Formulas From Calculus

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Clementine Little
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1 Formulas You Shoul Memorize (an I o mean Memorize!) S 997 Pat Rossi Formulas From Calculus. [sin ()] = cos () 2. [cos ()] = sin () 3. [tan ()] = sec2 () 4. [cot ()] = csc2 () 5. [sec ()] = sec () tan () 6. [csc ()] = csc () cot () Name Remar Note that if you now the erivatives of sin (), tan (), an sec (), then the erivatives of the corresponing co-functions cos (), cot (), an csc () are foun by: (a) Changing the sign, an (b) Replacing each factor of the erivative with its co-function. Eample [sec ()] = sec () tan () = [csc ()] = csc () cot () 7. sin () = cos () + C 8. cos () = sin () + C 9. tan () = ln sec () + C = ln cos () + C 0. cot () = ln sin () + C. sec () = ln sec () + tan () + C 2. csc () = ln csc () cot () + C 3. sec 2 () = tan () + C 4. csc 2 () = cot () + C 5. sec () tan () = sec () + C 6. csc () cot () = csc () + C 7. [eu ] = e u u 8. [ln (u)] = u u

2 9. e u u = e u + C 20. u = ln u + C u 2. u u = ln u + C 22. ln (u) u = u ln (u) u + C [ 23. sin (u) ] = u 2 u 24. [cos (u)] = u 2 u 25. [tan (u)] = u +u [cot (u)] = u +u [sec (u)] = u u u [csc (u)] = u u u 2 Remar 2 Note that if you now the erivatives of the inverse trig functions sin (), tan (), an sec (), the erivatives of the corresponing inverse co-functions can be foun by changing the sign. 29. a 2 u 2 u = sin ( u a) + C 30. u = ( a 2 +u 2 a tan u a) + C 3. ( u u = u u 2 a 2 a sec a ) + C Essential Trigonometric Ientities 32. sin 2 () + cos 2 () = 33. tan 2 () + = sec 2 () 34. cot 2 () + = csc 2 () Remar 3 Ientities are the so-calle Pythagorean Ientities. 35. tan () = sin() cos() = cot() 36. cot () = cos() sin() = tan() 37. sec () = cos() 38. csc () = sin() 39. sin (2) = 2 sin () cos () 40. cos (2) = cos 2 () sin 2 () = 2 sin 2 () = 2 cos 2 () 2

3 Right Triangle Trig. opp aj 4. sin (θ) = opp hyp 42. cos (θ) = aj hyp 43. tan (θ) = opp aj 44. cot (θ) = aj opp 45. sec (θ) = hyp aj 46. csc (θ) = hyp opp Logs an Eponential Functions 47. ln (y) = ln () + ln (y) ( 48. ln (a) log a (y) = log a () + log a (y) y ) = ln () ln (y) ( ) (a) log a = log y a () log a (y) 49. ln ( n ) = n ln () (a) log a ( n ) = n log a () 50. ln (e ) = (a) log a (a ) = 5. e ln() = (a) a log a () = 3

4 52. ln (e) = (a) log a (a) = 53. ln () = 0 (a) log a () = lim ln () = (a) lim log a () = 55. lim 0 + ln () = (a) lim 0 + log a () = y y = ln () e +y = e e y 57. e e y (a) a +y = a a y = e y (a) a a y 58. (e ) n = e n = a y (a) (a ) n = a n 4

5 59. lim e = (a) lim a = 60. lim e = 0 (a) lim a = 0 y y = e lim 0 ( + ) = e 62. lim 0 ( + ) = e (a) lim 0 ( ) = e ( 63. lim + ) = e ( ) 64. lim + = e ( ) (a) lim = e Remar 4 Note that 62.a an 64.a can be obtaine easily from 62 an 64 respectively, by plugging in in place of. 65. (y) n = n y n ( ) n 66. = n y y n Calculus of General Logs an Eponentials 67. [a ] = ln (a) a 68. [au ] = ln (a) a u u 69. a u u = ln(a) au + C 70. [log a ()] = ln(a) 7. [log a (u)] = u ln(a) u 5

