Math 409-502Harold P. [email protected] Club Meeting todayMonday, November 15 at 7:30 PMBlocker 156Speaker: Jeff Nash, Aggie former student, from Veritas DGCTitle: Math and Seismic ImagingFREE FOODMath 409-502 November 15, 2004 — slide #2Taylor’s theoremIf f is (n + 1) times differentiable on an interval, and if x and a are points of the interval, thenthere is some point c between x and a for which f (x) = f (a) + f0(a)(x − a) +12f00(a)(x − a)2+· · · +1n!f(n)(a)(x − a)n+1(n+1)!f(n+1)(c)(x − a)n+1.Example. Take f (x) = sin(x) and a = 0. Then f (0) = 0, f0(0) = 1, f00(0) = 0, f000(0) = −1,f(4)(0) = 0, f(5)(c) = cos(c). So there is some c between 0 and x for whichsin(x) = 0 + 1 · x1+ 0 · x2−13!x3+0 · x4+cos(c)5!x5.In particular, | sin(x) − (x −13!x3)| ≤15!|x|5. So (for instance) sin(0.1) ≈ 0.1 −13!(0.1)3with errorless than 10−7.Math 409-502 November 15, 2004 — slide #3Taylor seriesExample. Generalizing the preceding example, writesin(x) = x −13!x3+15!x5− · · · +(−1)n1(2n+1)!x2n+1+(−1)n+1cos(c)(2n+3)!x2n+3for some point c be-tween 0 and x.Because | cos(c)| ≤ 1 for every c, and limn→∞x2n+3/(2n + 3)! = 0, taking the limit givessin(x) =∞∑n=0(−1)nx2n+1(2n + 1)!, the Taylor series for sin(x) at 0.A function that can be represented by a Taylor series in powers of (x − a) is called analytic at a.The case a = 0 (illustrated above) is often called a Maclaurin series.Some examples of analytic functions are polynomials, sin(x), cos(x), and ex(for all x); also1/(1 − x) for |x| < 1.Math 409-502 November 15, 2004 — slide #4A strange examplef (x) =(e−1/x2, x 6= 0;0, x = 0.yxThis function is not analytic at 0. The Maclaurin series converges, but not to the function. Infact, every Maclaurin series coefficient is equal to 0.f0(0) = limx→0+e−1/x2x= limt→∞e−t21/t= limt→∞tet2l’Hˆopital= limt→∞12tet2= 0, and similarly for higher-orderderivatives.Math 409-502 November 15, 2004 — slide #5Homework1. Read section 17.4, pages 236–238.2. The third examination is scheduled for Wednesday, December 1.One of the problems on the exam will be to prove a version of l’Hˆopital’s rule selected from thefollowing eight possibilities:{00or∞∞}and {x → a or x → ∞}and {limf0(x)g0(x)= L or limf0(x)g0(x)= ∞}.Work on proofs of two cases (to discuss in class).Math 409-502 November 15, 2004 — slide