Now, we wish to find the instantenous rate of growth of the population. If we find it, it will be helpful to maybe compare it to former rates and form a pretty good impression of what the future holds. This is where e comes in handy. Now, the instanteneous rate of change represents how much the population will have grown in an infinitesimal amount of time. If we denote the infinitesimal interval of time to be "d"t and the effect it has on P be "d"P which is also an infinitesimal unit , instanteneous the rate of change will be.
Also, "d" is not a constant, but rather a symbol which declares that "d"P and "d"t are infinitesimals. Of course, e continues to appear in growth and decay situations, but let's change subject to Physics aswell as other curiosities. The role e has in Physics is somewhat complex.
As it is not really my domain, I'll just offer a brief introduction. This is all pretty complicated stuff, especially for a precalculus student. To simplify, let's say the system can only have 2 different states.
You can often find e in many probability questions and in game theory, a branch of Mathematics. However, for the example I'm going to give, we're going to talk about sticks, just to show how far from standard Math e can appear. Let's say we have a stick of lenght L. The net output rate is e 2. In this case we have the input rate how fast one crystal grows and want the total result after compounding how fast the entire group grows because of the baby crystals.
If we have the total growth rate and want the rate of a single crystal, we work backwards and use the natural log. My bank is generous and gives me the maximum possible compounding. How much will I have after 10 years? How much will I have after 3 years? We go a few months and get to 5kg. Half a year left? We wait a few more months, and get to 2kg. We get 1 kg, have a full year, get to.
As time goes on, we lose material, but our rate of decay slows down. If you want fancier examples, try the Black-Scholes option formula notice e used for exponential decay in value or radioactive decay. This article is just the start — cramming everything into a single page would tire you and me both.
Learn Right, Not Rote. Home Articles Popular Calculus. Feedback Contact About Newsletter. Math books and even my beloved Wikipedia describe e using obtuse jargon: The mathematical constant e is the base of the natural logarithm. And when you look up the natural logarithm you get: The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.
Understanding Exponential Growth Let's start by looking at a basic system that doubles after an amount of time. And it looks like this: Splitting in two or doubling is a very common progression. A Closer Look Our formula assumes growth happens in discrete steps. If we zoom in, we see that our bacterial friends split over time: Mr. Does this information change our equation?
Money Changes Everything But money is different. Blue The dollar Mr. Blue made Mr. Green The 25 cents Mr. Green made Mr. Month 4: Mr. Green, shoveling along 33 cents.
Month 8: Mr. Blue earns another 33 cents and gives it to Mr. Green, bringing Mr. Green up to 66 cents. This 11 cents becomes Mr. Month Things get a bit crazy. Blue earns another 33 cents and shovels it to Mr.
Specifically, he wanted to find a shortcut for exponents. He published his work, Mirifici Logarithmorum Canonis Descriptio , in It would be about another 70 years before this list of logarithms became associated with exponents. In , Swiss mathematician Jacob Bernoulli discovered the constant e while solving a financial problem related to compound interest.
He saw that across more and more compounding intervals, his sequence approached a limit the force of interest. Bernoulli wrote down this limit, as n keeps growing, as e. Because e is related to exponential relationships, the number is useful in situations that show constant growth. One common example, which Bernoulli explored, is related to compound interest—the interest you pay on a loan when you include both the initial principal the amount of the loan and accumulated interest over previous periods in the calculation.
Suppose you put some money in the bank, and the bank compounds that money annually at a rate of percent. The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.
Key Terms tangent : A straight line touching a curve at a single point without crossing it. Learning Objectives Identify some properties and uses of the natural logarithm.
Key Takeaways Key Points The natural logarithm is the logarithm with base equal to e. The number e and the natural logarithm have many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more. Key Terms natural logarithm : The logarithm in base e.
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