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Exponential Functions

Exponential Functions. The “I’m going to lie to you a bit” version. Exponential functions measure steady growth. If you really want to know what that means exactly, take differential equations ( a fter Calculus) Here’s the basic (lying) version

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Exponential Functions

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  1. Exponential Functions The “I’m going to lie to you a bit” version

  2. Exponential functions measure steady growth • If you really want to know what that means exactly, take differential equations (after Calculus) • Here’s the basic (lying) version • An exponential growth happens when something is making more of itself (in a “steady” way) • People, money, bacteria, etc…

  3. Example • One dollar makes one dollar every year. $1 $1 Year 0 Year 1

  4. Example • One dollar makes one dollar every year. $1 $1 $1 Year 0 Year 1

  5. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 Year 0 Year 1 Year 2

  6. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2

  7. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  8. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  9. Example • Every year I keep what I have and add what I have. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  10. Example • Every year I double my money $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  11. My dollars over time

  12. Example • Every year I double my money $1 y=1(2x) y=# of $ x=# of yrs $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  13. Approximation function ƒ(x)=2x

  14. Standard form(s) of an exponential • y=abx • a is the initial value (y-intercept) • b is the growth factor • y=aerx • a is the initial value (y-intercept) • r is the “per-capita rate” also called “exponential growth rate” • Conversion • b=er, r=ln(b)

  15. When to use an exponential model • Things making more things • People making more people, bacteria making more bacteria, money making more money (interest) • Lots of identical events happening at random times • Radioactive decay (atoms decay at random times), Heat transfer (atoms bump into each other randomly)

  16. Asymptotes I have to lie to you until you take Calculus edition

  17. Asymptote • An asymptote is a line that an equation gets close to but never reaches. • Every exponential y=abx has a horizontal asymptote at y=0.

  18. Growth and Decay I’m still lying to you a bit

  19. Simple Version • A growth function is an exponential function where y gets bigger when x gets bigger. • A decay function is an exponential function where y gets smaller when x gets bigger. • The easy way to tell is to graph it!

  20. Growth and Decay Exponential Growth Exponential Decay

  21. Non-calculator examples • y=3(4)x. • When x=0, y=3. When x=1, y=12. • When x gets bigger, y gets bigger. • Exponential Growth • y=2(1/3)x • When x=0, y=2. When x=1, y=2/3. • When x gets bigger, y gets smaller. • Exponential Decay

  22. Non-calculator examples • Y=7(2)-x. • When x=0, y=7. When x=1, y=7(2)-1=7/2. • When x gets bigger, y gets smaller. • Exponential Decay • Y=1(1/2)-x • When x=0, y=1. When x=1, y=(1/2)-1=2. • When x gets bigger, y gets bigger. • Exponential Growth

  23. 2) y=3-x decay. 1) y=3x growth. 4) y=(1/2)-x growth. 3) y=(1/2)x decay. d) 2&3 represent decay

  24. Logarithms Seriously, take calculus, please

  25. Example • Every year I double my money $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  26. Example • If I know what time it is and want to know how much money I have $1 a=1(2t) a=# of $ t=# of yrs $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  27. Example • What if I know money and want to know time? $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  28. Example • How long would it take me to get $1,000,000? $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  29. Flashback! I start 10 miles away from my house and drive away from my house at 30 mph. If I know how far I am from my house, how long have I been driving? d=number of miles away from my house t=number of hours I’ve been driving d=30t+10 (d-10)/30=t

  30. Flashback! I start 10 miles away from my house and drive away from my house at 30 mph. If I know how far I am from my house, how long have I been driving? d=number of miles away from my house t=number of hours I’ve been driving d=30t+10 (d-10)/30=t I need an inverse!

  31. Example • What’s the log27? • Get log(7) in your calculator ≈ 0.845098 • Get log(2) in your calculator ≈ 0.30103 • Log27 = Log(7)/Log(2) ≈ 0.845098/0.30103 • Log27≈2.80735

  32. Example problem • Find the domain of 2log7(4x-3)+7x-9 Whatever is inside the log has to be >0. I can find an answer whenever 4x-3>0 x>3/4

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