f a L Most reasonable functions are continuous, as seen in the following theorem:


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1 Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f f L L L The grph on the left is continuous t ; the other two grphs re not. More precisely, Definition. A function f : R R is continuous t if lim x f(x) = f(). f is clle continuous if it is continuous t every point in its omin. Most resonble functions re continuous, s seen in the following theorem: Theorem.2 Any function which is the sum, ifference, prouct, composition, n/or quotient of functions me up of constnts, powers of x, sines, cosines, rcsines, rctngents, exponentils n/or logrithms is continuous everywhere except where ny enomintor is zero. This theorem suggests tht to evlute most limits, you shoul strt by plugging in for x. If you get number, tht is usully the nswer. When you plug in for x n you get 0 in enomintor, you hve to work bit hrer to evlute limit. When you get nonzero 0 : nswer is ± (this inictes the presence of verticl symptote t x = ); check the signs of the top n bottom crefully to see whether the nswer is or. When you get 0 0 : nswer coul be nything Techniques for eling with 0 0 : Fctor n cncel Conjugte squre roots Cler enomintors of insie frctions L Hôpitl s Rule (see below)
2 Theorem.3 (L Hôpitl s Rule) Suppose f n g re ifferentible functions. either lim f(x) = lim g(x) = 0 or lim f(x) = lim g(x) = ±. x x x x Then: f(x) L f (x) lim = lim x g(x) x g (x). Suppose lso tht Limits t infinity: Suppose f : R R. To sy lim f(x) = L x mens tht s x gets lrger n lrger without boun, then f(x) gets closer n closer to L; equivlently this mens y = L is horizontl symptote of f. To evlute limit t infinity, plug in for x n use the following rithmetic rules with : { ± c = = if c > 0 = c = if c < 0 = ln = e = c = { if c > 0 0 if c < 0 c = 0 0 = ± c = ± so long s c 0 0 In the lst two situtions, you nee to nlyze the sign crefully to etermine whether the nswer is or. Wrning: The following expressions re ineterminte forms, i.e. they cn work out to be ifferent things epening on the context (these re evlute using L Hôpitl s Rule): Limits t infinity for rtionl functions cn be etermine immeitely by wy of the following theorem: Theorem.4 (Limits t infinity for rtionl functions) Suppose f is rtionl function, i.e. hs form Then: f(x) = mx m + m x m + m 2 x m x 2 + x + 0 b n x n + b n x n + b n 2 x n b 2 x 2 + b x + b 0.. If m < n (i.e. lrgest power in numertor < lrgest power in enomintor), then lim f(x) = 0. x 2. If m > n (i.e. lrgest power in numertor > lrgest power in enomintor), then lim x f(x) = ±. m 3. If m = n (i.e. lrgest powers in numertor n enomintor re equl), then lim f(x) = x b n.
3 2 Derivtives Definition 2. (Limit efinition of the erivtive) Let f : R R be function n let x be in the omin of f. If the limit f(x + h) f(x) lim h 0 h exists n is finite, sy tht f is ifferentible t x. In this cse, we cll the vlue of this limit the erivtive of f n enote it by f (x) or f y x or x. Differentible functions re smooth, i.e. re continuous n o not hve shrp corners, verticl tngencies or cusps. Assuming it exists, the erivtive f (x) computes:. the slope of the line tngent to f t x; 2. the slope of the grph of f t x; 3. the instntneous rte of chnge of y with respect to x; 4. the instntneous velocity t time x (if f is the position of the object t time x). We o not compute erivtives using the limit efinition given bove. We use ifferentition rules (given on the next two pges), which consist of list of functions whose erivtive we memorize n list of rules which tell us how to ifferentite more complicte functions me up of pieces whose erivtives we know. Derivtives hve mny pplictions. The most importnt (for Mth 230 purposes) is tht given ifferentible function f, you cn pproximte vlues of f ner using the tngent line to f t : Definition 2.2 Given ifferentible function f n number t which f is ifferentible, the tngent line to f t is the line whose eqution is y = f() + f ()(x ) (.k.. L(x) = f() + f ()(x )). For vlues of x ner, f(x) L(x); pproximting f(x) vi this proceure is clle liner pproximtion. You cn lso conclue mny things bout the grph of f from looking t its erivtive n its higherorer erivtives, s efine below: Definition 2.3 Let f : R R be function. The zeroth erivtive of f, sometimes enote f (0), is just the function f itself. The first erivtive of f, sometimes enote f () or y x, is just f. The secon erivtive of f, enote f or f (2) or 2 y x 2, is the erivtive of f : f = (f ). More generlly, the n th erivtive of f, enote f (n) or n y x n,is the erivtive of f (n ) : f (n) = ((((f ) ) )
