Advanced Calculus I (Math 4209) Martin Bohner


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1 Advnced Clculus I (Mth 4209) Spring 2018 Lecture Notes Mrtin Bohner Version from My 4, 2018 Author ddress: Deprtment of Mthemtics nd Sttistics, Missouri University of Science nd Technology, Roll, Missouri Emil ddress: URL:
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3 Contents Chpter 1. Preliminries Sets Functions Proofs 3 Chpter 2. The Rel Number System The Field Axioms The Positivity Axioms The Completeness Axiom The Nturl Numbers Some Inequlities nd Identities 9 Chpter 3. Sequences of Rel Numbers The Convergence of Sequences Monotone Sequences 14 Chpter 4. Continuous Functions 17 Chpter 5. Differentition Differentition Rules The Men Vlue Theorems Applictions of the Men Vlue Theorems 23 Chpter 6. Integrtion The Definition of the Integrl The Fundmentl Theorem of Clculus 28 iii
4 iv CONTENTS 6.3. Applictions Improper Integrls 29 Chpter 7. Infinite Series of Functions Uniform Convergence Interchnging of Limit Processes 32
5 CHAPTER 1 Preliminries 1.1. Sets Definition 1.1 (Cntor). A set is collection of certin distinct objects which re clled elements of the set. Nottion 1.2. The following nottion will be used throughout this clss: x A (or x A): x is n element (or is not n element) of the set A; A B: A is subset of B, i.e., if x A, then x B (or: x A = x B); A = B A B nd B A; : empty set. We hve A for ll sets A; A = {, b, c}: A consists of the elements, b, nd c; A = {x : x hs the property P }: A consists of ll elements x tht hve the property P ; A B := {x : x A or x B}: union of A nd B; A B := {x : x A nd x B}: intersection of A nd B; A \ B := {x : x A nd x B}: difference of A nd B; A B := {(, b) : A nd b B}: Crtesin product of A nd B (with (, b) = (c, d) = c nd b = d); A c := {x : x A}: complement of A (with respect to given set X); P(A) := {B : B A}: power set of A. Nottion 1.3 (Quntifiers). We use the following quntifiers throughout this clss: universl quntifier: for ll ; existentil quntifier: there exists. unique existentil quntifier:! there exists exctly one. Lemm 1.4. The opertions nd stisfy the commuttive, ssocitive, nd distributive lws. 1
6 2 1. PRELIMINARIES 1.2. Functions Definition 1.5. Let X nd Y be nonempty sets. A function (or mpping) f from X to Y is correspondence tht ssocites with ech point x X in unique wy y Y ; we write y = f(x). X is clled the domin of f while Y is clled the rnge of f. We write f : X Y. Remrk 1.6. Two functions f : X Y nd g : U V re clled equl (we write f = g) iff X = U nd Y = V nd f(x) = g(x) for ll x X. Nottion 1.7. Let f : X Y be function. Then we cll (i) G(f) := {(x, f(x)) : x X} X Y the grph of f; (ii) f(a) := {f(x) : x A} Y the imge of set A X; (iii) f 1 (B) := {x X : f(x) B} X the inverse imge of set B Y. Lemm 1.8. Let f : X Y be function nd A X, B Y. Then we hve (i) y f(a) x A : f(x) = y; (ii) x A = f(x) f(a); (iii) x f 1 (B) f(x) B. Definition 1.9. Let f : X Y be function. Then f is clled (i) onetoone if x 1, x 2 X with x 1 x 2 lwys implies f(x 1 ) f(x 2 ); (ii) onto if f(x) = Y ; (iii) invertible if f is both onetoone nd onto. Remrk (i) f : X Y is function iff x X!y Y : f(x) = y; (ii) f : X Y is onto iff y Y x X : f(x) = y; (iii) f : X Y is invertible iff y Y!x X : f(x) = y. Definition Let f : X Y nd g : Y Z be functions. Then the function g f : X Z with (g f)(x) = g(f(x)) for ll x X is clled the composite function of f nd g. Proposition A function f : X Y is invertible iff there exists exctly one function g : Y X stisfying (f g)(y) = y y Y nd (g f)(x) = x x X.
