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Exponential models

This article explores exponential models, focusing on their standard forms and key components. The first form is represented as ( y = a b^x ), where ( a ) signifies the initial value (y-intercept) and ( b ) the growth factor. The second form is ( y = a e^{rx} ), with ( r ) indicating the per-capita rate, also known as the exponential growth rate. We will also discuss the conversion between the two forms, with ( b = e^r ) and ( r = ln(b) ), providing a comprehensive understanding of exponential growth.

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Exponential models

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  1. Exponential models

  2. 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)
  3. 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)
  4. Bacteria A colony of bacteria weighs 1mg at 12:00 and 3mg at 2:00. How much will it weigh at 6:00? t=# of hours since 12:00 n=# of mg of bacteria (t,n) points are (0,1) and (2,3) Bacteria make more bacteria, so this is an exponential model. n=abt When t=0, Initial value (n) is 1. n=(1)bt When t=2, n=3. 3=1b(2) b=√3 n=(1)(√3) t At 6 hours, I have n=(1)(√3)6 27mg of bacteria
  5. 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)
  6. What is a percent rate? Imagine that you have an investment earning 3% per year. Per-cent means “for each one hundred.” 3% per year means “for each group of 100 dollars you have in your account, you earn 3 dollars per year.” An investment of $100 earns 3 dollars per year. An investment of $200 earns 6 dollars per year, An investment of $300 earns 9 dollars per year, etc.
  7. What is compounding? Now imagine that you have $100 in your account. After a year, you have more than $103 dollars! Why is that? Well those $3 per year weren’t paid at the end of the year, they were paid in installments. Each time you were paid money, your had more money in your account, so your rate went up 3% of 100.25 is more than 3% of 100. So even after a month, when you have earned a quarter you are now earning money faster This is because not just your $100 are earning money, but also the children of your $100 (that 25¢)are earning money. And the children of the 25¢ (the grandchildren of your $100 will also earn money for you.)
  8. What is APR? Because of compounding, the percent earned on a 3%/year investment after a year is actually more than 3% of what you started with. The percent you actually earn after a year is called your “annual percent yield” (APY) and it is higher than your “annual percentage rate” (APR) Credit cards always tell you APR because it is always the smaller number. It’s not what you really pay (APY).
  9. What is continuous compounding? Now imagine that you have $100 in your account. After a year, you have more than $103 dollars! Why is that? Well those $3 per year weren’t paid at the end of the year, they were paid in installments. However, in magical happyland (not the real world), you are not paid in installments, you are paid all the time. Imagine walking 3 miles per hour. You don’t just wait an hour and then teleport three miles, you’re gaining distance all the time, as you’re walking. Continuous compounding is the same idea: You are gaining money all the time, as you are waiting. This means that every little tiny bit of money you have is earning 3% per year all the time. Every day, every hour, every minute you earn new money and calculate your 3% again.
  10. Compound interest formula
  11. Compounding continuously We don’t know how to do this…. Yet.
  12. Compounding continuously Using a trick from calculus
  13. Transitioning to per-capita rate Imagine that you have an investment earning 3% per year. Per-cent means “for each one hundred.” 3% per year means “for each group of 100 dollars you have in your account, you earn 3 dollars per year.” But the number that actually goes in the equation is 3/100 When compounding continuously, the math is easier if we talk about groups of 1 instead of groups of 100.
  14. Transitioning to per-capita rate Imagine that you have an investment earning 3% per year. 3% per year means “for each group of 100 dollars you have in your account, you earn 3 dollars per year.” 3% also means “for each dollar I have, I earn 0.03 dollars per year.” 1 dollar earns 3 cents per year 10 dollars earns 30 cents per year 100 dollars earns 3 dollars per year
  15. What is a per-capita rate? Imagine that you have an investment earning 3% per year. 3% also means “for each dollar I have, I earn 0.03 dollars per year.” The per-capita rate is 0.03 dollars per year per dollar. Sometimes people cancel the dollars and write this as 0.03/year When compounding continuously, this is the number that goes in the r part of y=aerx
  16. Bacteria A colony of bacteria has been measured as growing at an exponential rate of 0.033 bacteria per bacterium per hour. At 6 hours, there were 4.266 mg of bacteria. How much did the colony weigh at 0 hours? t=# of hours n=number of mg of bacteria Exponential rate means use n=aert Looking for the initial value… find a. 4.266=ae0.033(6) a≈3.5
  17. General Strategy Define your variables (x,y) Write down your equation y=abx or y=aert Fill in the parts you know Values of a, b, or r Values of x and y at specific times Solve for the parts you don’t know A missing value of a, b, or r. Value of x or y at a different time.
  18. Suppose that at the start of the century, there are 25 teachers in a certain small town. Assuming the number of teachers grows exponentially with an exponential growth rate of .005/yr, what is the exponential growth model for this situation? P(t)=.005e25t P(t)=e25t P(t)=25e.005t P(t)=25et None of the above
  19. Suppose that at the start of the century, there are 25 teachers in a certain small town. Assuming the number of teachers grows exponentially with an exponential growth rate of .005/yr, what is the exponential growth model for this situation? t=# of years P(t)=# of teachers at time t P(t)=aert a=initial number of teachers =25. r=per-capita rate = 0.005 P(t)=25e0.005t C
  20. Doubling time problems A colony of bacteria has been measured as growing at an exponential rate of 0.033 bacteria per bacterium per hour. How long will it take my colony to double in size? t=# of hours n=number of mg of bacteria Exponential rate means use n=aert Looking for a time…. Find t. I started with a, when my colony doubled, I had 2a. 2a=ae0.033t 2=e0.033t Ln(2)=0.033t t=ln(2)/0.033 or approx 21 hours.

  21. How long will take an initial investment of $P to double in value under continuous compounding if the interest rate is 6%?

    a) About 4 years b) About 9 years c) About 12 years d) About 25 years e) About 42 years

  22. How long will take an initial investment of $P to double in value under continuous compounding if the interest rate is 6%?

    t=number of years A=amount after t years Continuous compounding, used A=Pert “How long” means find t. I started with P, when I doubled I had 2P. A=2P. 2P=Pe0.06t 2=e0.06t ln(2)=0.06t t=ln(2)/0.06 approx 11.6 years C about 12 years.
  23. Radioactive decay Carbon-14 has a half life of about 5750 years. What is the corresponding exponential rate? t=# of years m=# of grams of carbon-14 m=aert Find r. After 5750 years, I have half as much carbon-14. I started with a, so I have 0.5a left. 0.5a=aer5750 0.5=er5750 ln(0.5)=5750r r=ln(0.5)/5750 or approx -0.00012 atoms per year per atom. Bonus: the equation for carbon-14 decay is m=ae-0.00012t
  24. The half life of Cesium-137 is 30 years.What is the exponential decay rate, r, of this element? r= -.0552 r= -.0231 r= -.0431 r= -.0131 None of the above
  25. The half life of Cesium-137 is 30 years.What is the exponential decay rate, r, of this element? t=#of years m=#of grams of Cesium-137 m=aert Find r. I started with a. after 30 years (t=30), I had half as much (0.5a) 0.5a=aer30 0.5=er30 ln(0.5)=30r r=ln(0.5)/30 or approx -0.0231 atoms per year per atom B
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