Definitions & Theorems

Transcription

1 Definitions & Theorems Math 147, Fall 2009 December 19, 2010 Contents 1 Logic Sets The Peano axioms Properties of numbers Functions The absolute value Sequences Limits of sequences Series Subsequences Cauchy sequences Limits and continuity Limits of functions One-sided limits Continuity Compositions of functions Discontinuity Value theorems Differentiation Derivatives Inverse functions Mean value theorem Concavity Taylor s Theorem Big O notation c J. Lazovskis Professor: B. Forrest

2 1 Logic 1.1 Sets Definition A set is a collection of objects. The universal set is X. Definition Let the empty set be the set that contains no elements. Denote this by. Definition Let A, B X. The set difference of B and A is defined as B \ A = {x B : x A}. Definition Let A, B X. The union of A and B is defined as A B = {x : x A or x B}. Definition Let A, B X. The intersection of A and B is defined as A B = {x : x A and x B}. If {A α } α I is a collection of sets, then A α = {x : x A α for some α I} α I A α = {x : x A α α I} α I Definition Let A X. The complement of A is defined as A c = A = {x X : x A}. Theorem de Morgan s Laws Let A( X. With respect to the above definitions, A α)c = ( α I α I α I A α)c = α I A c α A c α Definition Define the following basic sets: N : natural numbers = {1, 2, 3,... } Z : integers = {..., 2, 1, 0, 1, 2,... } Q : rational numbers = { a b : a Z, b N, gcd(a, b) = 1} R : real numbers Definition A subset S R is termed an interval. For every x, y S, if z R is such that x z y, then z S. Note that the empty set is an interval. 1.2 The Peano axioms Axiom Principle of Mathematical Induction If S N is such that 1) 1 S 2) If n S, then n + 1 S Then S = N. Axiom Principle of Strong Mathematical Induction If S N is such that 1) 1 S 2) If {1,..., n} S, then n + 1 S Then S = N. Axiom Well-Ordering Principle If S N is non-empty, then S has a least element. 2

3 1.3 Properties of numbers Definition Let S R. An element α R is an upper bound for S if x α x S. If S has an upper bound, then S is bounded above. An element β R is a lower bound for S if x β x S. If S has a lower bound, then S is bounded below. If S is bounded above and below, then S is bounded. Axiom Least upper bound property Every non-empty subset S R that is bounded above has a least upper bound. Corollary Greatest lower bound property Every non-empty subset S R that is bounded below has a greatest lower bound. Theorem Archimedean property I N is not bounded above. Theorem Archimedean property II Let ɛ > 0. Then there exists an n N such that 0 < 1 n < ɛ. Corollary If x R, then there exists some z Z such that z < x z + 1. Corollary If x, y R with x < y, then there exists some r Q and s Q such that r, s (x, y). 1.4 Functions Definition A function is a rule that assigns to each element in a set X a single value y in a set Y. Definition A function f from a set X to a set Y is represented by f : X Y. A function f : X Y is 1 1 is for every x 1, x 2 X with x 1 x 2, we have f(x 1 ) f(x 2 ). A function f : X Y is onto if for every y Y there exists an x X such that = y. A function that is both 1 1 and onto is termed a bijection. Definition Two sets X, Y are termed equivalent if there exists a bijection f : X Y. expressed X Y. Then f is termed an isomorphism. This is Remark Q N. This is given by f : Q N defined by f( m n ) = 2n 3 m. Definition A set X is finite if X {1, 2,..., n} for some n N, or X. A set X is infinite if it is not finite. 1.5 The absolute value Definition The absolute value is a function x = Remark The absolute value has the following properties: 1. x = x 2. x 0 and x = 0 if and only if x = 0 3. xy = x y Theorem Triangle Inequality For every x, y, z R, x y x z + z y. Corollary For every x, y R, x + y x + y. { x if x 0 x if x < 0 3

