In this series of posts, I consider how two different definitions of the number are related to each other. The number is usually introduced at two different places in the mathematics curriculum:
- Algebra II/Precalculus: If dollars are invested at interest rate for years with continuous compound interest, then the amount of money after years is .
- Calculus: The number is defined to be the number so that the area under the curve from to is equal to , so that
These two definitions appear to be very, very different. One deals with making money. The other deals with the area under a hyperbola. Amazingly, these two definitions are related to each other. In this series of posts, I’ll discuss the connection between the two.
I should say at the outset that the second definition is usually considered the true definition of . However, compound interest usually appears earlier in the mathematics curriculum than definite integrals, and so an informal definition of is given at that stage of the curriculum.
In yesterday’s post, I presented an informal derivation of the continuous compound interest formula from the discrete compound interest formula . In today’s post, I’d like to give the more formal derivation using calculus.
What does it mean for something to compound continuously? In a nutshell, the rate at which the money increases should be proportional to the amount currently present. In other words, should earn ten times as much interest as . Since is the rate at which the money increases and is the current amount, that means
for some constant of proportionality . This is a differential equation which can be solved using standard techniques. We divide both sides by and then integrate:
(Technically, a better solution would use an integrating factor [see also MathWorld], but I find that the above derivation is much more convincing to students who are a few semesters removed from a formal course in differential equations.) When presenting this in class, I’ll sometimes lazily write in place of , with the understanding that to an arbitrary constant is just an arbitrary positive constant. Also, on the last line, plus or minus an arbitrary constant is just an arbitrary constant (which I’ll usually write as instead of ).
To solve for the missing constant , we use the initial condition :
Replacing by , we have arrived at the continuous compound interest formula .