7 1.3. PROOFS 3 Nottion Let f : X Y. The unique function g : Y X from Proposition 1.12 is clled the inverse function of f nd denoted by f Proofs Remrk 1.14 (Proof Techniques). Let P nd Q be two sttements. (i) To prove the impliction P = Q, i.e., the theorem with ssumption P nd conclusion Q: We sy if P is true, then Q is true. Or P is sufficient for Q. Or Q is necessry for P. The impliction P = Q is logiclly equivlent to the contrposition Q = P (negtions). () direct proof: ssume P nd show Q; (b) indirect proof: do direct proof with the contrposition; (c) proof by contrdiction: ssume P nd Q nd derive sttement tht contrdicts true sttement. (ii) To prove the equivlence P Q, show P = Q nd Q = P. Another possibility is to introduce intermedite sttements P 1, P 2,..., P n nd to prove P P 1 P 2... P n Q. Tble 1.15 (Truth Tble). P T T F F Q T F T F P F F T T Q F T F T P Q T F F F P Q T T T F P = Q T F T T Q = P T T F T P Q T F F T
8 4 1. PRELIMINARIES
9 CHAPTER 2 The Rel Number System 2.1. The Field Axioms Definition 2.1. Let K be set with t lest two elements, nd let +, : K K K be two functions tht we cll ddition nd multipliction. We sy tht K is field provided the following field xioms re stisfied: (F 1 ) + b = b +, b K (commuttivity of +); (F 2 ) ( + b) + c = + (b + c), b, c K (ssocitivity of +); (F 3 ) 0 K : + 0 = 0 + = K (dditive identity); (F 4 ) K b K : + b = 0 (dditive inverse); (F 5 ) b = b, b K (commuttivity of ); (F 6 ) ( b) c = (b c), b, c K (ssocitivity of ); (F 7 ) 1 K : 1 = 1 = K (multiplictive identity); (F 8 ) K \ {0} b K : b = 1 (multiplictive inverse); (F 9 ) ( + b) c = c + b c, b, c K (distributive property). Nottion 2.2. b := b, 2 :=, + b + c := ( + b) + c, bc := (b)c, b from (F 4 ) is denoted s, b from (F 8 ) is denoted s 1, : b = b := b 1 if b 0, b := + ( b). Proposition 2.3. Let K be field nd, b K. Then (i)!x K : + x = b, nmely x = b ; (ii) ( ) = ; (iii) ( + b) = b. Proposition 2.4. Let K be field nd, b K with 0. Then (i)!x K : x = b, nmely x = b ; (ii) ( 1 ) 1 = ; (iii) (b) 1 = b 1 1 provided b 0. 5
10 6 2. THE REAL NUMBER SYSTEM Proposition 2.5. Let K be field nd, b, c, d K. Then (i) b = 0 = 0 or b = 0; (ii) ( 1) = nd ( )( b) = b; (iii) b + c d = d+bc bd (iv) b : c d = d bc nd b c d = c bd provided bcd 0. provided bd 0; Proposition 2.6. In field we lwys hve The Positivity Axioms Definition 2.7. Let K be field. We cll K ordered if there exists set P K tht stisfies the positivity xioms: (P 1 ), b P = + b, b P; (P 2 ) K either P or P or = 0 (trichotomy). Nottion 2.8. We write > b if b P, < b if b P, b if > b or = b, b if < b or = b. Proposition 2.9. Let K be n ordered field nd, b, c, d K. Then (i) 2 > 0 if 0; (ii) 1 > 0; (iii) > 0 = 1 > 0; (iv) < b nd b < c = < c (trnsitivity); (v) < b nd c > 0 = c < bc; (vi) < b nd c < 0 = c > bc; (vii) 0 < < b = 0 < 1 b < 1 ; (viii) > 0 nd b > 0 = b > 0; < 0 nd b < 0 = b > 0; > 0 nd b < 0 = b < 0; (ix) = b b nd b ; (x) 0 < < b = 0 < 2 < b 2. Remrk There is no ordered field K such tht 2 = 1 for some K. Proposition Let K be n ordered field nd, b K with < b. Then there exists c K with < c < b, e.g., c := +b 2, where 2 :=
11 2.3. THE COMPLETENESS AXIOM 7 Nottion If K is n ordered field nd, b K, then we put (, b) := {x K : < x < b}, [, b] := {x K : x b}, (, b] := {x K : < x b}, nd [, b) := {x K : x < b}. Definition Let K be n ordered field nd T K with T. An m T is clled minimum (or mximum) of T provided m t (or m t) for ll t T. We write m = min T (or m = mx T ). Proposition Let K be n ordered field nd T K with T. If min T (or mx T ) exists, then it is uniquely determined. Exmple For T = (0, 1] we hve mx T = 1 nd min T does not exist The Completeness Axiom Definition Let K be n ordered field nd T K. We cll (i) s K n upper (or lower) bound of T if t s (or t s) for ll t T ; (ii) T bounded bove (or bounded below) if it hs n upper (or lower) bound; (iii) T bounded if it is bounded bove nd below; (iv) s = sup T the supremum of T (nd the infimum inf T nlogously) if () s is n upper bound of T nd (b) s s for ll upper bounds s of T. Proposition Let K be n ordered field nd T K. Then m = mx T exists s = sup T exists nd s T, nd then s = m. Theorem Let K be n ordered field nd T K. Then t T : s t nd s = sup T ε > 0 t T : t > s ε.