4 Corollary For every x, y, z R, x y x y. Remark The absolute value has the following inequalities: 1. x a < δ = x (a δ, a + δ) 2. 0 < x a < δ = x (a δ, a + δ) \ {a} 3. x a δ = x a δ, a + δ 2 Sequences Definition A sequence is an infinite ordered list of real numbers, denoted {a n }. 2.1 Limits of sequences Definition A number L is the limit of a sequence {a n } if for every ɛ > 0, there exists an N N such that if n N, then a n L < ɛ. Definition If a sequence has such a limit, then the sequence converges. If no such L exists, then the sequence diverges. Theorem Let {a n } be a sequence. Let L, M R such that lim an = L and lim an = M. Then M = L. Theorem Every convergent sequence is bounded. Definition A sequence {a n } is monotonic if and only if it satisfies any one of the following: 1. {a n } is increasing if a n < a n+1 for all n N 2. {a n } is non-decreasing if a n a n+1 for all n N 3. {a n } is decreasing if a n > a n+1 for all n N 4. {a n } is non-increasing if a n a n+1 for all n N Theorem Monotone Convergence Theorem If {a n } is non-decreasing and bounded above, then {a n } converges. Remark The Least upper bound property and the Monotone convergence theorem are equivalent. Corollary A monotonic sequence {a n } converges if and only if it is bounded. Remark If {a n } is non-decreasing, then either 1. {a n } is bounded and hence converges. 2. {a n } diverges to. Definition Given a sequence {a n } and an N N, the set {a N, a N+1, a N+2,... } is termed a tail. Remark The following are equivalent: 1. lim an = L 2. For every ɛ > 0, the open interval (L ɛ, L + ɛ) contains a tail of {a n }. 3. For every ɛ > 0, The open interval (L ɛ, L + ɛ) contains all but finitely many terms of {a n }. 4. Every open interval (a, b) containing L contains a tail of {a n }. 5. Every open interval (a, b) containing L contains all but finitely many terms of {a n }. 4

5 2.2 Series Definition Let {a n } be a sequence. Then a series with terms given by the sequence {a n } is a formal sum of the form a 1 + a 2 + a a n + = Definition For each k N, S k = n=1 a n is termed the k th partial sum of the series n=1 a n. The series converges if and only if {S k } converges as k. Theorem If {S k } diverges, then {a n } diverges. Definition Let r R. A geometric series with radius r is a series of the form 1 + r + r r n + = Theorem If r 1, then the geometric series with radius r will diverge. Theorem The series ar n converges to n=0 a 1 r Theorem Comparison Test Suppose that a n b n for all n N. 1. if n=0 a n diverges, then n=0 b n diverges. 2. If n=0 b n converges, then n=0 a n converges. n=1 n=0 r n a n if and only if r < Subsequences Definition Given a sequence {a n } and an increasing sequence n 1 < n 2 < of the natural numbers, the sequence b k = a nk = {a n1, a n2,... } is termed a subsequence of {a n }. Theorem If {a n } converges to L, then so does every subsequence of {a n }. Definition A point L is termed a limit point of a sequence if there exists a subsequence {a nk } of {a n } such that {a nk } converges to L. The set of all limits points of a sequence {a n } is denoted by Lim{a n }. Definition Let {a n } be a sequence. An element n N is termed a peak point of {a n } if a n < a n for all n > n. Theorem Peak Point Lemma Every sequence {a n } has a monotonic subsequence. Theorem Bolzano-Weierstrass Theorem Every bounded sequence {a n } has a convergent subsequence. Theorem Suppose that for sequences {a n }, {b n } we have lim 1. lim c an = c L for every c R 2. lim an + b n = L + M 3. lim an b n = L M 1 4. lim = 1 for M 0 b n M an 5. lim = L for M 0 M b n an = L and lim bn = M. Then 5

