By David V. Widder
Vintage textual content leads from easy calculus into extra theoretic difficulties. exact method with definitions, theorems, proofs, examples and routines. issues contain partial differentiation, vectors, differential geometry, Stieltjes essential, limitless sequence, gamma functionality, Fourier sequence, Laplace remodel, even more. a number of graded workouts with chosen solutions. 1961 edition.
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Post yr notice: First released March 2d 1979
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Extra resources for Advanced Calculus (2nd Edition)
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.