# Talk:Exponential Growth

### From Math Images

## Contents |

# General Comments

Chris 10:26, 27 June 2012 (EDT): Chengying, this page is definitely accessible for high school students, and it moves nicely from Algebra 1 through Algebra 2 and ventures into Calculus. I like your ideas for integrating animations into this so that the reader can more easily grasp the limit concept.

The focus in my editing is on clarity of writing.

# Basic Description

Chris 10:26, 27 June 2012 (EDT):

Your first sentence could describe any number of functions. What distinguishes an exponential function from other functions?

Since you are describing both growth and decay and your first example 2^x involves growth, it might make sense to have your second example focus on decay.

Table: Consider organizing the six values in ascending or descending order numerically.

Sentence after table: Isn't what determines an exponential function the presence of a variable in the exponent? Are you talking about situations in which the formula isn't given?

-- ALL CHECK

# General Formula

## For the Discrete Case

Chris 10:26, 27 June 2012 (EDT):

- For the Discrete Case: clearly written; I have one minor suggestion:

Switch "interest compounding problem" to "compounded interest problem "

- Exponential Growth: Compound Interest

Sentence #2 (S2): Change "Assume that you are not going to withdraw this $1000 in the middle of any year and the interest rate is kept constant. How much money will you end up with after 1 years?" to "Assuming no other deposits or withdrawals and a constant interest rate, what will be the value of the account after t years?"

The "Each year..." is the beginning of your answer, but it appears to be located in the same paragraph as the problem. I don't think it's necessary.

- Exponential Decay: Elimination of Drug from the Body

S2: Change "Suppose the initial amount of a drug is in the body 200mg" to "Suppose the initial amount of a drug in the body is 200mg"

-- ALL CHECK

Swu2 11:12, 6 July 2012 (EDT)"x" to "t"; Also maybe you wanna say "function value" instead of "function size" in the middle paragraph? --CHECK

## For the Continuous Case

Chris 10:26, 27 June 2012 (EDT):

Change "interest rage compounding problem" to "compounded interest rate problem".

I believe the formula for x(t) should have n*t, not t/n in the exponent. Since n is the number of times the bank compounds interest annually, it gets multiplied by t, not divided into it.

Table: I would go to at least two decimal points since money is calculated in dollars and cents. I would also consider changing the numbers in the "number of times interest is compounded per year" column so that there are much larger numbers than 10 in the column next to e.

e mouseover: Change "approximately equals to 2.71828" to "approximately equals 2.71828."

A strange thing happens when you use the online calculator. We should talk about it directly.

-- ALL CHECK

# Half-Life Decay

Chris 10:26, 27 June 2012 (EDT):

After deriving the formula for half-life time t, would it make sense to write the numerical value for ln(1/2) to the nearest three or so decimal places?

-- CHECK

# Doubling Time and the Rule of 70

Chris 10:26, 27 June 2012 (EDT):

To "If the percentage rate is x%, the percentage number is just x", add "not .01x."

Math Behind section: Instead of "alternatively," say "We can then multiply..."

-- ALL CHECK

# Exponential vs. Polynomial Growth

Chris 10:26, 27 June 2012 (EDT):

Do you want to refer to calculus here to support your statements about rate of change?

-- I added a short paragraph about calculus. Anything else I should expand to??

King's Problem link goes to the Bedsheet Problem.

-- CHECK

# References

# Things I Might Want in the Future

Chris 10:26, 27 June 2012 (EDT):

I recommend developing or having the Drexel/RPI folks develop an animation showing how the compounded interest formula approaches e^r as n increases.

-- I agree.

