Functions of Several Variables

Transcription

1 Functions of Severl Vribles Sketching Level Curves Sections Prtil Derivtives on every open set on which f nd the prtils, 2 f y = 2 f y re continuous. Norml Vector x, y, 2 f y, 2 f y n = ± (x 0,y 0) (x 0,y 0) y 1 Error Estimtion Chin Rules f(x 0 + x, y 0 + y) f(x 0, y 0 ) + Exmple, z = f(x, y) with x = x(t), y = y(t) [ ] [ ] (x 0, y 0 ) x + y (x 0, y 0 ) y f x t y t df dt = dx dt + y dy dt Integrtion Integrtion by Prts Integrte the chin rule, u(x)v(x) = u (x)v(x) dx + v (x)u(x) dx 1

2 Integrtion of Trig Functions For sin 2 x dx nd cos 2 x dx remember tht, cos 2x = cos 2 x sin 2 x Integrls of the form cos m x sin n x dx, when m or n re odd, you cn fctorise using cos 2 x + sin x = 1 nd then using, sin k x cos x dx = 1 k + 1 sink+1 x + C cos k x sin x dx = 1 k + 1 cosk+1 x + C Reduction Formule... Prtil Frctions Exmple, ssume 2x 1 (x + 3)(x + 2) 2 = A x Bx + C (x + 2) 2 Now multiply both sides by (x + 2)(x + 3) nd equte coefficients. ODE s Seprble ODE Seprte then integrte. Liner ODE hs solution, dy + f(x)y = g(x) dx y(x) = 1 [ u(x) ] u(x)g(x) dx + C u(x) := e R f(x) dx Exct ODE or s, is exct when, Assume solution is of the form, with, dy y) = M(x, dx N(x, y) M(x, y)dx + N(x, y)dy = 0 M = F M y = N F (x, y) = c N = F y Integrte to find two equtions equl to F (x, y), then compre nd find solution from ssumed form. 2

3 Second Order ODE s y + y + by = f(x) For the homogeneous cse (f(x) = 0) the chrcteristic eqution will be λ 2 + bλ + c = 0 If the chrcteristic eqution hs, Two Distinct Rel roots, (replce the λ s with the solutions to the chrcteristic eqn.) Repeted Rel roots, Complex roots, For the For the homogeneous cse, y = Ae λx + Be λx y = Ae λx + Bxe λx y = e αx (A cos βx + B sin βx) λ = α ± βi y = y h + y p y = solution to homogeneous cse + prticulr solution Guess something tht is in the sme form s the RHS. If f(x) = P (x) cos x(or sin) guess for y p is Q 1 (x) cos x + Q 2 (x) sin x Tylor Series Tylor Polynomils For differentible function f the Tylor polynomil of order n t x = is, Tylor s Theorem Sequences P n (x) = f() + f ()(x ) + f () 2! f(x) = f() + f ()(x ) + f () 2! (x ) f (n) () (x ) n n! (x ) f (n) () (x ) n + R n (x) n! R n (x) = f (n+1) (c) (x )n+1 (n + 1)! lim f(x) = L = lim x n = L n essentilly sys tht when evluting limits functions nd sequences re identicl. A sequence diverges when lim n = ± or lim n does not exist. n n Infinite Series Telscoping Series Most of the terms cncel out. 3

4 n-th term test (shows divergence) n diverges if lim n fils to exist or is non-zero. n Integrl Test Drw picture. Use when you cn esily find the integrl. p- series The infinite series Comprison Test Compre to p-series. Limit form of Comprison Test n 1 converges if p > 1 nd diverges otherwise. np Look t lim where b n is usully p-series. n b n If = c > 0, then n nd b n both converge or both diverge. If = 0 nd b n converges, then n converges. If = nd b n diverges, then n diverges. Rtio Test The series converges if ρ < 1, the series diverges if ρ > 1 or ρ is infinite, nd the test is inconclusive if ρ = 1. n+1 lim = ρ n n Alternting Series Test The series, converges if, ( 1) n+1 u n = u 1 u 2 + u 3 u The u n s re ll > 0, 2. u n u n+1 for ll n N for some integer N, nd 3. u n 0. Absolute Convergence If n converges, then Tylor Series n converges. Tylor Polynomils consist of dding finite number of things together, wheres Tylor Series is n infinite sum. The Mclurin series is the Tylor series t x = 0. 4

5 Power Series More Clculus Averge Vlue of Function b f(x) dx b Arc Length Arc length over [, b] = b 1 + f (x) 2 dx s = b x (t) 2 + y (t) 2 dt (prmetric) Speed ds dt = x (t) 2 + y (t) 2 Surfce Are of Revolution 2π b 2π b f(x) 1 + f (x) 2 dx y(t) x (t) 2 + y (t) 2 dt (prmetric) 5