12 8 2. THE REAL NUMBER SYSTEM Definition 2.19 (Definition of R). (i) An ordered field K is clled complete if sup T K exists whenever T K is nonempty set tht is bounded bove (completeness xiom). (ii) An ordered field tht is complete is clled the field of the rel numbers. We denote it by R. Theorem If S R is bounded below, then inf S exists. Theorem If S, T R nd s S t T : s t, then sup S inf T. Theorem In R we hve c 0!s 0 : s 2 = c. Nottion The s from Theorem 2.22 is denoted by c. Definition 2.24 (Definition of N). () 1 M nd (b) x M = x + 1 M The Nturl Numbers (i) A set M R is clled inductive if (ii) The intersection of ll inductive subsets of R is clled the set of the nturl numbers. We denote it by N. Also, we put N 0 = N {0}. Proposition 2.25 (Properties of N). (ii) N M whenever M R is inductive; (iii) if A N nd A is inductive, then A = N; (iv) min N = 1. (i) N is inductive; Theorem 2.26 (Principle of Mthemticl Induction). Let S(n) be some sttement for ech n N. If (i) S(1) is true nd (ii) S(k) is true = S(k + 1) is true k N, then S(n) is true for ll n N. Exmple n = n(n+1) 2 holds for ll n N. Proposition Let m, n N. Then m + n N nd m n N. Theorem 2.29 (The WellOrdering Principle). Let A N. Then min A exists.
13 2.5. SOME INEQUALITIES AND IDENTITIES 9 Theorem 2.30 (The Archimeden Property). c R n N : n > c. Corollry ε > 0 n N : 1 n < ε. Definition 2.32 (Definition of Z nd Q). We define the following sets. (i) Z := { : N 0 or N 0 } is clled the set of integers. { } (ii) Q := p q : p Z nd q Z \ {0} is clled the set of rtionl numbers. (iii) The set R \ Q is clled the set of irrtionl numbers. Proposition Let T Z be nonempty. (i) If T is bounded bove, then mx T exists; (ii) If T is bounded below, then min T exists. Theorem R \ Q, more precisely, 2 R \ Q Some Inequlities nd Identities Nottion Let m, n Z nd k R for k Z. We put 0 if n < m 1 if n < m n n k = m if n = m nd k = m if n = m ( k=m n 1 k=m n 1 ) k + n if n > m k n if n > m. k=m The following rules re cler: (i) (ii) (iii) (iv) (v) n k=m n k=m n k=m n k=m n k=m k = k = k = c k = c Exmple n ν=m n+p ν ; k=m+p n k=m n k=m k p for ll p Z; n+m k ; k ; 1 = n m + 1 if n m. n k=m k=m k is clled telescoping sum, where k := k+1 k is clled the forwrd difference opertor. We hve n k = n+1 m. k=m
14 10 2. THE REAL NUMBER SYSTEM Definition Let n N 0 nd α R. Then we define n! (red n fctoril ) nd the binomil coefficient ( α n) (red α choose n ) by n n ( ) (α + 1 k) α k=1 n! := k nd :=. n n! k=1 Proposition Let m, n N 0 nd α R. Then (i) ( ) ( α n + α ) ( n+1 = α+1 n+1) ; (ii) ( ) m n = m! n!(m n)! if m n (nd 0 if m < n); (iii) ( ) ( m n = m if m n. Definition Let R. We define 0 = 1, 1 =, nd n+1 = n for ech n N. If n N, then we put n = ( 1 n. ) Proposition Let, b R \ {0} nd p, q Z. Then (i) p q = p+q ; (ii) ( p ) q = pq ; (iii) (b) p = p b p. Theorem 2.41 (The Binomil Formul). Let, b R nd n N 0. Then n ( ) n ( + b) n = k b n k. k Exmple k=0 n ( n k) = 2 n for n N 0 nd n k=0 k=0 ( n ) k ( 1) k = 0 for n N. Theorem 2.43 (Finite Geometric Series). Let, b R nd n N 0. Then n k = n+1 1 n if 1 nd n+1 b n+1 = ( b) k b n k. 1 k=0 Theorem 2.44 (Bernoulli s Inequlity). Let n N 0 nd x 1. Then we hve (1 + x) n 1 + nx. Definition Let x R. Then the bsolute vlue of x is defined by Proposition Let, b R. Then (i) = ; x := mx{x, x}. k=0
15 2.5. SOME INEQUALITIES AND IDENTITIES 11 (ii) 0; nd = 0 = 0; (iii) b = b ; (iv) = 0 < ε ε > 0. Theorem 2.47 (Tringle Inequlities). If, b R, then b + b + b. Remrk Define d(x, y) := x y for x, y R. Then (i) d(x, y) = d(y, x); (ii) d(x, y) 0; nd d(x, y) = 0 x = y; (iii) d(x, z) d(x, y) + d(y, z).
