Overview of Calculus


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1 Overview of Clculus June 6, Limits Clculus begins with the notion of limit. In symbols, lim f(x) = L x c In wors, however close you emn tht the function f evlute t x, f(x), to be to the limit L (usully clle ɛ), we cn etermine now close x must be to c (usully clle δ) to meet the emn. A function f is clle continuous t c if lim f(x) = f(c). x c f is clle continuous if it is continuous for ll x in its omin. 2 Slopes n Derivitives Lines hve slopes. If then the slope f(x) = mx + b, y f(x + ) f(x) (m(x + ) + b) (mx + b) = = = m. (x + ) x Other thn lines the ifference quotient f(x + ) f(x), clle the slope of secnt line, is not constnt. If this quotient hs limit t 0, lim 0 f(x + ) f(x), then we sy tht f is ifferentible t x. We write the limit f (x), f x (x), Df(x), or y x. 1
2 n sy the erivtive of f t x. f is clle ifferentible if it is ifferentible for ll x in its omin. for exmple, for x ner vlue x 0, we hve the liner pproximtin f(x) f(x o )f (x 0 )(x x 0 ). If f (x) is positive (negtive), the we sy tht f is incresing (ecresing) t x. We use severl phrses to provie n intuitive mening to the erivtive. For exmple, the erivtive f (x) is the instntneous rte of chnge of f t x. the slope of the tngent line of f t x. For exmple, if x(t) is the position of n object t time t, then v(t) = x (t) is the velocity n (t) = v (t) is the ccelertion t time t. 3 Rules for Differentition Let f n g be ifferentible functions n b n b be constnts The linerity of the erivtive (Sum rule) x (f(x) + bg(x)) = f (x) + bg (x) Prouct rule Quotient rule Chin rule x (f(x) g(x)) = f(x)g (x) + g(x)f (x) f(x) x g(x) = g(x)f (x) f(x)g (x) g(x) 2 (f g) (x) = x (f(g(x)) = f (g(x))g (x) 4 Derivtives of Common Functions Powers For ny rel number p, Nturl logrithm x xp = px p 1. x ln x = 1 x. 2
3 Exponentil functions For ny rel number b > 0 In prticulr if b = e, Euler s constnt, Trigonometric functions x bx = b x ln b, x ex = e x, sin x = cos x, x 5 Definite Integrls cos x = sin x, x 1 tn x = sec x = x cos 2 x. We will let f be continuous n boune function on the intervl [, b], the gol is to efine the integrl b f(x)x in such wy tht its intuitive mening for positive functions is tht the integrl is the re below the function f, bove the xxis, to the fight of the verticl line x = n to the right of the line x = b. (The function tht ppers in the integrl is clle the integrn.) To efine these sums, first ivie the intervl into n contiguous subintervls of length For x i = b n. in the ith subintervl, i = 1, 2,..., n, Riemnn sum f(x i ). We focus on two such types of sums, nmely, upper n lower Riemnn sums. To efine these, choose the minimum vlue m i n the mximum vlue M i for the ith subintervl, i = 1, 2,..., n. Then the lower Riemnn sum f(m i ). is the re of n rectngles tht sit on the yxis unerneth the grph of f. The upper Riemnn sum f(m i ). is the re of n rectngles tht sit on the yxis bove the grph of f. Thus, the lower Riemnn sum is n unerestimte of the integrl n the upper Riemnn sum is n overestimte. However, s we refine the estimtes, the lower Riemnn sum increses n the upper Riemnn ecreses. Moreover, the ifference between these two sums converges to 0 in the limit s 0. 3
4 6 Funmentl Theorem of Clculus We nee prcticl wy to compute the integrl tht is more efficient thn computing Riemnn sums. We begin by efining function A(x) = x f(t)t, the integrl up to vlue x. Next, let s exmine the chenge in F over smll intervl of length. A(x + ) A(x) = x+ x f(t)t f(x), n A(x + ) A(x) f(x). This pproximtion becomes n equlity in the limit s 0. Thus, A A(x + ) A(x) (x) = lim = f(x). 0 We cll A n ntierivtive of f. If F is nother ntierivitive of f, then F (x) A (x) = f(x) f(x) = 0. The only function tht re zero re the constnt functions. In this cse, the constnt n c = F () A() = F (). F (b) F () = A(b) = b f(x)x.. Written in this wy, we cll this efinite integrl. Antierivtives re lso clle inefinite integrls n re written f(x)x = F (x) + c The ition of the constnt c is written to remin us tht ntierivtives re fmily of functions, ech one iffers from nother by constnt function. 7 Techniques for Integrtion Not surprisingly, rules for tking erivtives provies techniques for integrtion, Agin, let f, g, u, v, n w be continuous functions n b n b be constnts The linerity of the integrl (f(x) + bg(x)) = f(x)x + b g(x)x + c 4
5 Integrtion by prts u(x)v (x)x = u(x)v(x) v(x)u (x)x + c w substitution f(w(x))w (x)x = f(w)w + c 8 Exmples Integrtion by prts 1. xe x x = x( e x ) ( e x )(1)x = xe x e x + c = (x + 1)e x + c u(x) = x u (x) = 1 v(x) = e x v (x) = e x 2. w substitution ln xx = x ln x x 1 x = x ln x x u(x) = ln x v(x) = x u (x) = 1 x v (x) = 1 1x = x ln x x = x(ln x 1) + c 1. 2 cos x sin xw = 2ww = w 2 + c = sin 2 x + c or 2. w(x) = sin x 2 cos x sin xw = w (x) = cos x 2ww = w 2 + c = cos 2 x + c w(x) = cos x w (x) = sin x Notice tht the two ntierivtives sin 2 x ( cos 2 x) iffer by constnt, nmely c = 1. 5
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