[MUSIC] In this final example, we'll look at a rule of thumb used by bankers and investors to estimate how many years it takes for a sum of money, the principal, to double in value when invested with a given compounding annual interest rate. This is called the Rule of Seventy. It estimates the number of years for the principle to double is 70 divided by the interest rate. For example, if the interest rate is 1%, then you expect to take 70 divided by 1, which is 70 years for the principle to double. If the interest rate is 2%, then you expect to take 70 divided by 2 which is 35 years to double. If the interest rate is 7%, then you expect to take 70 over 7 which is 10 years to double and so on. The rule of 70 is only a good estimate for small interest rates. Now why does it work? Though the rule itself is very simple, the underlying reason why it works is profound and relies ultimately on a property of Euler's number e. Recall that the importance of e stems from the fact that the slope of the tangent line to the curve, y equals e to the x at the y intercept is 1. So that when we reflect in the line y=x to form the inverse function, this tangent then becomes a tangent line to the curve y=log(x) and its x intercept also of slope one. The equation of this tangent line to the curve of the natural logarithm, you can check quickly is y=x-1. And it's a very good approximation for the curve near x=1. Thus lnx is approximately equal to x-1 for x close to 1. Recall that the derivative of lnx is 1 over x, so when x equals 1, the value of the derivative is exactly 1, the slope of the tangent line. In terms of differentials, if we multiply through by dx, we get d of lnx = dx. This becomes an approximation, delta ln of x is approximately equal to delta x, which we expect to be a good estimate when x is close to 1. Making room to continue, delta (ln x), the change of ln x is log of 1 plus the change in x take way the log of 1, which is just ln (1 + delta x), because log of 1 is 0. Combined with the above, this gives ln (1+Delta x) is approximately equal to Delta x. As a simple abstract statement, this just says that ln (1+Delta)is approximately equal to Delta for any small numbers Delta. If we want to estimate ln(1+Delta) when Delta is small, we can just replace this expression by Delta itself! What on Earth has this got to do with the rule of 70? Let P denote the initial principle being invested and i denote the interest rate. Let y=y(x) be the value of the investment after x years. So P is the value of the investment when x = 0 so that y(0) = p, and the general formula is y = p (1+i / 100) to the power x, which uses an exponential growth model as the investment uses compound interest. We want the number of years x, after which the original principal, P, has doubled to 2P. So by the formula for y of x we get this expression for P. We can divide through by P and then take natural logs of both sides. But remember, taking logs brings the exponent down to the front. So the right hand side becomes x times ln of 1 plus i divided by 100, and then dividing through, x becomes this complicated looking fraction. Now comes the magic. Recall, log of 1 plus delta is approximately equal to delta for small delta. Here, take delta to be i on 100, which is a small number. And then we can replace ln of 1 + i over 100 simply by i on a 100, up to an approximation. Thus x becomes approximately the simple fraction ln of 2 over i divided by 100, which is 100 ln of 2 divided by i. The upshot is that to estimate the number of years, we just divide 100 ln of 2 by the interest rate. But 100 ln of 2 is 69.3 to one decimal place, which is close to 70. So the number of years is approximately 70 divided by the interest rate, which explains the rule of 70. Now 100 ln of 2, in fact, rounds down to 69. So the number of years is also approximately 69 divided by the interest rate, known as the rule of 69, which is slightly more accurate then the rule of 70. And 69 is also not too far off 72, which is a nice number, and we get the so called rule of 72, which is less accurate, but often used as a rule of thumb if the interest rate is an integer that divides 72 exactly. For example, if the interest rate is 6%, then 72 divided by 6 is 12, so expect about 12 years for the principle to double. It's slightly awkward to divide 6 into 70 or into 69, and in any case both answers round up to 12. This is the final video for module three, and we've come so far so quickly in transition from precalculus. We're well set up now to launch into the main techniques and applications of differential calculus in module four, and integral calculus in module five. We began this module by discussing average rates of change. And especially focused on interpretations as average speed or velocity in the context of distance or displacement functions. What you see on the speedometer of a car is a physical representation of what we think of as instantaneous velocity, which in mathematical terms is the limiting behavior of average velocity when the time intervals become vanishingly small. All of this has a general interpretation in terms of slopes of tangent lines to curves, which we saw were described as limits of slopes of secants. The slope of the tangent line is called the derivative. This leads naturally into limit definitions of the derivative and several natural equivalent formulations, including Leibniz notation, which expresses the derivative in terms of differentials. We discussed a variety of possible behaviors of curves of functions, captured succinctly using variations of limit notation, including examples of asymptotic behavior. We also discussed the important notion of continuity. Now today we exploited the notation of Leibnitz for the derivative to see how to translate idealized equations involving differentials, into approximations involving small changes in the variables expressed by delta notation. This has some surprising applications such as explaining the rule of Seventy. Please read the notes, and when you're ready, please attempt the exercises. Thank you very much for watching, and I look forward to seeing you in module four. [MUSIC]
