Math 118: Honours Calculus II Winter, 2005 List of Theorems. L(P, f) U(Q, f). f exists for each ǫ > 0 there exists a partition P of [a, b] such that

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1 Mth 118: Honours Clculus II Winter, 2005 List of Theorems Lemm 5.1 (Prtition Refinement): If P nd Q re prtitions of [, b] such tht Q P, then L(P, f) L(Q, f) U(Q, f) U(P, f). Lemm 5.2 (Upper Sums Bound Lower Sums): Let f be bounded on [, b]. If P nd Q re ny prtitions of [, b], then L(P, f) U(Q, f). Lemm 5.3 (Lower Integrls vs. Upper Integrls): Let f be bounded on [, b]. Then f Theorem 5.1 (Integrbility): f exists nd equls α there exists sequence of prtitions {P n } n=1 of [, b] such tht f. lim L(P n, f) = α = lim U(P n, f). n n Theorem 5.2 (Cuchy Criterion for Integrbility): Suppose f is bounded on [, b]. Then f exists for ech ǫ > 0 there exists prtition P of [, b] such tht U(P, f) L(P, f) < ǫ. Corollry (Piecewise Integrtion): Suppose < c < b. Then f Furthermore, when either side holds, c f nd c f. f = c f + c f. Theorem 5.3 (Drboux Integrbility Theorem): f exists nd equls α for ny sequence of prtitions P n hving subintervl widths tht go to zero s n, ll Riemnn sums S(P n, f) converge to α. 1

2 Theorem 5.4 (Linerity of Integrl Opertor): Suppose f nd g exist. Then (i) (f + g) = f + g (ii) (cf) = c f for ny constnt c R. Theorem 5.5 (Integrl Bounds): Suppose (i) f, (ii) m f(x) M for x [, b]. Then m(b ) f M(b ). Corollry (Preservtion of Non-Negtivity): If f(x) 0 for ll x [, b] nd f exists then f 0. Corollry (Continuity of Integrls): Suppose f exists. Then the function F(x) = x f is continuous on [, b]. Theorem 5.6 (Integrbility of Continuous Functions): If f is continuous on [, b] then f exists. Theorem 5.7 (Integrbility of Monotonic Functions): If f is monotonic on [, b] then f exists. Lemm 5.4 (Fmilies of Antiderivtives): Let F 0 (x) be n ntiderivtive of f on n intervl I. Then F is n ntiderivtive of f on I F(x) = F 0 (x)+c for some constnt C. Theorem 5.8 (Antiderivtives t Points of Continuity): Suppose (i) f exists; (ii) f is continuous t c (, b). Then f hs the ntiderivtive F(x) = x f t x = c. Corollry (Antiderivtive of Continuous Functions): If f is continuous on [, b] then f hs n ntiderivtive on [, b]. Theorem 5.9 (Fundmentl Theorem of Clculus [FTC]): Let f be integrble nd hve n ntiderivtive F on [, b]. Then f = F(b) F(). 2

3 Corollry (FTC for Continuous Functions): Let f be continuous on [, b] nd let F be ny ntiderivtive of f on [, b]. Then f = F(b) F(). Theorem 5.10 (Men Vlue Theorem for Integrls): Suppose f is continuous on [, b]. Then for some number c [, b]. f = f(c)(b ) Theorem 7.1 (Chnge of Vribles): Suppose g continous on g([, b]). Then is continuous on [, b] nd f is f(g(x))g (x) dx = g(b) g() f(u) du. Theorem 7.2 (Integrtion by Prts): Suppose f nd g re continuous functions on [, b]. Then fg = [fg] b f g. Lemm 7.1 (Polynomil Fctors): If z 0 is root of polynomil P(z) then P(z) is divisible by (z z 0 ). Lemm 7.2 (Liner Prtil Frctions): Suppose tht P(x)/Q(x) is proper rtionl function such tht Q(x) = (x ) n Q 0 (x), where Q 0 () 0 nd n N. Then there exists constnt A nd polynomil P 0 with deg P 0 < deg Q 1 such tht P(x) Q(x) = A (x ) + P 0 (x) n (x ) n 1 Q 0 (x). Lemm 7.3 (Qudrtic Prtil Frctions): Let x 2 + γx + λ be n irreducible qudrtic polynomil (i.e. γ 2 4λ < 0). Suppose tht P(x)/Q(x) is proper rtionl function such tht Q(x) = (x 2 + γx + λ) m Q 0 (x), where Q 0 (x) is not divisible by (x 2 + γx + λ) nd m N. Then there exists constnts Γ nd Λ nd polynomil P 0 with deg P 0 < deg Q 2 such tht P(x) Q(x) = Γx + Λ (x 2 + γx + λ) + P 0 (x) m (x 2 + γx + λ) m 1 Q 0 (x). 3

