Calculus: Analytic Geometry and Calculus, with Vectors

306 Functions, graphs, and numbers To prove the first part of this theorem, let f'(xo) = 0 and let f"(xo) = p, where p is a posi ...

5.3 Second derivatives, convexity, andfexpoints 307 calculate f'(x) and f"(x). Show that the graph of f has exactly one flexpoin ...

(^308) Functions, graphs, and numbers supposing that n > 1, show that the graph has a flexpoint at the point (xn,yn) for whic ...

5.3 Second derivatives, convexity, andfexpoints 309 and hence that the line having the equation y = x is an asymptote of G. grap ...

(^310) Functions, graphs, and numbers Figure 5.392 Tell why it is true that, as Figure 5.392 indicates, (P is increasing over th ...

5.3 Second derivatives, convexity, and fiexpoints 311 17 Make a sketch showing the points (x,y) on the graph of the equations x= ...

(^312) Functions, graphs, and numbers where h(x) = xf'(x) - f(x). We will know that g is decreasing over the interval 0 < x & ...

5.4 Theorems about continuous and differentiable functions 313 and is decreasing over the interval < x < 1, and why B4(x) ...

314 Functions, graphs, and numbers course in calculus to explore these matters thoroughly. In order to understand the proofs, it ...

5.4 Theorems about continuous and differentiable functions 315 If(x) I < f(b) + 1 when b - S < x < b. Since b - & b ...

316 Functions, graphs, and numbers If f(a) = M or f(b) = M, we set x2 = a or x2 = b. Otherwise, assuming that f(a) < M and f( ...

5.4 Theorems about continuous and differentiable functions 317 and is positive at x2, and if f is continuous over the closed int ...

318 Functions, graphs, and numbers for which If(x) 1 5 M when -1 5 x 5 1. Let x = 1/ lam. Then x F-- 0 and-1-<-<x<_1,bu ...

5.4 Theorems about continuous and differentiable functions 319 and setting y = 0 gives xn.F.I = xn AX-) f,(xn) When the method i ...

320 Functions, graphs, and numbers 14 Prove the Archimedes property of numbers: if e > 0 and a > 0, then there is an integ ...

5.4 Theorems about continuous and differentiable functions^321 18 Prove that if x is a number, then there exist an integer N and ...

322 Functions, graphs, and numbers Hence (5) shows that d2 is a digit and (11) d2<102(B-ip)<d2+1. Dividing by 102 and tran ...

5.4 Theorems about continuous and differentiable functions 323 Observe that if n is a positive integer and x = 1/-\/2na, then I ...

324 Functions, graphs, and numbers constant c for which f(x) = c when a < x < b? Does the hypothesis that Gf"(x) = 0 when ...

5.5 The Rolle theorem and the mean-value theorem 325 Figure 5.55 joining the points (a, f(a)) and (b, f(b)) of Figure 5.55. The ...