6 Theorem Suppose that for sequences {a n }, {b n }, the limit lim Then lim an = 0. an b n exists, and lim bn = 0. Theorem Squeeze Theorem for Sequences Suppose that {a n }, {b n }, {c n } are sequences with a n b n c n for all n N. Also suppose that lim an = L = lim cn. Then lim bn exists and lim bn = L. 2.4 Cauchy sequences Definition A sequence {a n } is Cauchy if for every ɛ > 0, there exists an N N such that if n, m N, then a n a m < ɛ. Theorem Every convergent sequence is Cauchy. Theorem Every Cauchy sequence is bounded. Theorem Suppose {a n } is Cauchy. Suppose that {a nk } is a subsequence of {a n } with a nk L. Then a n L. Theorem Completeness Theorem Every Cauchy sequence {a n } R converges. Theorem The following are equivalent in R: 1. Least upper bound property 2. Monotone convergence theorem 3. Bolzano-Weierstrass theorem 4. Completeness theorem 3 Limits and continuity 3.1 Limits of functions Definition A number L is the limit of a function as x a, if for every ɛ > 0, there exists a δ > 0 such that if 0 < x a < δ, then L < ɛ. Then we denote the limit by lim = L. Theorem Sequential Characterization of Limits The following are equivalent: 1. lim = L 2. Whenever {x n } is a sequence with x n a and x n a, then f(x n ) L. Theorem Suppose that for function, g we have lim = L and lim g = M. Then 1. lim c = c L for every c R 2. lim + g = L + M 3. lim g = L M 1 4. lim = 1 for M 0 g M 5. lim = L for M 0 g M 6

7 Theorem Suppose that for functions, g, the limit lim Then lim = 0. exists, and lim g = 0. g Definition A rational function is a function = p q, where p, q are polynomials. Theorem Algorithm for Rational Functions Let = p q. Then lim p 1. If q(a) 0, then lim = = p(a) q q(a) = f(a) lim 2. If q(a) = 0 and p(a) 0, then lim does not exist. 3. If q(a) = 0 and p(a) = 0, then p q = (x a)p1 (x a)q 1, and repeat 1. with p1 q 1. Theorem Squeeze Theorem for Functions Suppose that, g, h are functions with g h for all x in an open interval I containing x = a, except possibly at x = a. Also suppose that lim = L = lim h. Then lim g exists and lim g = L. 3.2 One-sided limits Definition A number L is the limit of a function as x a from above (the right), if for every ɛ > 0, there exists a δ > 0 such that if 0 < x a < δ, then a < ɛ. This implies that x (a, a + δ). This limit is denoted by lim = L. + Definition A number L is the limit of a function as x a from below (the left), if for every ɛ > 0, there exists a δ > 0 such that if 0 < x a < δ, then a < ɛ. This implies that x (a δ, a). This limit is denoted by lim = L. Theorem The following are equivalent: 1. lim = L 1. lim = lim = L + Remark One-sided limits have the same arithmetic rules, sequential characterization, and satisfy the squeeze theorem just as the two-sided limits. Definition A function f : R R is termed even if = f( x) The graph of f is symmetric about the y-axis. Definition A function f : R R is termed odd if = f( x) The graph of f is symmetric about the origin. Remark With respect to the above definitions, If is even, then lim exists if and only if x 0 If is odd, then lim x 0 exists if and only if Theorem Fundamental Trigonometric Limit sin lim = 1 x 0 x lim exists or + lim = lim. + lim exists. 7

8 Corollary If x is small, i.e. x 1, then sin x tan. Definition lim = if for every m > 0, there exists a δ > 0 such that if 0 < x a < δ, then > m. + lim = if for every m < 0, there exists a δ > 0 such that if 0 < x a < δ, then < m. + lim = if for every m > 0, there exists a δ > 0 such that if 0 < a x < δ, then > m. lim = if for every m < 0, there exists a δ > 0 such that if 0 < a x < δ, then < m. Definition A function has a vertical asymptote at x = a if either one of the following hold: lim = ± or lim = ± Continuity Definition A function is continuous at x = a if 1. lim exists 2. lim = f(a) Otherwise is termed discontinuous. Definition A function is continuous at x = a if for every ɛ > 0, there exists a δ > 0 such that if x a < δ, then f(a) < ɛ. Theorem Sequential Characterization of Continuity The following are equivalent: 1. is continuous at x = a 2. Whenever {x n } is a sequence with x n a, then f(x n ) f(a). Theorem Suppose that, g are continuous at x = a. Then 1. c is continuous at x = a for all c R 2. + g is continuous at x = a 3. g is continuous at x = a 4. g is continuous at x = a if g(a) 0 Definition A function is continuous on a, b if 1. is continuous at each x a, b 2. lim + = f(a) 3. lim x b = f(b) Definition Let f : S R with S R. Then is continuous on S (relative to S) if whenever {x n } S with x n x for x S, then f(x n ) f(x ). 3.4 Compositions of functions Definition If the range of a function f is the subset of the domain of a function g, then define the composition of f and g by h = g(). If f : X Y and g : Y Z, then h : X Z. Theorem Suppose that is continuous at x = a, and that g(y) is continuous at y = f(a). Then the function g() = g f is continuous at x = a. Remark If lim = L and lim g(y) = M, then it is not necessarily true that lim g = M. y L However, if lim = L and g(y) is continuous at y = L, then lim g = g(l). 8

