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# s_notes04 - V-l CALCULUS OF ONE-VARIABLE SCALAR FUNCTIONS...

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Unformatted text preview: V-l CALCULUS OF ONE-VARIABLE SCALAR FUNCTIONS Deﬁnition of limit. Let f(x) be a function deﬁned for all x near 0 except possibly at 0. We deﬁne the concept: lirnxno f(x) = L. (This deﬁnition will refer only to values of f(x) for x 7': 0. The value of f(x) for x = 0 doesn’t matter.) See picture on board, or ﬁgures [2-1a] and [2-lb] on p. 93. The deﬁnition: there exists a ﬁmnelﬁmctton e(t) for 0<t<d for some d (ﬁmnel means that e(t) decreases steadily to 0 as t—>0) such that 0 < ltl < d :3 lL—f(t)| < e(t). Derivative of f(x) at x=c (if it exists): Let D(h) = (f(c+h) ~— f(c)) / h. [D(h) is deﬁned for h¢0.] Then the derivative df/dx at c is limbo D(h), if this limit exists. The derivative ﬁmction f ’(x) is deﬁned by f '(c) = thdx at c as c varies. Continuity of f(x). Deﬁnition: f(x) is continuous at x=c if lichﬂx) = f(c). Differentiation rules. (f+g)’ = f ’ + g’; (fg)' = f 'g + fg'. Indeﬁnite integrals. An indeﬁnite integral of f(x) is a function F(x) whose derivative is f. We write jf(x)dx = F(x) + C with arbitrary constant C. V-2 Deﬁnite integral. Of f(x) over interval a _<_ x5 b. Notation: f f(x)dx. Deﬁnition: for a given (small) 8, divide interval [a,b] into small non-overlapping pieces, each of length < 8. Let n be the number of pieces. Let the piece lengths be Ax1,...,Ax.,. In each piece, choose an arbitrary sample point; let these points he x1*,. . . ,xn*. The quantity R85 == f(x1*)Ax1 + + f(xn*)Axn. R83 is called a Riemann sum of mesh 8 for f(x) on [a,b]. Consider an inﬁnite sequence of Riemann sums for which the successive meshes approach the limit 0. If, for every such sequence, the RS values in that sequence approach a limit, and if this limit has the same value L for each such sequence, no matter how the pieces and sample points are chosen, we say that the deﬁnite integral of f(x) on [a,b] exists and has the value L. Basic theorem: If f is a continuous function on [a,b], then the deﬁnite integral of f on [a,b] exists. Laws for deﬁnite integrals. (i) If the integral for f on [a,b] has the value L, then the integral for of (c a constant) on [a,b] exists and has the value cL. (ii) If the integrals for f and g on [a,b] exist and have the values L1 and L2, then the integral for f + g on [a,b] exists and has the value L; + Ll. Finding the value of a deﬁnite integral. There are two principal direct methods (i) getting an approximate value by using a computer to compute the values of Riemann sums for successively smaller meshes. (ii) ﬁnding an indeﬁnite integral F(x) for f(x), in which case the value of the deﬁnite integral is F(b) — F(a). This last fact is known as the ﬁmdamental theorem of integral calculus. V-3 LIMITS AND DERIVATIVES FOR VECTOR FUNCTIONS OF ONE REAL VARIABLE Let in) be a vector function of one variable. We deﬁne limit by using our deﬁnition of scalar limit above. We say that lim._,c§(t) = f, if and only if lichl i2 — 110)] = 0. We deﬁne derivative by using vector operations in close analogy to our deﬁnition of derivative for a scalar function. We let 15(11): (i/h)[R(c+h)—1i(c)] and then deﬁne dii/dt as limb.» 13(h). A graphical picture of this derivative is sometimes helpful. Such a picture is given as [3-1] on p. 95. In this picture, we view Using Cartesian coordinates and an iJ,k representation for a vector function is often helpful in computations. We get ﬁn) = f(t)i + g(t)I( + mm}. It follows from the deﬁnition for derivative that dﬁ /dt = f'(t)i + g'm} + rm). We deﬁne indeﬁnite integral in an exactly analogous way. In getting a ﬁnal general expression for an indeﬁnite integral, the arbitrary constant appears as an arbitrary constant for each coordinate or as a single arbitrary constant vector. For the deﬁnite integral, we have an analogous deﬁnition with the scalar operations in a Riemann sum becoming vector operations of multiplication by a scalar and addition of vectors, and with the ﬁnal limit becoming a vector limit. Note that in evaluating a deﬁnite integral, it will usually be simplest to use the i,i,k representation and evaluate the three resulting scalar integrals. V~4 Diﬂerentiation rules for vector operations. Sec (4.1) through (4.4) on page 98, and (4.9) On page 99. THEOREM ON UNIT VECTOR FUNCTIONS This theorem appears on pages 100-101. Note that a unit-vector function is a vector function of t where the value of the function is always a unit vector. The theorem tells us that at any point t = c, (i) the vector derivative at 0 must be orthogonal to the unit vector at c, and (ii) the magnitude of the vector derivative must equal the instantaneous angular rate of change with respect to t (at c) of the direction of the unit vector. This rate is measured in radians per unit increase in t. ...
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s_notes04 - V-l CALCULUS OF ONE-VARIABLE SCALAR FUNCTIONS...

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