16 12 2. THE REAL NUMBER SYSTEM
17 CHAPTER 3 Sequences of Rel Numbers 3.1. The Convergence of Sequences Definition 3.1. If x : N R is function, then we cll x sequence (of rel numbers). Insted of x(n) we rther write x n, n N. The sequence s defined by s n = n k=1 x k, n N, is lso known s series. Exmple 3.2. (i) n = 1 + ( 1) n ; (ii) n = mx{k N : k n 3 }; (iii) x 0 = 1 nd x n+1 = 2x n for ll n N 0 ; (iv) f 0 = f 1 = 1 nd f n+2 = f n+1 + f n for ll n N 0 ; (v) n = n k=1 1 k. Definition 3.3. A sequence is sid to be convergent if α R ε > 0 N N n N : n α < ε. We write α = lim n n or n α (s n ). A sequence is clled divergent if it is not convergent. Exmple 3.4. (i) n = 2n 4n+3 ; (ii) n = ( 1) n. Proposition 3.5. Any sequence hs t most one limit. Proposition 3.6 (Some Limits). We hve (i) If n = α for ll n N, then lim n n = α; (ii) lim n 1 n = 0; (iii) if x < 1, then lim n x n = 0; (iv) if x < 1, then lim n n k=0 xk = 1 1 x. 13
18 14 3. SEQUENCES OF REAL NUMBERS Definition 3.7. A sequence is clled bounded (or bounded bove, or bounded below) if the set { n : n N} is bounded (or bounded bove, or bounded below). Proposition 3.8 (Necessry Conditions for Convergence). Let be convergent sequence. Then (i) is bounded; (ii) stisfies the Cuchy Condition, i.e., ε > 0 N N : m, n N n m < ε. Remrk 3.9. n α implies n+1 n 0, 2n n 0. Exmple (i) n = ( 1) n ; (ii) n = n k=1 1 k (the hrmonic series). Theorem Suppose n α nd b n β s n. Then (i) n α ; (ii) n + b n α + β; (iii) c R : c n cα; (iv) n b n αβ; (v) n b n α β if β 0. Exmple (i) n α, m N = m n α m ; (ii) n 2 3 2n 2 +3n 1 2 s n. Theorem Suppose n α, b n β, c n R. Then (i) K R n N : n K = α K; (ii) n N : n b n = α β; (iii) α = β nd n N : n c n b n = lim n c n = α Monotone Sequences Definition A sequence is clled monotoniclly incresing (or monotoniclly decresing, strictly incresing, strictly decresing) provided n n+1 ( n n+1, n < n+1, n > n+1 ) holds for ll n N. We write n (,, ). The sequence is clled monotone if it is either one of the bove.
19 3.2. MONOTONE SEQUENCES 15 Theorem 3.15 (The Monotone Convergence Theorem). A monotone sequence converges iff it is bounded. Exmple (i) 1 = 2 nd n+1 = n+6 2 for ll n N; (ii) s n = n k=1 1 k ; (iii) s n = n k=1 1 k2 k ; (iv) n = ( n) n. We denote the limit of this sequence by e. Definition Let n be sequence nd let n k be sequence of nturl numbers tht is strictly incresing. Then the sequence b k defined by b k = nk clled subsequence of the sequence n. Theorem Every sequence hs monotone subsequence. for k N is Theorem 3.19 (Bolzno Weierstrß). Let, b R with < b. Every sequence in [, b] hs convergent subsequence tht hs its limit in [, b]. Theorem 3.20 (Cuchy). A rel sequence converges iff it is Cuchy sequence. Proposition Let n be convergent sequence with lim n n = α. Then every subsequence nk of n converges with lim k nk = α. Exmple (i) n α = 2n α, n+1 α; (ii) ( ) 2n, ( ) n n 2 n ; 2 (iii) ( 1) n ( n). Theorem 3.23 (The Nested Intervl Theorem). Let n, b n R with n < b n for ll n N, put I n = [ n, b n ], nd ssume I n+1 I n for ll n N nd b n n 0 s n. Then n N I n = {α} with α R nd lim n n = lim n b n = α exist.
20 16 3. SEQUENCES OF REAL NUMBERS
21 CHAPTER 4 Continuous Functions Definition 4.1. A function f : D R is sid to be continuous t (or in) x 0 D provided {x n : n N} D, lim n x n = x 0 = lim n f(x n) = f(x 0 ). Also, f is clled continuous if it is continuous t ech x 0 D. Exmple 4.2. (i) f(x) = x 2 + 3x 2, x R; (ii) f(x) = x, x 0; (iii) f = χ [0,1] ; (iv) f = χ Q is clled the Dirichlet function. Nottion 4.3. For two functions f, g : D R we define the sum f + g : D R nd the product f g : D R by (f +g)(x) = f(x)+g(x) nd (f g)(x) = f(x)g(x) ( ) for x D. If g(x) 0 for ll x D, then f g : D R is defined by f g (x) = f(x) g(x) for x D. Theorem 4.4. Let f, g : D R be continuous functions. Then f +g, f g : D R re continuous. If g(x) 0 for ll x D, then f g : D R is continuous. Corollry 4.5. Let m N, c k R (0 k m), nd p : R R be defined by p(x) = m k=0 c kx k, i.e., p is polynomil with degree m if c m 0. Then p is continuous. Also, if p, q re both polynomils nd D = {x R : q(x) 0}, then the rtionl function p q : D R is continuous. Theorem 4.6. If f : D R, g : U R re functions with f(d) U such tht f is continuous t x 0 D nd g is continuous t f(x 0 ) U, then g f : D R is continuous t x 0 D. Exmple x 2, x [ 1, 1]. 17
