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A Tour of the Calculus by David Berlinski

By David Berlinski

In its greatest element, the calculus services as a celestial measuring tape, in a position to order the countless expanse of the universe. Time and area are given names, issues, and boundaries; possible intractable difficulties of movement, progress, and shape are lowered to answerable questions. Calculus used to be humanity's first try and signify the area and maybe its maximum meditation at the subject of continuity. Charts and graphs all through.

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Sample text

Xn = 2/ ( Λ " υ . , Xn). Theorem 2 implies that for every point t0, y0, y0, . . , 2/ow"1} of Γ there exists a solution y = ψ(ί) of equation (5) which satisfies the initial condi­ tions Λ<ο) = io\ k = 0, 1,. , n - 1, or, as it is called, a solution with initial conditions h, 2/0,2/0, · · · , tfS1"1^ (8) Further, any two solutions with initial values (8) coincide on the common part of their intervals of definition. , if / is linear with respect to the variables y, y, . . , 2/ ί η ~ υ , and if its coefficients are defined and continuous on the interval q\ < t < q2, then for any initial values t0, y0, y0, .

Xn, we obtain xk = y{k\ xn = y(n\ k = 1, . . , n - 1, (9) (10) Replacing the right-hand sides of (9) on the basis of (6) and the right-hand side of (10) on the basis of (5), which is satisfied by y, we obtain (7). Let us assume, conversely, that the functions xlf . . , xn satisfy (7); we shall then take x1 for y and show that y satisfies (5). Setting x1 = y in the first of the equations of (7), we obtain x2 = y. Substituting y for x2 in the second equation of (7), we obtain x3 = y. Continuing this con­ struction further, we arrive at the relations (6).

N, we arrive at the problem of determining the solution of (5) under the initial conditions

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