Applied Calculus GS1001

Transcription

1 1 Applied Calculus GS1001 January 24, Numbers, sets 0.1 numbers natural numbers N (+) (0),1,2,3,... integers Z 3, 2, 1,0,... rational numbers m/n m,n Z n 0 real numbers R complex numbers C = {a+ib : a,b R} 0.2 inequalities real line a < b a less than b a left from b on real line a > b: a greater than b less/greater than or equal to a < b,b < c a < c transitive a > 0: a is positive. a < 0: a is negative. a 0: a is non-negative. a 0: a is non-positive. addition a < b, c < d a+c < b+d

2 2 Multiplication a < b, c > 0 ac < bc a < b, c < 0 ac > bc (additive inverse) a < b a > b multiplicative inverse & inequalities ab 0, a < b 1 a < 1 if ab < 0, b 1 a > 1 if ab > 0 b 1 ex ab > 0 1.a = 3 b = 2 3 < a = 3 b = < 1 3 ab < 0 3.a = 2 b = < 0 < Sets set= {objects with some property} set= {elements} {1,2,1}= {1,2} {elements : property} {1,2} = {2,1} empty set subset A B (my convention includes = ; otherwise proper subset!) B A A B (superset) intersection A B = {x A : x B} union A B = {x A or x B} difference A\B = {x A : x / B} indexed intersection / union Let {A i } i I be a family of sets, with index i in an index set I. indexed intersection i I A i = {x : x A i for all i I} indexed union i I A i = {x : x A i for some i I} product set A B = {(x,y) : x A, y B} A A = A n Ex. R R = R 2 number of elements A number of elements of A Ex {1,1} = 1, = 0, N =. Intervals Let a,b R with a b (a,b) = {x R : a < x < b} open i l (a,b] = {x R : a < x b} half open i l [a,b) = {x R : a x < b} half open i l

3 3 [a,b] = {x R : a x b} closed i l infinity > a a R negative infinity < a a R (a, ) = {x R : a < x} (,a) = {x R : x < a} [a, ) = {x R : a x} (,a] = {x R : x a} Note : [,...] and..., ] make no sense since, R Ex. Let I = R x, A x = [x, ). Then x I A x =, x I A x = R minimum, maximum, supremum, infimum Let A R, A. x = mina if x A, y A : y x. minimum x = maxa if x A, y A : y x. maximum M is upper bound for A if x A : x M. m is lower bound for A if x A : x m. x = supa, supremum of A, if x is an upper bound for A and y < x, y is not an upper bound for A (i.e., x is smallest upper bound). the supremum always exists in R { }! x = infa, infimum of A, if x is a lower bound for A and y > x, y is not a lower bound for A (i.e., x is largest lower bound). the infimum always exists in R { }! If x = supa A, then x = supa = maxa. If x = infa A, then x = infa=mina. Ex. A = [0,2]. mina = infa=0, supa = maxa = 2 A = [0,2). mina = infa=0, supa = 2, maxa does not exist A = (,2). mina does not exist, infa =, supa = 2, maxa does not exist 1 Functions and Limits 1.1 General properties Def : A,B sets. f : A B function from A to B assigns for each a A exactly one b B, b = f(a)

4 4 A domain 정의역 B target 공역 f(a) = {f(a) : a A} range치역 Def: A,B sets f : A Graph of f graph(f) = {(x,f(x)) : x A} A B S R 2 is = graph(f) with each vertical line 1 point Convention: If domain not specified, max domain ( 최대정의역 ) Ex. f(x) = 1 x 2 g(x) = 1 x 2 x x 2 x 2 x 2 x 0 x 2 0 x > 2 x 0,1 dom(f) = (2, ) dom(g) = R\{0,1} functions are equal only if their maximal domains are equal f(x) = x+3 (x+3)(x+1), g(x) = x+1 (x+1) 2, h(x) = (x+3)(x2 1) (x+1)(x 2 1), k(x) = (x+3)(x 2 +1) (x+1)(x 2 +1) k = f = g h dom(f) = dom(g) = R\{ 1} = dom(k) dom(h) = R\{±1}

5 5 Piecewise defined functions f(x) = 1 x if x 1 x 2 x > 1 conditious musn t overlap or for overlapping values x, same f(x) absolute value function { x x 0 x = x x < 0 Ex. 3 2 = = 2 0 = 0 Ex. {x R : x < 2} = ( 2,2) inequality x+y x + y Σx i Σ x i Even/odd Function Def f : A R even if x A : x A and f(x) = f( x) odd f(x) = f( x) graphs of even functions have y-axis symmetry graphs of odd functions have origin symmetry Ex. f(x) = x 2 on R, f(x) = f( x) even f(x) = x 2 on R + = (0, ) x > 0( dom(f)) x < 0(/ dom(f)) not even Increasing and Decreasing Function Def. f : A R, I A interval f increasing on I if f(x i ) < f(x j ) for x i < x j, x i I f decreasing on I if f(x i ) > f(x j ) for x i < x j, x i I (f increasing/decreasing if A =interval & I = A)

6 6 Ex. f(x) = x 2 decreasing on (,0], increasing on (0, ). Periodic Function Def. f : A R has period d if d > 0 and x A,x±d A and f(x±d) = f(x). f is periodic if it has a period. Usually minimal period, smallest d with such property, is meant. Ex. f(x) = sinx, d = 2π f(x) = tanx, d = π 1.2 Types of functions Linear functions f(x) = mx+b m slope quadratic functions f(x) = ax 2 +bx+c a 0 polynomial functions f(x) = a n x n +a n 1 x n 1 + +a 0 a i R Power functions f(x) = x a a R 1. a = n N + dom = R { 2.a = 1/n, n N + rootfunction n [0, ) n even x. dom(f) = R n odd

7 7 3. a = 1 reciprocal function 4. a = 0 dom(f) = (0, ) 5. a Q, a = m/n, n > 0 dom = [0, ) n even m > 0 (0, ) n even m < 0 R n odd m > 0 R\{0} n odd m < 0 6. a R\Q dom = (0, ) x a < 0 dom = [0, ) a > 0 x a := sup{x a : a Q, a < a} for x > 1,a > 0 x a := inf{x a : a Q, a < a} for (0 <)x < 1,a > 0 0 a := 0 for a > 0 x a := 1 x a for a < 0,x > 0. rational functions f(x) = P(x), P,Q polynomials Q(x) E.g. f(x) = 2x4 x 3 +1 x 3 +3 dom(f) = {x : Q(x) 0} algebraic functions +,,,

8 8 trigonometric functions sin(x),cos(x). dom = R tan(x) = sin(x) dom = R\{π/2+nπ : n Z} cos(x) are periodic minimal period of sin,cos = 2π minimal period of tan = π Exponential functions f(x) = a x, a > 0 constant special ex. a = e = e x = exp(x) a x := sup{a x : x < x,x Q} for a > 1 a x := inf{a x : x < x,x Q} for a < 1 log functions a > 0 a 1 f(x) = log a x if a = e, log e x = lnx natural log 1.3 Composition of functions f : A B g : B C g f(x) = g(f(x)) Def. g f(x) composition of f and g