Calculus: Analytic Geometry and Calculus, with Vectors

326 Functions, graphs, and numbers We now prove Theorem 5.27 the first part of which says that if f is continuous over an interv ...

5.5 The Rolle theorem and the mean-value theorem 327 S2, and Ix2 - xil < Ss. We may suppose thatxi 5 X. If xi and x2 both lie ...

328 Functions, graphs, and numbers where fg(x) dx stands for some particular function whose derivative is 9(x), and then so dete ...

5.5 The Rolle theorem and the mean-value theorem 329 (1) Theorem If f is such that f" exists over an interval containing a andx, ...

330 Functions, graphs, and numbers and the desired formula (2) follows from this. The generalized mean-value theorem has numerou ...

5.5 The Rolle theorem and the mean-value theorem 331 15 Supposing that n is a positive integer and x ; 1, differentiate 1 (1) ...

332 Functions, graphs, and numbers Choose x2 such that x2 > x, and g(x) > g(xi) when x > x2. It then follows from (7) t ...

5.5 The Rolle theorem and the mean-value theorem 333 Theorem If f is integrable (Riemann) over a =< x 5 b and if F'(x)= f(x) ...

334 Functions, graphs, and numbers people are prone to purchase only those that are required no matter what they cost, and being ...

5.6 Sequences, series, and decimals 335 represent felons in courts of law, but nevertheless our precious corpus of scientific in ...

336 Functions, graphs, and numbers If it happens that this sequence of partial sums converges to s, so that (5.621) n-- then we ...

5.6 Sequences, series, and decimals 337 The set S consisting of the numbers s1, s2, s3, is then nonempty and has the upper bound ...

338 Functions, graphs, and numbers bounded below by 0 and hence must have a limit. If we let L denote this limit, then L = Jim j ...

5.6 Sequences, series, and decimals 339 Proof of this fact can be based on the ordinary process by which "long division" is used ...

340 Functions, graphs, and numbers 2 Write an infinite decimal which converges to an irrational number between 0.43211 and 0.432 ...

5.6 Sequences, series, and decimals 341 (1646-1716) and others before Euler. A given series ul + u2 + u3 + is assigned the value ...

(^342) Functions, graphs, and numbers N where C = Z Isk - s4. If we choose Ni such that Ni > N and C/n < e/2 k=1 when n &g ...

5.6 Sequences, series, and decimals 343 Show that, for each n = 1, 2, 3, , (2) an < an+1 < bn+1 < b,. Tell why there mu ...

344 Functions, graphs, and numbers 0 < x < 1. It is now time to seek the moral of this story. If we are not sure whether t ...

5.7 Darboux sums and Riemann integrals 345 Figure 5.721 P. Figure 5.721 is available for inspection. For each choice of the poin ...