Unformatted text preview:

MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � � � � � � � � � � Lecture 38 18.01 Fall 2006 Lecture 38: Final Review Review: Differentiating and Integrating Series. ∞If f(x) = anx n, then n=0 � � n+1∞� ∞anxf�(x) = nanx n−1 and f(x)dx = C + n + 1 n=1 n=0 Example 1: Normal (or Gaussian) Distribution. x x e−t2 dt = 1 − t2 +(−2! t2)2 +(−3! t2)3 + ··· dt 0 0 x t4 t6 t8 = 1 − t2 + 2! − 3! + 4! − ... dt 0 x3 1 x5 1 x7 = x − 3 +2! 5 − 3! 7 + ... x 2Even though e−tdt isn’t an elementary function, we can still compute it. Elementary functions 0 are still a little bit better, though. For example: sin x = x − x3! 3 + x5! 5 − ··· = ⇒ sin π 2= π 2 − (π/3!2)3 +(π/5!2)5 − ··· But to compute sin(π/2) numerically is a waste of time. We know that the sum if something very simple, namely, πsin = 1 2 It’s not obvious from the series expansion that sin x deals with angles. Series are sometimes com-plicated and unintuitive. π πNevertheless, we can read this formula backwards to find a formula for . Start with sin = 1.2 2 Then, � 1 dx ��1 π π 0 √1 − x2 = sin−1 x� = sin−1 1 − sin−1 0 = 2 − 0 = 0 2 We want to find the series expansion for (1 − x2)−1/2, but let’s tackle a simpler case first: � �� � � �� �� �1 1 1 1 1 1 − 2 − 2 − 1 − 2 − 2 − 1 − 2 − 2 (1 + u)−1/2 = 1 + − 2 u + 1 2 u 2 + 1 2 3 u 3 + ··· · · · 1 1 3 1 3 5 = 1 − 2 u +2 · 4 u 2 − 2 · 4 · 6 u 3 + ··· · · · Notice the pattern: odd numbers go on the top, even numbers go on the bottom, and the signs alternate. 1� � � � � � � Lecture 38 18.01 Fall 2006 Now, let u = −x2 . 1 1 3 1 3 5(1 − x 2)−1/2 = 1 + 2x 2 +2 · 4 x 4 +2 · 4 · 6 x 6 + ··· · · · 1 x3 1 3 x5 1 3 5 x7 (1 − x 2)−1/2dx = C + x +2 3 +2 · 4 5 +2 · 4 · 6 7 + ··· · · · � 1 � � � �� � � �� � π 1 1 1 3 1 1 3 5 1 2= 0 (1 − x 2)−1/2dx = 1 + 2 3+2 · 4 5+2 · 4 · 6 7+ ··· · · · Here’s a hard (optional) extra credit problem: why does this series converge? Hint: use L’Hôpital’s rule to find out how quickly the terms decrease. The Final Exam Here’s another attempt to clarify the concept of weighted averages. Weighted Average A weighted average of some function, f , is defined as: � b w(x)f(x) dx aAverage(f) = � b w(x) dx a � b Here, w(x) dx is the total, and w(x) is the weighting function. a Example: taken from a past problem set. You get $t if a certain particle decays in t seconds. How much should you pay to play? You were given that the likelihood that the particle has not decayed (the weighting function) is: w(x) = e−kt Remember, � ∞ 1 e−kt dt = k0 The payoff is f(t) = t The expected (or average) payoff is ∞ f(t)w(t) dt ∞ te−kt dt0� = �0 ∞ w(t) dt ∞ e−kt dt0 0 = k ∞ te−kt dt = ∞(kt)e−kt dt 0 0 Do the change of variable: u = kt and du = k dt 2� � Lecture 38 18.01 Fall 2006 ∞ duAverage = ue−u k0 ∞On a previous problem set, you evaluated this using integration by parts: ue−u du = 1. 0 Average = � ∞ 0 ue−u du k = 1 k On the problem set, we calculated the half-life (H) for Polonium120 was (131)(24)(60)2 seconds. We also found that ln 2 k = H Therefore, the expected payoff is 1 H = k ln 2 where H is the half-life of the particle in seconds. Now, you’re all probably wondering: who on earth bets on particle decays? In truth, no one does. There is, however, a very similar problem that is useful in the real world. There is something called an annuity, which is basically a retirement pension. You can buy an annuity, and then get paid a certain amount every month once you retire. Once you die, the annuity payments stop. You (and the people paying you) naturally care about how much money you can expect to get over the course of your retirement. In this case, f(t) = t represents how much money you end up with, and w(t) = e−kt represents how likely your are to be alive after t years. What if you want a 2-life annuity? Then, you need multiple integrals, which you will learn about in multivariable calculus (18.02). Our first goal in this class was to be able to differentiate anything. In multivariable calculus, you will learn about another chain rule. That chain rule will unify the (single-variable) chain rule, the product rule, the quotient rule, and implicit differentiation. You might say the multivariable chain rule is One thing to rule them all One thing to find them One thing to bring them all And in a matrix bind them. (with apologies to JRR Tolkien).


View Full Document

MIT 18 01 - Final Review

Documents in this Course
Graphing

Graphing

46 pages

Exam 2

Exam 2

3 pages

Load more
Download Final Review
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Final Review and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Final Review 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?