The Art and Craft of Problem Solving

324 CHAPTER 9 CALCULUS For example, f(x) : = l/x is continuous on (0,5), but achieves neither maximum nor minimum values on this ...

9.2 CONVERGENCE AND CONTINUITY 325 Uniform continuity is just what we need to complete our proof of the FTC. Example 9.2.7 Show ...

326 CHAPTER 9 CALCULUS 9.2. 13 Let (an) be a (possibly infinite) sequence of positive integers. A creature like ao + -----;--- a ...

9.2.20 Draw two nonintersecting circles on a piece of paper. Show that it is possible to draw a straight line that divides each ...

328 CHAPTER 9 CALCULUS 00 9.2.34 (Putnam 1994) Let (an) be a sequence ofposi­ tive reals such that. for all n. an ::; a 2 n + a ...

9.3 DIFFERENTIATION AND INTEGRATION 329 that at relative minima, the second derivative is non-negative. Thus p"(a) 2 o. But p(a) ...

330 CHAPTER 9 CALCULUS as desired. • The tangent-line definition of the derivative stems from the formal definition of the deriv ...

4 (^3). 5 3 2.5 2 (^1). 5 (^0). 5 (^0). 5 9.3 DIFFERENTIATION AND INTEGRATION 331 y =j(x) slope =<X (^1). 5 2 Therefore, the ...

332 CHAPTER 9 CALCULUS a u b Rolle's theorem has an important generalization, the mean value theorem. If f(x) is continuous on [ ...

v ,�,.�� � ·.:·: •• J'/OA . . . ..... .. .. .... �.�:,f 9.3 DIFFERENTIATION AND INTEGRATION 333 y = f( x) (fantasy) . .. .• • ...

334 CHAPTER 9 CALCULUS compute the values of the derivatives f(k) ( (^0) ), k = 1,2,3, .... (We are using the nota­ tion f(k) fo ...

9.3 DIFFERENTIATION AND INTEGRATION 335 n Example 9.3.6 Logarithmic D ifferentiation. Let f(x) = 1] (x+k). Find f'(I). Solution: ...

336 CHAPTER 9 CALCULUS for all x 2 o. This certainly implies that [(x)^2 � [(Of + [' (0)^2. Thus there is a constant C := J [(0) ...

9.3 DIFFERENTIATION AND INTEGRATION 337 If k ranges from 1 to n, then k^2 / n^2 ranges from (^1) / n^2 to 1, which suggests that ...

338 CHAPTER 9 CALCULUS Symmetry and Transformations Problem 3.1 .26 on page 73 asked for the evaluation of the preposterously na ...

9.3 DIFFERENTIATION AND INTEGRATION 339 Notice that x � I /(I-x) maps 1 / 11 and 1 / 101 respectively to 11 / 10 and 101/ 100. T ...

340 CHAPTER 9 CALCULUS four distinct points. 9.3. 16 Convert the statement "the tangent line is the best linear approximation to ...

y �--------------------�----� x 9.3.28 (Putnam 2(02) Let k be a fixed positive inte­ ger. The nth derivative of I/(J!< -I) ha ...

342 CHAPTER 9 CALCULUS where a> I. 9.3.44 The Schwarz Inequality. The Cauchy-Schwarz inequality has many generalizations. Her ...

9.4 POWER SERIES AND EULERIAN MATHEMATICS 343 provided that this limit exists. Note that the an (x) are all defined for all real ...