9 3.5 Discontinuity Definition A function has a discontinuity at x = a of is defined on an open interval I containing a except possibly at x = a, and is not continuous at x = a. Let D(f) denote the collection of all points of discontinuity of f. Definition There are two types of discontinuities: 1. If lim exists, but either f(a) is not defined, or f(a) = lim, then x = a is termed a removable discontinuity. 2. If a D(f)such that lim Jump discontinuity Vertical asymptote discontinuity Oscillatory discontinuity does not exist, then a is termed an essential discontinuity of f. These are the types of essential discontinuities Remark A function with a removable (non-essential) discontinuity may be redefined at the point of discontinuity to create a continuous function, but a function with an essential discontinuity cannot be made continuous by redefining the function at the point of discontinuity. 3.6 Value theorems Theorem Intermediate Value Theorem Suppose that is continuous on a, b such that f(a) < 0 and f(b) > 0. Then there exists some c (a, b) such that f(c) = 0. Corollary Suppose that is continuous on a, b. c (a, b) such that f(c) = α. If f(a) < α < f(b), then there exists some Corollary Suppose that is continuous on an interval I. interval. Then f(i) = { : x I} is an Definition A function is termed uniformly continuous on S R if for every ɛ > 0, there exists a δ > 0 such that for x a, x 2 S if x 1 x 2 < δ, then f(x 1 ) f(x 2 ) < ɛ. Theorem Sequential Characterization of Uniform Continuity Let S R and f : S R. Then the following are equivalent: 1. is uniformly continuous on S. 1. If {x n }, {y n } are sequences in S, and lim xn y n = 0, then lim f(xn ) f(y n ) = 0. Theorem Suppose that is uniformly continuous on S R. If {x n } S is Cauchy, then {f(x n )} is Cauchy. Theorem Suppose that is continuous on a, b. Then is uniformly continuous on a, b. Theorem Suppose that is continuous on (a, b). Then is uniformly continuous on (a, b) if and only if lim and lim exist. + x b Theorem Extreme Value Theorem Suppose that is continuous on a, b. Then there exist c, d a, b such that f(d) for all x a, b. Corollary If is continuous on a, b, then f(a, b) = {; s a, b} is a closed interval. This is equal to f(c), f(d). 9

