Download e-book for iPad: Calculus the Easy Way by Douglas Downing

By Douglas Downing

ISBN-10: 0812091418

ISBN-13: 9780812091410

Longtime favorites for either lecture room and self-teaching support, Barron's effortless manner sequence titles evaluate a wide selection of topics, featuring primary innovations in transparent, easy-to-understand language and examples. Calculus the simple manner covers all of the necessities of a first-year calculus direction, together with derivatives, integrals, trignometric services, normal logarithms, exponential features, and an creation to differential equations.

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Dx y (2-8) EQUATIONS WITH SEPARABLE VARIABLES 2-3] becomes after multiplication l)y y dx, X dx The equation is now + i/ equation of form (2-7) tinuous), for where the integrals are any rfu = dy = (2-9) 0. c (c > (2-10) 0). exact (except where is = = jXdx+ X or Y is 1^ dx ox + ^^dy ay solutions of (2-7) are given equation y' = JYdy, discon- (2-11) indefinite integrals of the functions. jX{x) dx An ij we can choose u Hence the + and we obtain the general solution as exact, x^ An 37 + = Xdx+ Y dy.

Conversely, ii y f(x) and z = = = = (1-48) then, necessarily, fix) that ls,f"(x) = = g'ix) = F[x,f{x),g(x)]; and y = f(x) satisfies Eq. (1-47). g(x), F[x,f(x),f'(x)], In general, a pair of equations ^= is Gix, ^= y, z), is a pair of functions y = f(x), z = interval of x, which satisfy (1-49) identically. = y e —X 3x -re, I z — e (1-49) y, z) called a system of two first order differential equations. a system is F{x, A solution of such g(x), defined in the same For example, —X — 3x e a solution of the system dy dx ^ ^ dz ""' dx "" since -e-^ + -e-^ - 36^"^ = e-^ + e^^ - 2(e-^ - e^^), Se^"^ = e-^ - e^^ - 2(6"^ + e^"").

Ri, y)dy = Q{xu / Jo y) dy, -2/1 uixi, yi) = u{xi, 0) + Q{xi, y) dy / (2-38) P{x,Q)dx+ Q{x,,y)dy. ^0 Similarly, from (2-37) = u{xi, yi) ^(O, yi) + P{x, y^) dx / (2-39) P(x, yi) dx + Q{0, y) dy. r, y) dx dy [P{x, / 2/i) - P{x, ^0 rx\ rxi \ Jo - rvi Jo Jo ~ oy (x, y) dy dx.

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Calculus the Easy Way by Douglas Downing

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