4 The first erivtive of function mesures its tone (when f > 0, the function is incresing; when f < 0, the function is ecresing). The secon erivtive of function mesures its concvity (when f > 0, the function is concve up; when f < 0, the function is concve own). The secon erivtive of function which gives the position of n object gives the ccelertion of the object. Derivtives of functions tht you shoul memorize: Constnt Functions Power Rule Specil cses of the Power Rule: Trigonometric Functions Exponentil Function Nturl Log Function Inverse Trig Functions x (c) = 0 x (xn ) = nx n (so long s n 0) x (mx + b) = m x ( x) = 2 x ( ) x x = x 2 x (x2 ) = 2x x (sin x) = cos x x (cos x) = sin x x (tn x) = sec2 x x (cot x) = csc2 x x (sec x) = sec x tn x x (csc x) = csc x cot x x (ex ) = e x x (ln x) = x x (rctn x) = x 2 + x (rcsin x) = x 2 Rules tht tell you how to ifferentite more complicte functions: Sum Rule Difference Rule Constnt Multiple Rule Prouct Rule Quotient Rule Chin Rule (f + g) (x) = f (x) + g (x) (f g) (x) = f (x) g (x) (kf) (x) = k f (x) for ny constnt k (fg) (x) = f (x) g(x) + g (x) f(x) ( ) f g (x) = f (x) g(x) g (x) f(x) [g(x)] 2 (f g) (x) = f (g(x)) g(x)
5 3 Integrls Definition 3. Given function f : [, b] R, the efinite integrl of f from to b is b f(x) x = lim P 0 k= where the expression insie the limit is Riemnn sum for f. n f(c k ) x k, Note: In Mth 230, the limit bove lwys exists (but it oesn t lwys exist for crzy functions f... tke Mth 430 for more on tht). The efinite integrl of function is number which gives the signe re of the region between the grph of f n the xxis. Are bove the xxis is counte s positive re; re below the xxis is counte s negtive re. (A efinite integrl lso gives the isplcement of n object from time to time b, if the object s velocity t time x is the integrn f(x).) As with erivtives, we o not compute integrls with this efinition. We use the Funmentl Theorem of Clculus, which tells us to evlute integrls using ntierivtives: Definition 3.2 Given function f, n ntierivtive of f is function F such tht F = f. Every continuous function hs n ntierivtive (lthough you my not be ble to write its formul own); ny two ntierivtives of the sme function must iffer by constnt (so if you know one ntierivtive, you know them ll by ing +C to the one you know). Definition 3.3 Given function f, the inefinite integrl of f, enote f(x) x, is the set of ll ntierivtives of f. Theorem 3.4 (Funmentl Theorem of Clculus Prt II) Let f be continuous on [, b]. Suppose F is ny ntierivtive of f. Then b f(x) x = F (x) b = F (b) F (). Despite the similr nottion, f(x) x n b f(x) x re very ifferent objects. The first object is set of functions; the secon object is number.
6 Theorem 3.5 (Integrtion Rules to Memorize) 0 x = C M x = Mx + C x n x = xn+ + C (so long s n ) n + x = ln x + C (I on t cre so much bout the ) x sin x x = cos x + C cos x x = sin x + C sec 2 x x = tn x + C csc 2 x x = cot x + C sec x tn x x = sec x + C csc x cot x x = csc x + C e x x = e x + C x 2 x = rctn x + C + x x = rctn x + C x = rcsin x + C x 2 Theorem 3.6 (Linerity of Integrtion) Suppose f n g re integrble functions. Then: b b b [f(x) + g(x)] x = [f(x) g(x)] x = [k f(x)] x = k b b f(x) x + f(x) x b b b g(x) x; g(x) x; f(x) x for ny constnt k; (n the sme rules hol for inefinite integrls). Here re some other bsic properties of integrls: Definition 3.7 Let < b n let f : [, b] R be integrble. Then f(x) x = b b f(x) x n f(x) x = 0.