10 4 Differentiation 4.1 Derivatives Definition Given and a point x = a R, the quantity f(a) x a is termed Newton s quotient for centered at x = a. f(a) Definition A function is termed differentiable at x = a if lim exists. Then it is x a f(a) expressed as f (a) = lim. If the limit does not exist, then the function is not differentiable at x a x = a. Definition If a function is differentiable at x = a, then the line y = f(a) + f (a)(x a) is termed the tangent line to the graph of through (a, f(a)). This is also termed the linear approximation of at x = a, and denoted by L a. Theorem If is differentiable at x = a, then is continuous at x = a. Theorem Suppose that, g are differentiable at x = a. Then 1. c is differentiable at x = a for all c R with d dx c = c d dx 2. If h = + g, then h is differentiable at x = a with h (a) = f (a) + g (a) 3. If h = g, then h is differentiable at x = a with h (a) = f (a)g(a) + f(a)g (a) Theorem Chain Rule Suppose is defined upon an open interval I containing x = a, and g(y) is defined upon an open interval J with f(i) J. Suppose that is differentiable at x = a, and that g(y) is differentiable at y = f(a). Then h = g is differentiable at x = a with h (a) = g (f(a))f (a). Definition For a function, an element x = c is termed a local maximum if there exists an open interval I = (a, b) containing c such that f(c) for all x (a, b). SImilarly, an element x = d is termed a local minimum if there exists an open interval I = (a, b) containing d such that f(d) for all x (a, b). The two together are termed local extrema. Theorem Suppose that x = c is either a local extremum for. If f (c) exists, then f (c) = 0. Definition A point x = c is termed a critical point of if is defined upon an open interval I containing c, and either 1. f (c) = 0 2. is not differentiable at x = c. Remark If is continuous on a, b, then each of the global extrema (max/min) will be either at 1. an endpoint 2. a critical point in (a, b), i.e. an interior critical point 4.2 Inverse functions Definition Suppose that f : X Y is a bijection. Define a function g : Y X by g(y) = x if and only if y = g. Then the function g(y) is termed the inverse of and is denoted by f 1. Definition If f : S R with S R, then is invertible on S if f is one-to-one on S. Then define g : f(s) S by g(y) = x if and only if x S and = y. Definition A function f : S R is said to be (strictly) increasing on S if whenever x 1, x 2 S with x 1 < x 2, then ( f(x 1 ) < f(x 2 ) ) f(x 1 ) f(x 2 ). Similarly, f is (strictly) decreasing on S if whenever x 1, x 2 S with x 1 < x 2, then ( f(x 1 ) > f(x 2 ) ) f(x 1 ) f(x 2 ). If f is either (strictly) increasing or (strictly) decreasing on S, then f is monotonic on S. 10

11 Remark If is strictly increasing/decreasing on S, then f is 1 1 on S, and hence invertible. Theorem Suppose that is continuous on an interval I and also 1 1 on I. Then either 1. is strictly increasing on I 2. is strictly decreasing on I Thus continuous functions tend to be precisely 1 1 when they are monotonic. Theorem Suppose that is strictly increasing/decreasing on a, b. Then is continuous on a, b if and only if f(a, b) = f(a), f(b), i.e. if and only if f(a, b) is an interval. Remark Suppose that f : S R is strictly increasing/ decreasing. Let T = f(s). Then f is invertible on S. Let g : T S be the inverse of f. Then g(y) is strictly increasing/decreasing. Corollary Let I be an interval and f : I R be strictly increasing/decreasing with J = f(i). Then is continuous on I if and only if g : J I (where g = f 1 ) is also continuous. Remark The graph of and its inverse are essentially the same, as (x, ) (g(y), y). To graph g(y) in the usual orientation, we switch x and y. Theorem Inverse Function Theorem Suppose that is defined on a continuous and open interval I containing x. Suppose that is either strictly increasing or strictly decreasing on I with inverse g : J = f(i) I. Then if is differentiable at x with f (x ) 0, then g(y) is differentiable at y = f(x ) with g (y ) = 1 f (x ) e n 1 Definition Let e be the unique base such that lim = 1. Then = e x = f 1. n 0 n 4.3 Mean value theorem Theorem Rolle s Theorem If is continuous on a, b and differentiable on (a, b) with f(a) = f(b), then there exists a c (a, b) with f (c) = 0. Theorem Mean value theorem If is continuous on a, b and differentiable on (a, b), then there exists a c (a, b) with f (c) = f(b) f(a). b a Theorem Suppose that, g are continuous on some closed interval a, b and differentiable on (a, b). If f = g for all x (a, b), then there exists some m R such that g = + m for all x a, b. Theorem Increasing Function Theorem Suppose that, g are continuous on some closed interval a, b and differentiable on (a, b) with f > 0 for all x (a, b). Then is strictly increasing on a, b. Remark The increasing function theorem holds on the interval a, b if there exist at most finitely many points x 1,..., x n a, b such that f(x i ) = 0 for 1 i n. Theorem Suppose that, g are continuous on a, b and differentiable on (a, b). Also suppose that m f M for all x (a, b). Then f(a) + m(x a) f(a) + M(x a) for all x (a, b). Theorem Suppose that is defined on an interval I with M for all x I. Then is uniformly continuous on I. 11

