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Timothy Foster, Bryan Johnson, Travis Eckstrom, Andrew Sutter • Indicated by the Function 푦 = 푓−1 푥 , Inversely Related to 푦 = 푓(푥)

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Timothy Foster, Bryan Johnson, Travis Eckstrom, Andrew Sutter • Indicated by the Function 푦 = 푓−1 푥 , Inversely Related to 푦 = 푓(푥) Timothy Foster, Bryan Johnson, Travis Eckstrom, Andrew Sutter • Indicated by the function 푦 = 푓−1 푥 , inversely related to 푦 = 푓(푥). • Domain of the inverse corresponds to the Range of the original function, and vice versa. • Can be shown as a reflection across 푦 = 푥 • Indicated by the Function 푓 푥 = 푎푥 such that 푎 is a positive number. • Used for modeling Exponential Growth/Decay, or Continuous Compound Interest • Inverse of exponential function; 푥 if 푓 푥 = 푎 , 푥 = 푙표푔푎푓(푥) • Logarithm with base 푒 known as “natural logarithm”, represented by 푙푛 The idea that a mathematical inverse to exponential growth and decay existed had been understood even by the earliest of mathematicians. However, it was not until the early 1600s that the principles of logarithms were published by John Napier of Scotland. The availability of Logarithms opened up new frontiers in mathematics, while simplifying old formulas. This advancement took place due to the fact that previously difficult multiplications and divisions could be represented by the addition or subtraction of logarithms due to the law of exponents. • Key term: One-to-one Function- A function in which there are no repeated x values for any given y value. • Given a graph, the best way to determine whether or not a function is one-to-one is to perform the “horizontal line test,” which is similar to the “vertical line test” for determining if a graph represents a function. An example of the horizontal line test is shown on the next slide: 2 y=(x-2) y=[(x-3)3+8]/4 4 4 3 3 2 2 1 1 0 Y 0 0 1 2 3 4 Y 0 1 2 3 4 X X Since the graph intercepts the For each y value, there is only horizontal line at multiple one corresponding x value; the points, the function is not function graphed is one-to- considered one-to-one. one. If 푓(푥) is a one-to-one function such that 푓 푥 = 푦, then the inverse function is defined to be the function such that 푓−1 푦 = 푥. Therefore, 푓−1 푓 푥 = 푥. Conversely, if 푓 푥 is one-to-one such that 푓 푦 = 푥, then 푓−1 푥 = 푓(푦), and thus 푓 푓−1 푥 = 푥. Commonly, the inverse is found 푥 by solving the current equation for x and then interchanging the x’s and y’s. 푦 ! Students commonly confuse inverse notation with exponential notation. Remember: 1 푓−1(푥) ≠ 푓(푥) For instance, if 푓 푥 = 푥2, then: 1 1 푓−1 푥 = 푥 = 푓(푥) 푥2 1 푥 ≠ 푥2 To graph inverse functions, always remember that the graph of 푓−1 is a reflection of the graph of 푓 about the line 푦 = 푥. For instance, if point (A,B) belongs to function 푓, point (B,A) belongs to function 푓−1. f(x) and f-1(x) 8 6 (A,B) Y 4 2 (B,A) 0 0 2 4 6 8 X To find the derivative of the inverse of 푓(푥), let 푔 푥 = 푓(푥). Then, use the following equality: 1 푔′ 푥 = 푓′(푔 푥 ) At present, there are two main scales used for measuring temperature: the Fahrenheit scale and the Kelvin scale (which is the same as the Celsius scale, calibrated so that 0 is absolute zero). Given the temperature in degrees Fahrenheit, the Kelvin equivalent can be expressed with the formula below: 5 퐾 = 퐹 − 32 + 273 9 http://www.lightlabsusa.com/images/P/lab-thermometerjpg.jpeg Find the inverse function of the Fahrenheit to Kelvin conversion formula. Solution: 5 Solve for the 퐾 = 퐹 − 32 + 273 independent variable, 9 in this case F. 5 퐾 − 273 = 퐹 − 32 9 9 퐾 − 273 + 32 = 퐹 = 퐾−1 5 Now the temperature in Fahrenheit is a function of the temperature in Kelvins. Finding the inverse allows us to now determine the temperature in Fahrenheit if given Kelvins instead. The periodic table to the right displays the melting and boiling points of Sulfur. What are these temperatures in Fahrenheit? Solution: Since we are given Kelvins, we need to use the 퐾−1 formula. 