7/17/97 Final Exam: \Cheat Sheet" Z Z 1 2 6. csc xdx = cot x dx =lnjxj 1. x Z Z x a x 7. sec x tan xdx = sec x 2. a dx = ln a Z Z 8. csc x cot xdx = csc x 3. sin xdx = cos x Z Z 1 x dx 1 = tan 9. 4. cos xdx = sin x 2 2 x + a a a Z Z dx x 1 2 p = sin 10. 5. sec xdx = tan x 2 2 a a x Z Z d b [f y g y ] dy [f x g x] dx. In terms of y : A = 11. Area in terms of x: A = c a Z Z d b 2 2 [g y ] dy [f x] dx. Around the y -axis: V = 12. Volume around the x-axis: V = c a Z b 13. Cylindrical Shells: V = 2xf x dx where 0 a< b a Z Z b b b 0 0 14. f xg x dx = f xg x] f xg x dx a a a R m 2 2 n 2 2 15. HowtoEvaluate sin x cos xdx.:Ifm or n are o dd use sin x =1 cos x and cos x =1 sin x, 2 1 1 2 resp ectively and then apply substitution. Otherwise, use sin x = 1 cos 2x or cos x = 1 + cos 2x 2 2 1 or sin x cos x = sin 2x. 2 R m n 2 2 2 16. HowtoEvaluate tan x sec xdx.:Ifm is o dd or n is even, use tan x = sec x 1 and sec x = 2 1 + tan x, resp ectively and then apply substitution. 17. Table of Trigonometric Substitutions Expression Substitution Identity p 2 2 2 2 a x x = a sin , 1 sin = cos 2 2 p 2 2 2 2 < < a + x x = a tan , 1 + tan = sec 2 2 p 3 2 2 2 2 x a x = a sec 0 < or < sec 1 = tan 2 2 " n n1 X X x f x and T = f x +f x f x +2 18. M =x i n i n 0 n 2 i=1 i=1 x 19. S = [f x +4f x +2f x +4f x + +2f x + 4f x + f x ] n 0 1 2 3 n2 n1 n 3 3 3 5 K b a K b a K b a 2 2 4 20. jE j and jE j and jE j T M S 2 2 4 12n 24n 180n Z Z Z Z 1 t b b 21. f x dx = lim f x dx and f x dx = lim f x dx t!1 t!1 a a 1 t P 1 22. Test for Divergence. If lim a do es not exist or if lim a 6= 0, then the series a is n!1 n n!1 n n n=1 divergent. 23. Remainder Estimate for the Integral Test. If a converges by the Integral Test and R = s s , n n n Z Z 1 1 then f x dx R f x dx n n+1 n 10 P P 24. Limit Comparison Test. Supp ose that a and b are series with p ositive terms. a If n n a a n n = c>0, then either b oth series converge or b oth diverge. b If lim = 0 and lim n!1 n!1 b b P n P n P P a n b converges, then a also converges. c If lim = 1 and b diverges, then a also n n n!1 n n b n diverges. P n1 25. Alternating Series Estimation Theorem. If s = 1 b is the sum of an alternating series n that satis es a and b ab ove, then jR j = js s jb n n n+1 P 1 a n+1 a is absolutely .IfL exists and L<1, then the series 26. The Ratio Test Let L = lim n n!1 n=1 a n P 1 a is divergent. convergent. If L exists and L> 1 or is in nity, then the series n n=1 p P 1 n 27. The Ro ot Test Let L = lim j = L.IfL exists and L<1, then the series a is ja n!1 n n n=1 P 1 absolutely convergent. If L exists and L>1 or is in nity, then the series a is divergent. n n=1 P 1 n 28. Theorem. If f x= c x a has radius of convergence R> 0 then f x is di erentiable on n n=0 Z 1 1 n+1 X X x a 0 n1 , each a R; a + R and a f x= nc x a and b f x dx = C + c n n n +1 n=1 n=0 with radius of convergence R. n 29. Theorem. If f has a p ower series representation at a, that is, if f x=c x a for jx aj <R, n n f a then its co ecients are of the form n! 30. Taylor's Formula. If f has n + 1 derivatives in an interval I that contains the number a, then for x in I there is a number z strictly b etween x and a such that the remainder term in the Taylor series n+1 f z n+1 can b e expressed as R x= x a n n + 1! 1 1 1 2n+1 n X X X x 1 x n x 1 n 31. = x ,1; 1 ,1; 1 ,[1; 1] e = tan x = 1 1 x n! 2n +1 n=0 n=0 n=0 1 1 2n+1 2n X X x x n n ,1; 1 ,1; 1 sin x = cos x = 1 1 2n + 1! 2n! n=0 n=0 1 X k k n 32. The Binomial Series If k is any real numb er and jxj < 1, then 1 + x = x where n n=0 k k 1k n+1 k k = =1. for n 1 and n 0 n! dy dy =dt dx 33. = if 6=0. dx dx=dt dt 0 34. The Arc Length Formula. If f is continuous on [a; b], then the length of the curve y = f x, Z b p 0 2 a x b,is L = 1+ [f x] dx. a 35. Theorem. If a curve C is describ ed by the parametric equations x = f t, y = g t, t , 0 0 where f and g are continuous on [ ; ] and C is traversed exactly once as t increases from to , s Z 2 2 dx dy then the length of C is L = + dt. dt dt s Z 2 2 dx dy 2y + dt 36. Surface Area. S = dt dt y 2 2 2 . 37. Polar co ordinate conversion: x = r cos , y = r sin , r = x + y , and tan = x s Z Z 2 b b 1 dr 2 2 38. For p olar co ordinates: A = r d and L = r + d 2 d a a 11.