7 Theorem 3.8 (Aitivity property of integrls) Suppose f is integrble. Then for ny numbers, b n c, b f(x) x + c b f(x) x = c f(x) x Difficult integrls require more vnce techniques. usubstitutions re useful to integrte functions which re the prouct of two or more relte terms: Theorem 3.9 (Integrtion by u substitution  Inefinite Integrls) f(g(x)) g (x) x = f(u) u by setting u = g(x). Theorem 3.0 (Integrtion by u substitution  Definite Integrls) by setting u = g(x). b f(g(x)) g (x) x = g(b) g() f(u) u It is helpful to memorize the following ie, which comes from using usubstitution u = mx + b: Theorem 3. (Liner Replcement Principle) Suppose you know f(x) x = F (x) + C. Then for ny constnts m n b, f(mx + b) x = F (mx + b) + C. m Integrls whose integrns re proucts of unrelte terms re often compute using integrtion by prts: Theorem 3.2 (Integrtion by Prts (IBP) Formul) u v = uv v u. A lst technique useful to evlute some integrls is to fin the prtil frction ecomposition of the integrn; to use this technique, the enomintor of the integrn shoul be fctorble n the egree of the numertor shoul be less thn the egree of the enomintor.
8 4 Infinite Series An infinite series is n ttempt to n infinite list of numbers n = n= If the infinite list of numbers hs finite sum, we sy the series converges; otherwise we sy the series iverges. More formlly: Definition 4. Let n be n infinite series. For ech N, let S N be the N th prtil sum of the series; this is efine to be the sum of ll the n for which n N. Then:. If L is rel number such tht lim N S N = L, then we sy the infinite series converges (to L) n write n = L. In this setting L is clle the sum of the series. 2. If lim N S N = ± or if lim N S N DNE, then we sy the infinite series n iverges. Importnt: There is big ifference between sying n converges n sying n converges. Without the Σ, you ren t ing the numbers. Therefore, you shoul never omit the Σ when escribing whether or not n infinite series converges. We ivie convergent series into two clsses s follows: Definition 4.2 Let n be n infinite series. We sy the series is bsolutely convergent (or tht the series converges bsolutely) if n converges. If n converges but n iverges, then we sy n is conitionlly convergent (or tht the series converges conitionlly). The reson we cre whether series converges bsolutely or conitionlly is the following theorem: Theorem 4.3 (Rerrngement Theorem) Suppose n is n infinite series.. If n converges conitionlly, then the terms of tht series cn be rerrnge so tht the rerrnge series converges to ny number you like! (The series cn lso be rerrnge so tht the rerrnge series iverges.) 2. If n converges bsolutely to L, then no mtter how the terms of the series re regroupe or rerrnge, the rerrnge series still converges bsolutely to L. The mjor ppliction of series is to obtin lternte representtions of functions which cn be use for pproximtions: Definition 4.4 Suppose f is function which cn be ifferentite over n over gin t x = 0. The Tylor series (centere t 0) of f (.k.. Mclurin series of f is the power series f (n) (0) x n. n! If we truncte this series t the N th power term, we obtin prtil sum of the Tylor series clle the N th Tylor polynomil (centere t 0) of f. This polynomil is enote P N (x).
9 . P N (x) is polynomil of egree N; Generl properties of Tylor polynomils: 2. P 0 (x) is the constnt function of height f(0); 3. P (x) is the tngent line to f when x = 0; 4. P N (x) is the best N th egree polynomil pproximtion to f ner 0. The point of Tylor series n Tylor polynomils is tht if function f hs N th Tylor polynomil P N, then P N (x) f(x) (this pproximtion improves s N increses, but is pretty goo even for smll N). Theorem 4.5 (Uniqueness of power series) Suppose f is function which cn be ifferentite over n over gin t x = 0. Then if we write f s power series of the form then the coefficients n must stisfy f(x) = n x n, n = f (n) (0) n! for ll n. In other wors, the only power series of the form n x n which cn represent f is its Tylor series centere t 0. Commonly use Tylor series: e x = x n n! = + x + x2 2 + x3 3! +... (hols for ll x) sin x = cos x = ( ) n x 2n+ (2n+)! ( ) n x 2n (2n)! = x x3 3! + x5 5! x7 7! +... (hols for ll x) = x2 2 + x4 4! x6 6! +... (hols for ll x) x = x n = + x + x 2 + x (hols when x (, )) ln( + x) = n= rctn x = ( ) n+ n x n = x x2 2 + x3 3 x (hols when x (, ]) ( ) n 2n+ x2n+ = x x3 3 + x5 5 x (hols when x [, ])
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