22 18 4. CONTINUOUS FUNCTIONS Theorem 4.8. Let f : [, b] R be continuous, where, b R with < b. Assume f() < 0 nd f(b) > 0. Then α (, b) : f(α) = 0. Theorem 4.9 (The Intermedite Vlue Theorem). Let f : [, b] R be continuous, where, b R with < b. If f() < c < f(b) or f(b) < c < f(), then α (, b) : f(α) = c. Exmple (i) h(x) = x 5 + x + 1, x R, hs zero in ( 2, 0); (ii) h(x) = 1 1+x 2 x2, x R, hs zero in (0, 1); (iii) if I R is n intervl nd f : I R is continuous, then f(i) is n intervl. Theorem 4.11 (The Extreme Vlue Theorem). Let f : I = [, b] R be continuous, where, b R with < b. Then both mx f(i) nd min f(i) exist. Definition Let D R. The function f : D R is clled strictly incresing (or strictly decresing, incresing, decresing) if f(v) > f(u) (or f(v) < f(u), f(v) f(u), f(v) f(u)) holds for ll u, v D with u < v. We write f (,, ). Also, f is clled strictly monotone if it is either strictly incresing or strictly decresing. Theorem Let f : I f(i) be strictly monotone, where I is n intervl. Then f is invertible nd f 1 : f(i) I is continuous nd strictly monotone. Corollry Suppose I is n intervl nd f : I R is strictly monotone. Then f is continuous iff f(i) is n intervl. Theorem Let x 0 D R nd f : D R. Then f is continuous t x 0 iff ε > 0 δ > 0 ( x D : x x 0 < δ) f(x) f(x 0 ) < ε. Exmple (i) f(x) = x, f : [0, ) [0, ) is continuous t x 0 = 4; (ii) f(x) = x 3, f : R R is continuous t x 0 = 2; (iii) f from (ii) is continuous on D = [0, 20]; (iv) f(x) = 1 x, f : (0, 1) R. Definition Let D R nd f : D R. Then f is clled uniformly continuous (on D) if ε > 0 δ > 0 : ( u, v D : u v < δ) f(u) f(v) < ε.
23 4. CONTINUOUS FUNCTIONS 19 Theorem Let f : [, b] R be continuous, where, b R with < b. Then f is uniformly continuous.
24 20 4. CONTINUOUS FUNCTIONS
25 CHAPTER 5 Differentition 5.1. Differentition Rules Definition 5.1. (i) An x 0 R is clled limit point of D if there exists {x n : n N} D \ {x 0 } with lim n x n = x 0. (ii) We write lim x x0,x D f(x) = l provided x 0 is limit point of D nd lim n f(x n ) = l whenever {x n : n N} D \ {x 0 } with lim n x n = x 0. Exmple 5.2. (i) lim x 4 (x 2 2x + 3) = 11; (ii) lim x 1 x 2 1 x 1 = 2. Remrk 5.3. (i) Let x 0 D be limit point of D. Then f : D R is continuous t x 0 iff lim x x0 f(x) = f(x 0 ). (ii) If x 0 is limit point of D nd f, g : D R with lim x x0 f(x) = α R nd lim x x0 g(x) = β R, then (by Theorem 3.11) nd (if β 0) lim ((f + g)(x)) = α + β, x x 0 lim ((f/g)(x)) = α/β. x x 0 lim ((fg)(x)) = αβ, x x 0 Definition 5.4. Let x 0 (, b) = I. A function f : I R is clled differentible t (or in) x 0 provided f(x) f(x 0 ) lim x x 0 x x 0 exists, in which cse we denote this limit by f (x 0 ). Also, f is clled differentible (on I) if f (x) exists for ll x I. In this cse, f : I R is clled the derivtive of f. Exmple 5.5. (i) f(x) = 4x 5; (ii) f(x) = mx + b; (iii) f(x) = x 2 ; 21
26 22 5. DIFFERENTIATION (iv) f(x) = x. Proposition 5.6. Let m N. Let f : R R be defined by f(x) = x m for ll x R. Then f is differentible nd f (x) = mx m 1. Proposition 5.7. Let x 0 (, b) = I. If f : I R is differentible t x 0, then it is continuous t x 0. Theorem 5.8 (Rules of Differentition). Let x 0 (, b) = I. (i) If f, g : I R re differentible in x 0, then so is αf + βg for ll α, β R, fg, nd (if g(x 0 ) 0) f/g with (αf + βg) (x 0 ) = αf (x 0 ) + βg (x 0 ), (fg) (x 0 ) = f (x 0 )g(x 0 ) + f(x 0 )g (x 0 ) (f/g) (x 0 ) = f (x 0 )g(x 0 ) f(x 0 )g (x 0 ) (g(x 0 )) 2 Product Rule, Quotient Rule. (ii) If g : I g(i) is differentible in x 0 nd if f : J R with J g(i) is differentible in g(x 0 ), then f g : I R is differentible in x 0 with (f g) (x 0 ) = f (g(x 0 ))g (x 0 ) Chin Rule. (iii) If f : I f(i) is continuous nd strictly monotone nd differentible in x 0 with f (x 0 ) 0, then f 1 : f(i) I is differentible in y 0 = f(x 0 ) with (f 1 ) (y 0 ) = Exmple 5.9. (i) f(x) = n x; (ii) f(x) = x p/q. 1 f (f 1 (y 0 )) The Men Vlue Theorems Definition Suppose I = (, b) with < b nd f : I R. (i) An x 0 I is clled locl mximizer (or locl minimizer) of f, if there exists δ > 0 such tht f(x 0 ) f(x) (or f(x 0 ) f(x)) for ll x I with x x 0 < δ. (ii) An x 0 I for which f (x 0 ) exists is clled criticl point of f provided f (x 0 ) = 0.
