55 * 1000) 1000 + 550 = 1550Growth factors are always greater than one.
Decay Factors
A decay factor is a number that is multiplied with an "original" number to give a "new" number that is some percentage less than the "original" number.
Example: If the price of a
$5item is decreased by a factor of0.8, then the new price is$4.$5 * 0.8 = $4Decay factors are used in lieu of percent decreases.
Example: If the price of a
$5item is increased by20%, then the new price is$4.$5 is the "original" number (amount or value) $5 - (20% of $5) $5 - (0.2 * $5) $5 - $1 = $4 $4 is the "new" number (amount or value)In general, the decay factor for a
n%decrease is calculated as follows.1) subtract n% from 100% ... n% is the percent decrease ... 100% represents the original number 2) convert (100% - n%) to a decimal number ... (100% - n%) / 100Example: The decay factor for a
55%decrease is calculated as follows.In this example, n% is 55% 1) 100% - 55% = 45% 2) 45% = 45 / 100 = 0.45 the decay factor for a 55% decrease is 0.45 specific cases... 10 decreased 55% is 10 * 0.45 = 4.5 ...or... 10 decreased 55% is 10 - (55% of 10) (0.55 * 10) 10 - 5.5 = 4.5 33 decreased 55% is 33 * 0.45 = 14.85 ...or... 33 decreased 55% is 33 - (55% of 33) (0.55 * 33) 33 - 18.15 = 15.85 1000 decreased 55% is 1000 * 0.45 = 450 ...or... 1000 decreased 55% is 1000 - (55% of 1000) (0.55 * 1000) 1000 - 550 = 450Decay factors are always less than one and greater than or equal to zero. A decay factor of zero implies a 100% decrease.
From the Textbook
Complete the table found on page A-29, exercise #1.
percent increase: 5% __B__ 15% __D__ 100% 300% __G__ growth factor: __A__ 1.35 __C__ 1.045 __E__ __F__ 11.00 A: 100% + 5% = 105%; 105 / 100 = 1.05 B: 1.35 * 100 = 135; 135% - 100% = 35% C: 100% + 15% = 115%; 115 / 100 = 1.15 D: 1.045 * 100 = 104.5; 104.5 - 100 = 4.5% E: 100% + 100% = 200%; 200 / 100 = 2.0 F: 100% + 300% = 400%; 400 / 100 = 4.0 G: 11.0 * 100 = 1100; 1100 - 100 = 1000%Page A-29, exercise #6.
value of a home in 1967: $26,000 value of a home in 2005: $152,030 a) Find the inflation growth rate from 1967 to 2005.152030 / 26000 = 5.85 [units not important]b) Find the inflation rate (percent increase) from 1967 to 2005.5.85 * 100 = 585; 585 - 100 = 485% [units not important] ...or using the % increase formula... (152030 - 26000) / 26000 = 4.85; 4.85 * 100 = 485%c) In 2005, the home sells for $245,000. Find the profit in terms of 2005 dollars.$245000 - $152030 = $92,970 [units are important]When GDT did this exercise on the board, he first calculated the percent increase and then erroneously converted the percentage into the growth factor. GDT's error? He subtracted 100 from the percentage instead of adding 100. GDT discovered his answer was wrong when he multiplied the growth factor by the 1967 value and did not get the 2005 value.
$26000 * 5.8 = $152100.00 Note the effects of rounding: 152,100 ≠ 152,030 ______ 152030 / 26000 = 5.8473076923 was rounded to 5.85
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