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CSSS 505 Calculus Summary Formulas CSSS 505 Calculus Summary Formulas Differentiation Formulas d − dy dy du 1. (x n ) = nx n 1 17. = × Chain Rule dx dx dx dx d 2. ( fg) = fg′ + gf ′ dx d f gf ′ − fg′ 3. ( ) = dx g g 2 d 4. f (g(x)) = f ′(g(x))g′(x) dx d 5. (sin x) = cos x dx d 6. (cos x) = −sin x dx d 7. (tan x) = sec 2 x dx d 8. (cot x) = −csc2 x dx d 9. (sec x) = sec x tan x dx d 10. (csc x) = −csc x cot x dx d 11. (e x ) = e x dx d 12. (a x ) = a x ln a dx d 1 13. (ln x) = dx x d 1 14. (Arcsin x) = dx 1− x 2 d 1 15. (Arc tan x) = dx 1+ x 2 d 1 16. (Arcsec x) = dx | x | x 2 −1 Trigonometric Formulas 1. sin 2 θ + cos 2 θ = 1 sinθ 1 13. tanθ = = 2. 1+ tan 2 θ = sec2 θ cosθ cotθ 3. + 2 θ = 2 θ cosθ 1 1 cot csc 14. cotθ = = 4. sin(−θ ) = −sinθ sinθ tanθ 5. cos(−θ ) = cosθ 1 15. secθ = 6. tan(−θ ) = − tanθ cosθ + = + 1 7. sin(A B) sin Acos B sin B cos A 16. cscθ = 8. sin(A − B) = sin Acos B − sin B cos A sinθ + = − π 9. cos(A B) cos Acos B sin Asin B 17. cos( −θ ) = sinθ 2 10. cos(A − B) = cos Acos B + sin Asin B π 18. sin( −θ ) = cosθ 2 11. sin 2θ = 2sinθ cosθ 12. cos 2θ = cos 2 θ − sin 2 θ = 2cos 2 θ −1 = 1− 2sin 2 θ Integration Formulas Definition of a Improper Integral b ∫ f (x) dx is an improper integral if a 1. f becomes infinite at one or more points of the interval of integration, or 2. one or both of the limits of integration is infinite, or 3. both (1) and (2) hold. 1. ∫ a dx = ax + C 12. ∫ csc x dx = ln csc x − cot x + C n+1 x 13. sec2 x dx = tan x + C 2. x n dx = + C, n ≠ −1 ∫ ∫ n +1 14. sec x tan x dx = sec x + C 1 ∫ 3. dx = ln x + C 2 ∫ x 15. ∫ csc x dx = −cot x + C x = x + 4. ∫ e dx e C 16. ∫ csc x cot x dx = −csc x + C x 2 x a 17. tan x dx = tan x − x + C 5. a dx = + C ∫ ∫ ln a dx = 1 x + 6. ln x dx = x ln x − x + C 18. Arc tan C ∫ ∫ a 2 + x 2 a a 7. sin x dx = −cos x + C dx x ∫ 19. = Arcsin + C ∫ 2 2 8. ∫ cos x dx = sin x + C a − x a = + − + dx 1 x 1 a 9. tan x dx ln sec x C or ln cos x C 20. = Arcsec + C = Arc cos + C ∫ ∫ 2 2 x x − a a a a x 10. ∫ cot x dx = ln sin x + C 11. ∫ sec x dx = ln sec x + tan x + C Formulas and Theorems 1a. Definition of Limit: Let f be a function defined on an open interval containing c (except possibly at c ) and let L be a real number. Then lim f (x) = L means that for each ε > 0 there x → a exists a δ > 0 such that f (x) − L < ε whenever 0 < x − c < δ . 1b. A function y = f (x) is continuous at x = a if i). f(a) exists ii). lim f (x) exists x → a iii). lim = f (a) x → a 4. Intermediate-Value Theorem A function y = f (x) that is continuous on a closed interval [a,b] takes on every value between f (a) and f (b) . Note: If f is continuous on [a,b] and f (a) and f (b) differ in sign, then the equation f (x) = 0 has at least one solution in the open interval (a,b) . 5. Limits of Rational Functions as x → ±∞ f (x) i). lim = 0 if the degree of f (x) < the degree of g(x) x → ±∞ g(x) x2 − 2x Example: lim = 0 x → ∞ x3 + 3 f (x) ii). lim is infinite if the degrees of f (x) > the degree of g(x) x→±∞ g(x) x3 + 2x Example: lim = ∞ x → ∞ x2 − 8 f (x) iii). lim is finite if the degree of f (x) = the degree of g(x) x→±∞ g(x) 2x2 − 3x + 2 2 Example: lim = − x → ∞ 10x − 5x2 5 6. Average and Instantaneous Rate of Change i). Average Rate of Change: If (x , y ) and (x , y ) are points on the graph of 0 0 1 1 y = f (x) , then the average rate of change of y with respect to x over the interval f (x ) − f (x ) y − y ∆y []x , x is 1 0 = 1 0 = . 0 1 − − ∆ x1 x0 x1 x0 x () = ii). Instantaneous Rate of Change: If x0 , y0 is a point on the graph of y f (x) , then ′ the instantaneous rate of change of y with respect to x at x0 is f (x0 ) . f (x + h) − f (x) 7. f ′(x) = lim h → 0 h 8. The Number e as a limit n 1 i). lim 1+ = e n → +∞ n 1 n ii). lim 1+ n = e n → 0 1 9. Rolle’s Theorem If f is continuous on [a,b] and differentiable on (a,b) such that f (a) = f (b) , then there is at least one number c in the open interval (a,b) such that f ′(c) = 0 . 10. Mean Value Theorem If f is continuous on [a,b] and differentiable on (a,b), then there is at least one number c f (b) − f (a) in (a,b) such that = f ′(c) . b − a 11. Extreme-Value Theorem If f is continuous on a closed interval [a,b], then f (x) has both a maximum and minimum on []a,b . 12. To find the maximum and minimum values of a function y = f (x) , locate 1. the points where f ′(x) is zero or where f ′(x) fails to exist. 2. the end points, if any, on the domain of f (x) . Note: These are the only candidates for the value of x where f (x) may have a maximum or a minimum. 13. Let f be differentiable for a < x < b and continuous for a a ≤ x ≤ b , 1. If f ′(x) > 0 for every x in (a,b), then f is increasing on []a,b . 2. If f ′(x) < 0 for every x in (a,b), then f is decreasing on []a,b . .
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