The Art and Craft of Problem Solving

304 CHAPTER 8 GEOMETRY FOR AMERICANS how to construct an equilateral triangle, right? (If not, do Problems 8.2.28 and 8.3.3 1 !) ...

8.5 TRANSFORMATIONS 305 90 °. Given just the location of points A , B, e, show how to locate the points X, Y, Z. Solution: Begin ...

306 CHAPTER^8 GEOMETRY FOR AMERICANS Homotheties are a transfonnational way of thinking about similarity. Not all similari­ ties ...

8.5 TRANSFORMATIONS 307 of BX as X ranges freely about Cl. In other words, the centers are the image of Cl under the homothety w ...

308 CHAPTER 8 GEOMETRY FOR AMERICANS co, draw the two tangents to co. The two points of tangency determine a chord whose midpoin ...

(b) For any points X, Y , 8.5 TRANSFORMATIONS 309 X'y' = Xy r^2. OX·Oy (c) The image of any "circle" is a "circle." (d) The imag ...

310 CHAPTER 8 GEOMETRY FOR AMERICANS Actually, our proof is incomplete, since our diagram only handles the case when X is outsid ...

8.5 TRANSFORMATIONS 311 After the inversion, we still have four "circles," but the situation is easy to analyze. Circles c 2 and ...

312 CHAPTER 8 GEOMETRY FOR AMERICANS of CD, B' the midpoint of DE, C' the midpoint of EA, D' the midpoint of AB, and E' the midp ...

hinges, with fixed locations along FB and BP, respec­ tively, so that FA = Api and AB = piC = CPo At P and pi, pens (or etching ...

314 CHAPTER 8 GEOMETRY FOR AMERICANS 8.5.47 Using compass and ruler, draw a circle that is externally tangent to three given mut ...

Chapter 9 Calculus In this chapter, we take it for granted that you are familiar with the basic calculus ideas like limits, cont ...

316 CHAPTER 9 CALCULUS of y = f(x), the x-axis, and the vertical lines x = a and x = b. The right-hand side has a completely dif ...

9.2 CONVERGENCE AND CONTINUITY 317 since in a small interval of time .1t, the curtain's area will grow by approximately f(t}.1t. ...

318 CHAPTER 9 CALCULUS Convergence We say that the real-valued sequence (an) converges to the limit L if lim an =L. n->oo Wha ...

9.2 CONVERGENCE AND CONTINUITY 319 This forces Xn -t O. Conversely, if the tenus of a sequence are greater in ab­ solute value t ...

320 CHAPTER 9 CALCULUS To see why this is true, use the binomial theorem^3 to get 1+- = I +m- + -+ -+ -+ ... ( 1 ) m 1 (m) 1 (m) ...

7 s 3 2 9.2 CONVERGENCE AND CONTINUITY 321 y=x 2 � � 4 • 6 8 y=5/x Next, draw a horizontal line left from this midpoint until it ...

322 CHAPTER 9 CALCULUS Since the sequence is monotonic and bounded, it must converge. Now let us show that it converges to Va. S ...

9.2 CONVERGENCE AND CONTINUITY 323 1. First, show that V'n + 1 - Tn = O(n-^2 /^3 ). All that matters about O(n-^2 /^3 ) is that ...