A course of higher mathematics, vol. 2 by Smirnov V.I. PDF

By Smirnov V.I.

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2π 36. Replace Θ by -Θ in cos \^n - = sin Θ and sin l-^rr l·-'}- and use parity. a ± —7T | = 9 4 -10 te* -Η sin . (bxjaje is strictly decreasing. 18. 38. [~ - , where b is the positive factor of implies 1/H 7 3* 6" -180° 14. h/H J = bkp^R max 0 °° kHpr 7 "I* (0, exp page 183 2 2* = y^ -h/H since e bR oo - 1 SECTION 5, y e To maximize J, it suffices to minimize = 0 for h = h Also, -(b/a)x y kHpe = h/H + kHp e df/dh This is one way to find c. c, which is more natural. y However, if t-*», then x->0, so (b/a)x y = ce cos a, cos I a ± —IT | = + sin a 40 cos Θ, CHAPTER 4 40.

22. 26. a , 2 2, _ 2 2K fa - x ; dx = ja h W = p A 24. fX0 fX0 1 ^nh2 CHAPTER 7 SECTION 1 , 2. page 306 x - | ( 5 - 9) y + 1 4. , , r, 60 x = Zi_L2 , 3y - 4 y ^o y 3 du CHAPTER 7 3y1/3 - 4 27 ^1/3 _ ^ , y ^ T + 3 -2y 3 10. x - 14. x = 18. yes 20. Suppose y i s i n t h e domain of g. "7, y + 7 y * -7 have g f - y ; = g [ - f f x j ] 22. 24. 28. i2. x « ry - ι ; 16. no Then y = f(x) = g[f(-x)] where x - g(y) · = - x = -g(y) We , so g i s odd. = g W 3 + g W so gfxj 3 = -gfxj + x. x = f[g(x)) 26. 22 —.

Min y (-1/12) = 3)14; 14. ~ y(-~) Remark. 1612 2'7; 3 max y(t"3) min [ n ] (n-l) /n -A-- A _ , ... a n 2 2 = 12. n-l n n An' then G G G /A continuing, Gn/A = (2n 4 no maxs 1)] 1) (2n-l)/2/(2n)n This example is related to the following situation. Let L(x) denote the lateral area of a frustum of a cone having fixed height b, one fixed base of radius a, and one variable base of 2 2 radius x. Then L(x) = rr(a + x) Ib 2 + (a - x)2. If a ~ 2b , then L(x) is least for x = 0 (cone) and increases with x.

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