6 By-Passing U-Substitution, Given Simple Composite Functions Suppose that F () is an anti-erivative of f (). (i.e., suppose that f () = F () + C) Then if an c are constants, the following general principles hol: f () = F () + C an f ( + c) = F ( + c) + C) Some Cases in Point: sin () = cos () + C sin ( + c) = cos ( + c) + C cos () = sin () + C cos ( + c) = sin ( + c) + C e = e + C e +c = e+c + C = ln () + C = ln ( + c) + C +c Series an Sums Finite Series 72. n i=0 ri a = rn+ r Finite Geometric Series with ratio r. 73. n n+ i=0 (i + a) = (a 2 + a 2 ) Arithmetic Series with common ifference. 74. n i= a i = i= a i + n i=+ a i Infinite Series 75. If n= a n converges, then a n 0. Conversely, if a n oes not go to 0, then the series must iverge. 76. n= = n n 77. The series: is the Harmonic Series. It iverges. (a) ( ) n+ n= n (b) ( ) n n= n = ( )n+ n +... an = ( )n n +... are Alternating Harmonic Series. These series converge. 6

7 78. n=0 rn a = a + ra + r 2 a + r 3 a r n a +... is the Geometric Series with ratio r. (a) If r <, the series converges, an the sum is given by (b) If r, the series iverges. first term r. 79. The series: (a) n= ( )n+ a n = a a 2 + a 3 a ( ) n+ a n +... an (b) n= ( )n a n = a + a 2 a 3 + a ( ) n+ a n +... where:. a n 0 2. a n a n+ 3. lim n a n = 0 are Alternating Series. They converge. Also: ( ) n+ a n ( ) n+ a n a + n= n= (A similar statement can be mae for n= ( )n a n ) 80. Direct Comparison Test Let n= a n an n= b n be such that a n an b n are non-negative for all but finitely many terms. (a) If a n b n for all but finitely many terms an n= b n converges, then n= a n converges also. (b) If a n b n for all but finitely many terms an n= a n iverges, then n= b n iverges also. (c) If a n b n an n= a n converges, this tells us nothing about n= b n. () If a n b n an n= b n iverges, this tells us nothing about n= a n. 8. Limit Comparison Test Let n= a n an n= b n be such that a n an b n are nonnegative for all but finitely many terms. a (a) If lim n n bn = c, where c is non-zero an finite (i.e. 0 < c < ), then both n= a n an n= b n converge, or both iverge. a (b) If lim n a n bn = 0 or if lim n n bn =, we can conclue nothing. In this case, we have mae a poor choice for comparison. Choose another series for comparison. 82. p-series The series n= where p is a positive constant, is the p-series. This series n p converges if p > an iverges if p. 7

8 83. Ratio Test Consier n= a n. a (a) If lim n+ n a n <, the series n= a n converges. a (b) If lim n+ n a n >, the series n= a n iverges. a (c) If lim n+ n a n =, the Ratio Test is inconclusive. 84. The Integral Test Let n= a n be such that a n is non-negative for all but finitely many terms. If: (a) f () is a continuous function on the interval [, ), with the property that f (n) = a n for n =, 2, 3,... an (b) f () is ecreasing on [, ), then either both n= a n an b f () converge, or they both iverge. 85. The n th Root Test Consier n= a n. a (a) If lim n n a n <, then series n= a n converges. (b) If lim n n a n >, then series n= a n iverges. (c) If lim n n a n =, then the n th Root Test is inconclusive. 86. The series f (n) ( 0 ) n=0 ( n! 0 ) n is the Taylor Series of f () with center 0. Here, f () = n=0 Some Well-Known Taylor Series f (n) ( 0 ) n! ( 0 ) n 87. e = n n=0 = n +... for all. n! 2! 3! n! 88. ln ( + ) = n=0 ( )n n+ = n+ 2 3 ( )n n for < n+ 89. sin () = n=0 ( )n 2n+ = ( ) n 2n for all. (2n+)! 3! 5! (2n+)! 90. cos () = n=0 ( )n 2n = ( ) n 2n +... for all. (2n)! 2! 4! (2n)! 9. = n=0 n = n +... for <. 8

sin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations

sin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on