27 5.3. APPLICATIONS OF THE MEAN VALUE THEOREMS 23 Theorem Suppose I = (, b) with < b nd f : I R. Assume tht x 0 I is such tht f (x 0 ) exists. If x 0 is locl mximizer (or minimizer) of f, then it is criticl point. Theorem 5.12 (Rolle s Theorem). Suppose tht f : [, b] R with < b is continuous on [, b] nd differentible on (, b). Assume f() = f(b) = 0. Then there exists criticl point of f in (, b). Theorem 5.13 (The Lgrnge Men Vlue Theorem). Suppose tht f : [, b] R with < b is continuous on [, b] nd differentible on (, b). Then x 0 (, b) : f (x 0 ) = f(b) f(). b Theorem 5.14 (The Cuchy Men Vlue Theorem). Suppose tht f, g : [, b] R with < b both re continuous on [, b] nd differentible on (, b). Then x 0 (, b) : f (x 0 ) {g(b) g()} = g (x 0 ) {f(b) f()}. Exmple (i) f, g : [0, 3] R defined by f(x) = 3 x 2 nd g(x) = 9 x2 ; (ii) e(x) 1 + x for ll x R; (iii) Generlized Bernoulli inequlity Applictions of the Men Vlue Theorems Theorem 5.16 (The Identity Criterion). Let I R be n intervl nd suppose tht f : I R is differentible on I. Then f is constnt on I (i.e., there exists c R such tht f(x) = c for ll x I) iff f (x) = 0 for ll x I. Theorem Let I be n intervl nd f : I R be differentible on I. (i) If f (x) > 0 for ll x I, then f is strictly incresing on I. (ii) If f (x) < 0 for ll x I, then f is strictly decresing on I. Exmple (ii) f(x) = x+b cx+d. (i) e nd l re strictly incresing; Theorem 5.19 (L Hôpitl s Rules). Let I = [, b) R, < b, b R or b =, nd suppose tht f, g : I R re differentible on I with g (x) 0 for ll x I.
28 24 5. DIFFERENTIATION Assume tht α = lim x b,x<b f (x) g (x) then lim x b,x<b f(x) g(x) exists. If either lim f(x) = lim g(x) = 0 or lim g(x) =, x b,x<b x b,x<b x b,x<b exists nd is equl to α. l(1+x) Exmple (i) lim x 0 x = 1; l(1+x) x (ii) lim x 0 x = ; (iii) lim x x n e( x) = 0 for ll n N; (iv) lim x 0,x>0 xl(x) = 0; (v) lim x 0,x>0 A(x, x) = 1. Nottion If I is n intervl nd f : I R is differentible with f : I R, nd f : I R is lso differentible, then we write f = (f ) = f (2). If f (k) for k N is defined nd differentible, we put f (k+1) = ( f (k)). Also, we put f (0) = f. Theorem Let I be n intervl, n N, nd suppose f : I R hs n derivtives. If f (k) (x 0 ) = 0 for ll 0 k n 1 for some x 0 I, then, for ech x I \ {x 0 }, there exists point z strictly between x nd x 0 with f(x) = f (n) (z) (x x 0 ) n. n! Theorem Let I be n intervl nd suppose f : I R is such tht the below derivtives exist nd re continuous. Assume x 0 I is criticl point of f. (i) If f (x 0 ) > 0, then x 0 is locl minimizer of f. (ii) If f (x 0 ) < 0, then x 0 is locl mximizer of f. (iii) If f (x 0 ) = 0 nd f (x 0 ) 0, then x 0 is neither locl minimizer nor locl mximizer of f. (iv) If f (x 0 ) = 0 nd f (x 0 ) = 0 nd f (x 0 ) > 0, then x 0 is locl minimizer of f. (v) If f (x 0 ) = 0 nd f (x 0 ) = 0 nd f (x 0 ) < 0, then x 0 is locl mximizer of f. Theorem 5.24 (Lgrnge Reminder Theorem). Let I be n open intervl contining the point x 0 nd let n N 0. Suppose tht f : I R hs n + 1 derivtives. Then for ech x I \ {x 0 }, there exists z strictly between x nd x 0 such tht n f (k) (x 0 ) f(x) = (x x 0 ) k + f (n+1) (z) k! (n + 1)! (x x 0) n+1. k=0
29 5.3. APPLICATIONS OF THE MEAN VALUE THEOREMS 25 Definition Let I be n open intervl contining x 0 nd n N 0. Suppose tht f : I R hs n derivtives. f : I R t the point x 0 is defined s n f (k) (x 0 ) p n (x) = (x x 0 ) k. k! k=0 Exmple Find p 3 for f(x) = 1/x t x 0 = 1. The nth Tylor polynomil for the function Theorem Let I be n open intervl contining x 0 nd suppose f : I R hs derivtives of ll orders. Suppose there re positive numbers r nd M such tht [x 0 r, x 0 + r] I nd f (n) (x) M n for ll x [x 0 r, x 0 + r]. Then f (k) (x 0 ) f(x) = (x x 0 ) k if x x 0 r. k! k=0 Exmple e(x) = k=0 xk k!. Note lso tht e R \ Q.