12 Theorem First Derivative Test Suppose that c is a critical point of., and is continuous at x = c. 1. Suppose that there exists an open interval (a, b) containing c such that f 0 for all x (a, c) and f 0 for all x (c, b). Then c is a local min of. 2. Suppose that there exists an open interval (a, b) containing c such that f 0 for all x (a, c) and f 0 for all x (c, b). Then c is a local max of. 4.4 Concavity Definition Let I R be an interval. 1. A function is concave upward on I if for each a, b I with a < b, the secant line joining (a, f(a)) and (b, f(b)) sits above the graph of on a, b. 2. A function is concave downward on I if for each a, b I with a < b, the secant line joining (a, f(a)) and (b, f(b)) sits below the graph of on a, b. Definition Suppose that is differentiable on I with derivative function f. If a I and f differentiable at x = a, then the quantity f f (a) lim = f x a is termed the second derivative at x = a. In general, f ( n) := d dx f n 1. Theorem Suppose that is such that f > 0 for all x I. Then is concave upward on I. If f < 0 on I, then is concave downward on I. Theorem Second Derivative Test Suppose that c is a critical point of, and is continuous at x = c. 1. Suppose that f is continuous at x = c and f (c) > 0. Then c is a local min for. 2. Suppose that f is continuous at x = c and f (c) < 0. Then c is a local max for. Definition A point x = c is termed a point of inflection for is is continuous at x = c and if there exists an open interval (a, b) containing c such that either 1. is concave upward on (a, c) and concave downward on (c, b) 2. is concave downward on (a, c) and concave upward on (c, b) Definition An extended real number is an element in the set of numbers R = R {± }. Theorem L Hôpital s Rule Suppose that f, g : (a, b) R for extended real numbers a, b and a < b. Suppose that f, g are differentiable on (a, b) with g 0 and g 0 for all x (a, b). f 1. Suppose that lim = 0 = lim g. Then if lim g = L for L R, then lim = L + g f 2. Suppose that lim = 0 = lim g. Then if lim x b x b x b g = L for L R, then lim = L x b g 3. Suppose that lim = ± and lim g = ±. + + f Then if lim + g = L for L R, then lim = L + g 4. Suppose that lim = ± and lim g = ±. x b x b f Then if lim x b g = L for L R, then lim = L x b g Theorem Cauchy Mean Value Theorem Assume that, g are continuous on a, b and differentiable on (a, b) with g 0 for all x (a, b). Then there exists a c (a, b) such that f (c) g (c) f(b) f(a) =. b a 12

13 5 Taylor s Theorem Definition Suppose that is n times differentiable at x = a. Then the n-th degree Taylor polynomial for centered at x = a is P n,a = n k=0 f k (a) (x a) k k! Definition Suppose that is n times differentiable at x = a. Then the error term in approximating using the n-th degree Taylor polynomial is R n,a = P n,a Theorem Taylor s Theorem Suppose that is n times differentiable on an open interval I containing x = a. Then for every x I and x a, there exists some c (x, a) such that R n,a = P n,a = f (n+1) (c) (x a)n 1 n Big O notation Definition A function is big O of g as x a if there exists a δ > 0 and m > 0 such that M g for all x (a δ, a + δ), except possibly at x = a. This is expressed as = O(g) as x a. Definition For functions, g, the equation = g + O(x n ) holds if g = O(x n ). Theorem Suppose that = O(x n ) and g = O(x m ). Then 1. c = O(x n ) = c O(x n ) = O(x n ) 2. + g = O(x n ) + O(x m ) = O(x k ) such that k = min{n, m} 3. g = O(x n )O(x m ) = O(x n+m ) 4. x k = x k O(x n ) = O(x k+n ) 5. If n m, then g = O(x n ) 6. If n 1, then 1 x = 1 x O(xn ) = O(x n 1 ) Theorem Suppose that f (n+1) is continuous on δ, δ. Then = P n,a + O(x n+1 ). Theorem If f (n+1) is continuous on δ, δ and if p is a polynomial of degree n or less with = p + O(x n+1 ), then p = P n,0. Theorem If p is a polynomial of degree n for n N {0} and p = O(x n+1 ), then p = 0 for all x. 13