4 Theorem 7.3 (Liner Interpoltion Error): Let f be twice-differentible function on [0, h] stisfying f (x) M for ll x [0, h]. Let Then L(x) = f(0) + h 0 f(h) f(0) x. h L(x) f(x) dx Mh3 12. Corollry (Trpezoidl Rule Error): Let P be uniform prtition of [, b] into n subintervls of width h = (b )/n, nd f be twice-differentible function on [, b] stisfying f (x) M for ll x [, b]. Then the error En T. = T n f of the uniform Trpezoidl Rule stisfies T n = h n i=1 E T n nmh 3 12 f(x i 1 ) + f(x i ) 2 = M(b )3 12n 2. Theorem 8.1 (Pppus Theorems): Let L be line in plne. (i) If curve lying entirely on one side of L is rotted bout L, the re of the surfce generted is the product of the length of the curve times the distnce trvelled by the centroid. (ii) If region lying entirely on one side of L is rotted bout L, the volume of the solid generted is the product of the re of the region times the distnce trvelled by the centroid. Theorem 9.1 (Incresing Functions: Bounded Asymptotic Limit Exists): Let f be monotonic incresing function on [, ). Then f is bounded on [, ) lim x f exists. Corollry (Improper Integrls of Non-Negtive Functions): Let f be nonnegtive function tht is integrble on [, T] for ll T. If there exists bound B such tht T f B for ll T, then f converges. Corollry (Comprison Test): Suppose 0 f(x) g(x) nd T f nd T exist for ll T. Then (i) (ii) g C f C; f D g D. g 4

5 Corollry (Limit Comprison Test): Let f nd g be positive integrble functions stisfying f(x) lim x g(x) = L. (i) For 0 < L < we hve g C f C. (ii) When L = 0 we cn only sy g C f C. Theorem 9.2 (Cuchy Criterion for Improper Integrls): Let f be function. (i) Suppose t f exists for ll t (, b). Then f C ǫ > 0, δ > 0 such tht y x, y (b δ, b) f < ǫ; (ii) Suppose T f exists for ll T >. Then f C ǫ > 0, T such tht T2 T 2 T 1 T f < ǫ. T 1 x Theorem 9.3 (Cuchy Criterion for Infinite Series): The infinite series k converges if nd only if for ech ǫ > 0, there exists N N such tht m m > n N k < ǫ. Theorem 9.4 (Divergence Test): If k=n k C then lim n n = 0. Theorem 9.5 (Non-Negtive Terms: Convergence Bounded Prtil Sums): If k 0 nd S n = n k then k C {S n } n=1 is bounded sequence. Corollry (Comprison Test): If 0 k b k for k N then (i) (ii) b k C k C; k D b k D. 5

6 Corollry (Limit Comprison Test): Suppose k 0 nd b k > 0 for k N nd lim k k /b k = L. Then (i) if 0 < L < : k C b k C; (ii) if L = 0: b k C k C. Corollry (Rtio Comprison Test): If k > 0 nd b k > 0 nd for ll k N, then k+1 k b k+1 b k (i) (ii) b k C k D k C; b k D. Corollry (Rtio Test): Suppose k > 0 nd b k > 0. (i) If number x < 1 such tht k+1 k (ii) If number x 1 such tht k+1 k x for ll k N, then k C. x for ll k N, then k D. Corollry (Limit Rtio Test): Suppose k > 0 for ll k N nd Then k+1 lim = c. k k (i) 0 c < 1 k C, (ii) c > 1 k D, (iii) c = 1? 6

7 Theorem 9.6 (Integrl Test): Suppose f is continuous, decresing, nd non-negtive on [1, ). Then f(k) C f C. Theorem 9.7 (Absolute Convergence): An bsolutely convergent series is convergent. Theorem 9.8 (Rdius of Convergence): For ech power series k=0 c kx k there exists number R, clled the rdius of convergence, with 0 R, such tht Abs C if x < R, c k x k D if x > R, k=0? if x = R. Lemm A.1 (Complex Conjugte Roots): Let P be polynomil with rel coefficients. If z is root of P, then so is z. Theorem A.1 (Fundmentl Theorem of Algebr): Any non-constnt polynomil P(z) with complex coefficients hs complex root. Corollry A.1.1 (Polynomil Fctoriztion): Every complex polynomil P(z) of degree n 0 hs exctly n complex roots z 1, z 2,..., z n nd cn be fctorized s P(z) = A(z z 1 )(z z 2 )...(z z n ), where A C. Corollry A.1.2 (Rel Polynomil Fctoriztion): Every polynomil with rel coefficients cn be fctorized s P(x) = A(x 1 ) n 1...(x k ) n k (x 2 + γ 1 x + λ 1 ) m 1...(x 2 + γ l x + λ l ) m l. 1 7

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