If you thought that the only purpose of calculus was to confuse you, then think again. Continuing with my Why Study Calculus? series, I discuss yet another application of this branch of mathematics to numbers. Numbers and the operations performed on them are the linchpins to mathematics, and all higher branches are one way or another intimately linked to their inherent properties. One nice application of the Calculus to numbers is the approximation of cube roots. What is enticing is that this technique can be done without a calculator and without even a knowledge of the underlying theory.
The method to get the cube root is very similar to that discussed in my Square Root article. The theory behind this method hinges upon the derivative, which in calculus, is a special kind of limit. The differential, which is a quantity that approximates the derivative, particularly under certain conditions, is the mathematical tool that we employ to estimate our cube root.
The way the method works is as follows: Suppose I want to approximate the cube root of 65. This is the number which when multiplied by itself three times will give 65 exactly. Now 65 is not a perfect cube like 64. That is, there is no whole number cube root of 65. The cube root of 64 is 4, since 4*4*4 = 64, and 4 is obviously a whole number. (I have used the “*" symbol for multiplication here) The calculus, using the concept of the differential, tells us that the cube root of 65 will be approximately equal to the cube root of 64 plus 1 divided by three times the cube root of 64 squared. Whew! That’s quite a mouthful.
Let’s break this down and simplify. The cube root of 64 is 4. The difference between 64 and 65 is 1. That’s where the 1 comes from in the numerator of the second part of our equation. Now the cube root of 64 squared is 4^2 or 4*4 or 16; 3*16 = 48. Thus according to this method, the cube root of 65 will be 4 + 1/48 or 4.0208 to the ten-thousandths place. (You can get 1/48 by doing the division by hand, or by observing that 1/48 is a little bigger than 1/50 which is 0.02. So if you used 4.02, you wouldn’t be far off either.
Take out your calculator and compute the cube root of 65. You will see that the result is 4.0207. The approximation is off by less than one ten-thousandth! If you need to be any closer than that—well then, take out your calculator!
See more at www.mathbyjoe.com/catalog/item/2924777/3351424.htm Wiz Kid Calculus. Also see here for more articles (see here www.mathbyjoe.com/page/page/2908884.htm (Square Roots without a Calculator). |