30 26 5. DIFFERENTIATION
31 CHAPTER 6 Integrtion 6.1. The Definition of the Integrl Definition 6.1. Let f : [, b] R with < b be function. If = x 0 < x 1 < x 2 < < x n = b, then Z = {x 0, x 1,, x n } is clled prtition of the intervl [, b] with gp Z = mx{x k x k 1 : 1 k n}, nd if ξ k [x k 1, x k ] for ll 1 k n, then we cll ξ = (ξ 1, ξ 2,, ξ n ) intermedite points of the prtition Z. The sum n S(f, Z, ξ) = f(ξ k )(x k x k 1 ) k=1 is clled Riemnn sum. If ξ is such tht f(ξ k ) = inf f([x k 1, x k ]) for ll 1 k n (or f(ξ k ) = sup f([x k 1, x k ]) for ll 1 k n), then we cll L(f, Z) = S(f, Z, ξ) the lower Drboux sum (or U(f, Z) = S(f, Z, ξ) the upper Drboux sum). Definition 6.2. A function f : [, b] R with < b is sid to be Riemnn integrble if lim n S(f, Z n, ξ n ) exists for ny sequence of prtitions Z n with lim n Z n = 0 nd with intermedite points ξ n. Remrk 6.3. If f : [, b] R is Riemnn integrble, then, no mtter wht sequences Z n nd ξ n we tke, the limit of S(f, Z n, ξ n ) s n is lwys the sme. We then cll this limit f(x)dx = f. Exmple 6.4. f(x) = x, I = [, b]. Proposition 6.5. Let < b nd I = [, b]. (i) If f, g : I R re Riemnn integrble, then so is αf + βg for ll α, β R with (αf + βg) = α f + β (ii) If f(x) = c for ll x I, then f is Riemnn integrble with f = c(b ). 27 g.
32 28 6. INTEGRATION (iii) If f, g : I R re Riemnn integrble nd f(x) g(x) for ll x I, then f g. (iv) If f : I R is Riemnn integrble, then f is bounded on I nd inf f(i) f sup f(i). b (v) If f : I R is Riemnn integrble nd if g(x) = f(x) for ll x I but finite number of points x I, then g is Riemnn integrble nd f = g. (vi) If c (, b) nd f : I R nd f : [, c] R, f : [c, b] R re Riemnn integrble, then f = c Theorem 6.6. If f : [, b] R with < b is continuous, then it is Riemnn integrble. Nottion 6.7. If > b, then we put f = b f. We lso put f = 0. f The Fundmentl Theorem of Clculus Theorem 6.8 (Fundmentl Theorem of Clculus, First Prt). Suppose F : [, b] R is differentible on [, b] nd F on [, b]. Then c f. F = F (b) F (). : [, b] R is Riemnn integrble Definition 6.9. A function F : I R is clled n ntiderivtive of f : I R if F is differentible with F (x) = f(x) for ll x I. Remrk If f possesses n ntiderivtive F, then ny other ntiderivtive of f cn differ from F only by constnt. Exmple (i) 5 0 x3 dx = 54 4 ; (ii) 4 e = e(4) 1; 0 (iii) p 0 s = 1; (iv) 1 n 6 n k=1 k5 1 6 s n. Theorem 6.12 (Fundmentl Theorem of Clculus, Second Prt). Let f : I R be continuous on the intervl I R nd let I. Then F (x) := x f for ech x I
33 6.4. IMPROPER INTEGRALS 29 is n ntiderivtive of f. Remrk Continuous functions possess ntiderivtives. Proposition If f is Riemnn integrble on I, then F defined in the FTOC (Prt II) is continuous (even Lipschitz continuous) on I. Exmple (ii) x 0 dt 1+t 2. (i) x 1 dt t ; 6.3. Applictions Theorem Suppose f, g : I R re continuous, x 0 I, y 0 R. Then there exists exctly one continuously differentible function y with y(x 0 ) = y 0 nd y (x) = f(x)y(x) + g(x) for ll x I, nmely { x } y(x) = e(f (x)) y 0 + g(t)e( F (t))dt x 0 Exmple xy + 2y = 4x 2, y(1) = 2. with F (x) = x x 0 f(t)dt. Theorem 6.18 (Integrtion by Prts). Let f, g : [, b] R be continuously differentible. Then Exmple te(t)dt = 1. 0 f(x)g (x)dx = f(b)g(b) f()g() f (x)g(x)dx. Theorem 6.20 (Substitution). If g : [α, β] R is continuously differentible, f : g([α, β]) R continuous, then f(g(t))g (t)dt = g(b) g() Exmple e( x)dx = 2( 2 1)e( 2) + 2. f(x)dx Improper Integrls Definition Let < b nd f : (, b) R. (i) f is sid to be loclly integrble on (, b) if f is integrble on ech closed subintervl [c, d] (, b).
