Скачать или смотреть Where does "e" actually come from? Deriving a limit definition of e and more!

Where does "e" come from, and why do you see "e" everywhere in math, physics and engineering?

00:00 In this video, we derive the limit definition of e starting from the simple differential equation y'=y, then we add a few small generalizations to explain why "e" is so pervasive in math and math-rich subjects.

00:18 y'=y is a super important differential equation, and we start here in our quest to derive Euler's constant e! The geometric interpretation of this differential equation is that y is equal to its own derivative, or the solution is a function whose slope is equal to its y-coordinate at every point.

01:28 Exponential functions are great candidates for solving y'=y, but 2^x has slopes that are a little too shallow and 3^x has slopes that are a little too steep. We propose that the solution to y'=y is an exponential function b^x with the perfect base "b" so that the slope at every point is equal to the y coordinate.

03:17 Plug b^x into the differential equation y'=y. After applying the limit definition of the derivative and doing a bunch of limit math, we find the value of the base, which we how call "e": limit as h goes to zero of (1+h)^(1/h). We motivate convergence of this limit by plugging values of h in as h approaches zero.

07:21 Quick review of key facts: the limit definition of e, the approximate decimal value of e 2.71828... and the fact that e^x is the solution of y'=y, which is the same as saying e^x is the special function equal to its own derivative.

08:44 Generalizations: Ce^x is also a solution to y'=y, Ce^kx is a solution of y'=kx, which says the rate of change in a quantity is proportional to the amount - this is the key feature of growth models (population growth and compound interest) as well as decay models (nuclear decay and discharging capacitors). Finally, we see that ALL exponential functions can be written with a base of e, so we expect to see e everywhere.

#calculus #exponentials #limits