# Comments from the Past

Swu2 13:42, 21 May 2012 (EDT)http://instacalc.com/?d=UGx1ZyBpbiBkaWZmZXJlbnQgdmFsdWVzIG9mIG4gdG8gYXBwcm94aW1hdGUgZS4&c=biA9IDEwMHxlID0gKDEgKyAxL24pXm58fHx8fA&s=sssssss&v=0.9 the link i was talking about

Swu2 11:26, 23 May 2012 (EDT)

1. I suggest since you mentioned Euler’s constant, we can take it a step further after proving the formula for continuous growth (x=x0 e^rt) by saying something like “e is the base rate of growth shared by all continually growing processes.”, or “e represents the idea that all continually growing systems are scaled versions of a common rate”. This way the reader could gain a more intuitive understanding of what ‘e’ means.

This is a helper page to Euler's number, so I don't think it is very necessary to explain much about *e*. --Chengying

2. I have a small problem with the wording of the last sentence of the “math explanation” of continuous growth: instead of saying the formula “x=x0e^rt” is a better choice, I say it is the only choice since the growth is continuous. -- CHECK: well...the sentence is not there anymore...

3. I think exponential growth and continuous growth are closely related but different. Continuous growth is exponential growth with an infinitely small growth periods. So instead of saying “if the exponential growth is continuous”, we should say “if growth is continuous” because exponential growth means non-continuous, instantaneous growth. (Unless you say continuous exponential growth.)

I don't think exponential growth has to be non-continuous and instantaneous. --Chengying

4. After “half-life” decay, I thought we should expand it to general decay at a given rate, whose equation is just x=x0 e^(-rt). So maybe we could name that section “Decay” and have the famous example “half-life decay” be a special case for “general decay” (existing as two subsections?)

I added in the "for the continuous case" section to indicate that for general decay cases, follow the formula and set *r* negative. Discrete decay case is already illustrated through drug elimination in body. I don't think it is that necessary to have one more section about decays. (Chengying)

5. Minor issue: a few words in the names of section 2.3 and 4 should be capitalized. (‘continuous’, ‘cases’, and ‘time’) -- CHECK

6. Great job! I like all the improvements!! Just remember that there has to be an *n* in the exponent of the compounded interest formula where *n* is the number of times compounded per year Cool page!! Jorin 16:10, 29 May 2012 (EDT) --CHECK. Thanks!

Awesome! The progression of ideas in the page is much clearer now, and the introduction is fantastic! Most of my suggestions are really minor, so don't be concerned by the length of the list:

- First couple sentences of the second paragraph should be, "An example of such a function... Every time [omit "when"]
*x*increases..." - I think instead of an interactive applet in the intro (though that would be cool -- I have nothing against it), it would be sufficient to just offer a couple more columns to your table showing different
*r*values. - Under "For the Discrete Case," the first sentence should end, "... the value after
*t*periods of time." - In the fourth paragraph of that same section, it would be good to specify, "...scenarios of discrete exponential growth/decay..."
- The "Exponential Growth: Compound Interest" section seems to end rather abruptly with, "... you had the year before." Despite this being followed by the table and graph, it's a bit odd. Since you have both a graph and a table, it would be good to add that, along with, "One way to look at this problem is to use a table," another way to look at it is using a graph. Also, it would be helpful to say something about what we're seeing in that table and that graph -- the increasing differences between
*x*and_{t}*x*_{t}_{+1}as*t*increases, the growth by a factor of 1.5, etc. You don't have to go into a lot of detail, but some explanation would be helpful. - In the "Exponential Decay: Elimination of Drug from Body" section, we have the same issue as in the previous section. Just a few more sentences helping describe what's going on in the table and graph and what the reader should be focusing on would be helpful.
- In the first sentence of "For the Continuous Case," it should be "amount of interest," not "amount of interests."
- In the first bullet point under that, you should specify that you'd get $1050 at the end of one year.
- The "note" in this section is pretty long, which means there's a lot going on between "... different frequencies:" and the actual table. I'd suggest putting the whole "note" section in a box or at least indenting it to make it more visually apparent what's going on here.
- For the last sentence before the table, try, "...functions and will be explained soon."
- The second sentence below the table, use: "... or even more frequently."
- First sentence of the next paragraph, use: "...Euler's number,
*e*, is closely connected..." - Before stating the general formula for continuous exponential growth, it would be good to remind the reader of the general formula for discrete growth to make it more clear to the reader how you got from the limit state above to this formula.
- This: "This is because as time passes, e is raised to a higher power if the growth rate is positive, which leads the function to increase more. However, if the growth rate is negative, e is raised to a more negative power. In other words, 1/e is raised to a higher power." is confusing. It would help if instead of "higher" and "more negative," you used phrases like "increasingly large" and "increasingly negative," or something.
- When you give the mathematical explanation of the continuous function, you need to use
*t*in the place of*x*all the way through. - Also in that section, you need: "Solving for
*y*, the equation..." - In that same section, it would be helpful to number the initial equation so that you can refer the reader back to it when you say that
*x*= 0 implies*y*=*y*_{0}. - In "Half-Life Decay," I'm pretty sure you're misusing the carbon-14 example. It's been a long time since I took Chemistry, but I think the half-life of a particle is the amount of time it takes for half of a group of such particles to change into stable particles by releasing electrons and/or protons. That's analogous to the amount of time it takes for the probability of one such particle having decayed to it's stable state to be 1/2. DON'T QUOTE ME, but I'm pretty sure that's roughly how half-life decay works. I do know that it doesn't mean that the amount of the substance changes, though it changes into a new element.
- In "Exponential vs. Polynomial Growth," it would be good to talk about what you mean by "relative rate of change" and "absolute rate of change."

