Download e-book for iPad: A Second Course in Calculus by Harley Flanders

By Harley Flanders

ISBN-10: 148323102X

ISBN-13: 9781483231020

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Given n, each of the equal rectangles ly* either meets one of the division lines, or lies entirely inside some J t . The sum An decomposes accordingly: t=l (Ji) where X7 sums the terms where ly* meets a boundary of some J\, and ]£"(J t -) is taken over all ly* in the interior of J{. We make crude, but adequate, esti­ mates. Each side of a J\- can intersect at most 2n of the ly*, so the boundary intersects at most 8n of them. « I n21—4 n2 | n Each term 2 " is almost the integral of / over J\. The error comes from at most \n boundary terms, so y) (Ji) f(x^y^)\ljk\ - // Ji fdxdy <4(4nB).

12. 13. A = C = neither Pick Pick fxx(0, fw(0, 4. the the 0), 0), min 6. min 8. max 10. neither curve x = t, y = 0: g(0) = fxx(0, 0) > 0, etc. curve x = [B/(A2 + B2)l'2]t, y = [-A/(A2 + B2)v% where 2 2 B = fxy(Qy 0). Then g(0) = A(AC - B )/(A + B2), where etc. 14. £(3) = fxxxx* + 3fxxyx2y + Sfxvyxy2 + fyyyyz + 3[/**xx + /xy(# + xy) + fyyyy] + fxx™ + fyyW 15. If A 7* 0, then A# = (Ax + ify) 2 by completing the square. If A = C = 0, then B = 0 since AC - £ 2 = 0, # = 0. Section 5, page 348 2. 1 4 1 4 1 4 (2" ' , 2 ' , 2- / ) 4.

3 1 1 0 = 0, Sample Test V "l . 2, 2, a 0 ; -l, a 0 32. 0, a 2 i-i 30. 3, a 11 0 ro 29 1 V ; 3, a 0 + b 1 0 lj 34. (Ay')-x' = xAy' = yAx' = y(Xx') = Xyx' = Xy'-x' = 0 36. - | A | = | - A | = |0 - A\ = (0 - X)(0 - M) (0 - v) = -XMv. l 38. 6, A = PDP~ , where D has X, /x, J/ on the diagonal and 0 elsewhere. Clearly A = 0 if and only if D = 0. 40. Suppose Ax' = Xx', with x' ^ 0. Then A 2 x' = A (Ax') = A(Xx') = X(Ax') = X(Xx') = XV. Hence X2 is a characteristic root of A2. SAMPLE TEST 1. 2. 3. 4. If A and B are 3 X 3 matrices and AB = 0, does it follow that either A = 0 or £ = 0?

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A Second Course in Calculus by Harley Flanders


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