34 30 6. INTEGRATION (ii) f is sid to be improperly integrble on (, b) if f is loclly integrble on (, b) nd if f(x)dx := lim c +,d b d c f(x)dx exists nd is finite. This limit is clled the improper Riemnn integrl of f over (, b). Exmple (i) 1 0 (ii) 1 1 x 2 dx = 1. 1 x dx = 2; Theorem If f, g re improperly integrble on (, b) nd α, β R, then αf + βg is improperly integrble on (, b), nd (αf + βg)(x)dx = α f(x)dx + β g(x)dx. Theorem 6.25 (Comprison Theorem). Suppose f, g : (, b) R re loclly integrble. If 0 f(x) g(x) for ll x (, b), nd if g is improperly integrble on (, b), then f is improperly integrble on (, b) with f(x)dx g(x)dx. Exmple (i) s(x)/ x 3 is improperly integrble on (0, 1]; (ii) l(x)/ x 5 is improperly integrble on [1, ). Definition Let < b nd f : (, b) R. (i) f is sid to be bsolutely integrble on (, b) if f is improperly integrble on (, b). (ii) f is sid to be conditionlly integrble on (, b) if f is improperly integrble but not bsolutely integrble on (, b). Theorem If f is loclly nd bsolutely integrble on (, b), then f is improperly integrble on (, b), nd f(x)dx f(x) dx. Exmple s(x)/x is conditionlly integrble on [1, ).
35 CHAPTER 7 Infinite Series of Functions 7.1. Uniform Convergence Exmple 7.1. (i) lim x x0 lim n (1 + x/n) n = lim n lim x x0 (1 + x/n) n ; (ii) d dx lim n (1 + x/n) n d = lim n dx (1 + x/n)n ; (iii) 1 0 lim n (1 + x/n) n dx = lim n 1 0 (1 + x/n)n dx; (iv) f n (x) = nx/(1 + nx), n, x 0; (v) f n (x) = x n ; (vi) f n (x) = s(nx) n. Definition 7.2. Let f n : I R be functions for ech n N nd let f : I R. We sy tht the sequence f n converges (i) pointwise to f if lim n f n (x) = f(x) for ll x I; (ii) uniformly to f if ε > 0 N N : ( n N x I) f n (x) f(x) < ε. The pointwise or uniform convergence of the series k=0 g k is defined s bove with f n = n k=0 g k. Exmple 7.3. Let f n (x) = x n on [0, 1]. (i) f n converges uniformly on [0, 1/2]; (ii) f n does not converge uniformly on [0, 1]. Exmple 7.4. f n (x) = 2n 2 x/(1 + n 4 x 4 ) is not uniformly convergent on R. Theorem 7.5 (Cuchy Criterion). A sequence of functions f n : I R converges uniformly on I iff ε > 0 N N : ( m, n N x I) f n (x) f m (x) < ε. 31
36 32 7. INFINITE SERIES OF FUNCTIONS Theorem 7.6 (Weierstrß MTest). Suppose g k : I R stisfies g k (x) M k for ll x I nd for ll k N such tht k=1 M k is convergent. Then k=1 g k(x) is uniformly convergent. Exmple 7.7. c(k 2 x) k=1 k is uniformly convergent on R Interchnging of Limit Processes Theorem 7.8 (Continuity of the Limit Function). Let f n : I R be continuous on I for ll n N nd suppose tht f n f uniformly on I. Then f is continuous on I, i.e., lim lim f n(x) = lim lim f n (x) for ll x 0 I. n n x x 0 x x 0 Exmple 7.9. f(x) = s(kx) k=1 k is continuous on R. 2 Theorem 7.10 (Integrtion of the Limit Function). Let f n : I = [, b] R be Riemnn integrble on I for ll n N nd suppose tht f n f uniformly on I. Then f is Riemnn integrble on I with ( ) b lim f n(x)dx = n f(x)dx = lim n f n (x)dx. Corollry k=1 f k(x)dx = k=1 f k(x)dx if f k : [, b] R re Riemnn integrble for ll k N nd k=1 f k(x) is uniformly convergent on [, b]. Theorem 7.12 (Differentition of the Limit Function). Let f n : I = [, b] R be differentible on I for ll n N nd suppose tht f n g uniformly on I. Also suppose tht lim n f n (x 0 ) exists for t lest one x 0 I. Then f n converges uniformly on I, sy to f, nd f is differentible on I with f (x) = g(x), i.e., lim n d dx f n(x) = d dx lim f n(x). n d Corollry dx k=1 f k(x) = k=1 d dx f k(x) if f k : [, b] R re differentible for ll k N, k=1 f k (x) is uniformly convergent on [, b], nd k=1 f k(x 0 ) is convergent for t lest one x 0 [, b]. Exmple (i) k=0 k(x x 0 ) k ; (ii) s(kx) k=1 k ; 3 (iii) f n (x) = s(n2 x) n.
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