In general, really well done! You've put together a superb page. -Diana (13:36, 6/5/12)

"For the Discrete Case" -- I think naive users won't know what "discrete" means, and should be enlightened. They may not know "continuous" either. --CHECK. I added the explanation for "discrete" right after the formula. The explanation for "continuous" is after the table because I think that's when I introduced what continuous compounding looks like. What do you think of this arrangement? --Chengying

"Here is a link to an online calculator that shows this limit is true" -- you can't prove the limit is true with an online calculator, it can only show that it's reasonable. --CHECK.

This page is working nicely. I'll have a bit more to say. Gene.

Hi Chengying,

Sorry for taking so long to get back to you. I really like the latest edits you've made, and your addendum to the carbon-14 part makes things much clearer.

The relative vs absolute rates of change bit is still a bit confusing, but I like the parenthetical explanations. Here's what I suggest:

- "This is because though the growth rate, or relative rate of change (ratio between the quantity before the change and the quantity after the change), is constant over time, the absolute rate of change (the difference between the quantity before the change and the quantity after the change) becomes bigger as the function grows."

But does that seem like too much to put in parentheses? I'm not sure -- it could be better to put it in balloons, but that's your call. --CHECK. Put them into balloons.

Two minor points:

- Under "Exponential Growth: Compound Interest," you have:

- "One way to look at this problem is to use a table:
- Each year, the money in your bank account is 5% more than the amount you had the year before.
- One way to look at this problem is to use a table. Another way is to look at the graph."

- But you don't really need that first "One way to look at this problem is to use a table:" I'd take that line out. --CHECK.

- In the math section under "Doubling Time and the Rule of 70," you have "if r is given as a fraction," where it should be "if t is given as a fraction." --Here I meant if
*r*is given as a percentage (e.g. it is given 5% instead of 5). I guess it is not very clear... I changed the sentence to "if*r*is given as a percentage rate (some number between 0 and 1)"

This is looking great! I'm going to copy this email onto your discussion page to keep everything together.

-Diana 15:14, 11 June 2012 (EDT)

The pain Gene again:

If for equally spaced x values, the value of the function grows by a constant factor, then the function is growing exponentially.

- I think you want either "said to be growing exponentially" or "growing exponentially" in italics

Likewise, I think in "Here, discrete means ... ' that discrete should be in italics. In the next sentence I'd prefer "growth" to be in italics.

Exponential Decay: Elimination of Drug from THE Body

Please search for moeny and replace it with money

Rule of 70: " can be APPROXIMATELY found by dividing 70

-- All CHECK! Thanks! --Cwang3 15:57, 18 June 2012 (EDT)