9 퐾−1 = 퐹 = 퐾 − 273 + 32 5 9 9 퐹 = 388.36 − 273 + 32 퐹 = 717.8 − 273 + 32 5 5 퐹 = 239.65 퐹 = 832.6 The melting point of sulfur is 239.65oF, and the boiling point is 832.6oF. The basic exponential form is 푓 푥 = 푎푥 in which 푎 is a constant. It is useful to note the following (푎 > 1): • When 푥 < 0, 푓 푥 < 1 • When 푥 = 0, 푓 푥 = 1 • When 푥 > 0, 푓 푥 > 1 • 푓 푥 > 0 for all 푥 http://www.proteanservices.com/wp-content/uploads/2010/08/bacteriaCulture.jpg Occurs when 푎 > 1 푓 푥 = 2푥 Domain: −∞, ∞ 4.5 4 Range: 0, ∞ 3.5 푥 lim 푎 = ∞ 3 푥→∞ 2.5 푥 lim 푎 = 0 푥→−∞ 2 1.5 1 0.5 0 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Occurs when 푎 = 1 푓 푥 = 1푥 1.2 1 0.8 Domain: −∞, ∞ Range: {1} 0.6 0.4 0.2 0 Occurs when 0 < 푎 < 1 1 푥 푓 푥 = 2 Domain: −∞, ∞ 4.5 4 Range: 0, ∞ 3.5 푥 lim 푎 = 0 3 푥→∞ 2.5 푥 lim 푎 = ∞ 푥→−∞ 2 1.5 1 0.5 0 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 What is 푒? • 푒 is an irrational number that is approximately equal to 2.71828182845904523536 • It can be more accurately defined as 푛 1 푒 = lim 1 + 푛→∞ 푛 e has been found in the growth patterns of nature and has many applications in mathematics. Let’s apply 푒 to the concept of exponential growth: 푓 푥 = 푒푥 This graph is special because 푒 is the only value for 푎 in which the slope at 푥 = 0 is 1. Proof: 푓 0 + ℎ − 푓 0 푓′ 0 = lim ℎ→0 ℎ 푒ℎ − 푒0 = lim ℎ→0 ℎ 푒ℎ − 1 = lim = 1 ℎ→0 ℎ Derivative of 푓 푥 = 푒푥 using the Definition of the Derivative: 푓 푥 + ℎ − 푓 푥 푓′ 푥 = lim ℎ→0 ℎ 푒푥+ℎ − 푒푥 푒푥 ∙ 푒ℎ − 푒푥 = lim = lim ℎ→0 ℎ ℎ→0 ℎ 푒푥 푒ℎ − 1 = lim ℎ→0 ℎ 푒ℎ − 1 = 푒푥 lim = 푒푥 ∙ 1 = 푒푥 ℎ→0 ℎ What does this tell us? That 푒 also has special properties regarding its differentiation. 푒푥 ′ = 푒푥 Or, more formally using the Chain Rule: 푑 푒푢(푥) = 푒푢(푥) ∙ 푢′ 푑푥 When money is invested in the bank, it is compounded using the following formula: 푟 푛푡 퐴 = 푃 1 + 푛 Here, P is the principal amount, r is the interest rate, and n is the number of times compounded per year. If the money is compounded continuously (n approaches infinity), then the following is used instead: 퐴 = 푃푒푟푡 푟 푛푡 lim 푃 1 + 푛→∞ 푛 푟 푛푡 = lim ln 푃 1 + 푛→∞ 푛 푟 = lim ln 푃 + 푛푡ln 1 + 푛→∞ 푛 푟 ln 1 + 푛 1 = ln 푃 + lim 푛→∞ 푛푡 푟 푛2 푟 1+ 푛 (L’Hospital) -> = ln 푃 + lim − 1 푛→∞ − 푛2푡 푟푡 ln 푃 +푟푡 푟푡 = ln 푃 + lim 푟 = ln 푃 + 푟푡 → 푒 = 푃푒 푛→∞ 1 + 푛 You invest $5000 in a bank which offers a 0.5% fixed interest rate compounded monthly. How much more money will you make if you decide to wait 20 years rather than 10? Here… 푃 = 5000 푟 = 0.005 푛 = 12 푡 = {10, 20} http://i.istockimg.com/file_thumbview_approve/3294158/2/stock-photo-3294158-chart-of-coin-s-stacks.jpg Plugging in the values and using a calculator to estimate, we get: 12∙{10, 20} .005 퐴 = 5000 1 + 12 After 10 years, the amount of money in the bank is $5256. After 20 years, though, the amount is $5526. In 10 years, you make $256, but in twice the time, you make more than twice the profit. You research a different bank and find that it offers 0.55% fixed interest also compounded monthly. How long would it take for your $5000 to double at this bank? Now… 푃 = 5000 푟 = 0.0055 푛 = 12 푡 = ? 퐴 = 10000 Plugging in these values we get 12∙푡 .0055 10000 = 5000 1 + 12 Now solve for 푡 12푡 .0055 2 = 1 + 12 ln 2 = 12(푡) ln 1.0004583 ln 2 푡 = = 126.06 12 ln 1.0004583 Conclusion: At a fixed interest rate of .55% compounded monthly it would take 126 years and 24 days for your deposit to double. This is true for any starting amount at this interest rate. The Logarithm is simply the name of the inverse operation of exponentiation. That is… 푥 log푏 푏 = 푥 Whereas the exponential functions asks how many times to multiply a certain base, logarithms ask how many times was a base multiplied to get the result. The natural log, ln(x), is of the base e. Logarithms can be used to reverse exponentiation or make what are otherwise exponential models seems linear. Logarithms have very important and useful properties. If it were not for the following relationships, there would have been no need to adopt logarithms in the first place. 푥 푏 = 푦 → log푏 푦 = 푥 푚 log 푚푛 = log 푚 + log 푛 log = log 푚 − log 푛 푏 푏 푏 푏 푛 푏 푏 푥 ln 푥 log푏 푎 = 푥 log푏 푎 log 푥 = 푏 ln 푏 푥 log 푥 log푏 푏 = 푥 푏 푏 = 푥 The relative strength of earthquakes is measured by a logarithmic scale known as the Richter Scale.
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