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Calculus Cheat Sheets

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• 1. CSSS 505 Calculus Summary Formulas Differentiation Formulas dn dy dy du ( x ) = nx n −1 = × 1. 17. Chain Rule dx dx dx dx d ( fg ) = fg ′ + gf ′ 2. dx gf ′ − fg ′ df ( )= 3. g2 dx g d f ( g ( x)) = f ′( g ( x)) g ′( x) 4. dx d (sin x) = cos x 5. dx d (cos x) = − sin x 6. dx d (tan x) = sec 2 x 7. dx d (cot x) = − csc 2 x 8. dx d (sec x) = sec x tan x 9. dx d (csc x) = − csc x cot x 10. dx dx (e ) = e x 11. dx dx (a ) = a x ln a 12. dx d 1 (ln x) = 13. dx x d 1 ( Arc sin x) = 14. dx 1− x2 d 1 ( Arc tan x) = 15. 1+ x2 dx d 1 ( Arc sec x) = 16. dx | x | x2 −1
• 2. Trigonometric Formulas sin θ 1 sin 2 θ + cos 2 θ = 1 1. 13. tan θ = = cosθ cot θ 1 + tan 2 θ = sec 2 θ 2. cosθ 1 1 + cot 2 θ = csc 2 θ 3. 14. cot θ = = sin(−θ ) = − sin θ sin θ tan θ 4. 1 cos(−θ ) = cosθ 5. 15. secθ = cosθ tan(−θ ) = − tan θ 6. 1 sin( A + B ) = sin A cos B + sin B cos A 7. 16. cscθ = sin θ sin( A − B) = sin A cos B − sin B cos A 8. π cos( A + B) = cos A cos B − sin A sin B 9. − θ ) = sin θ 17. cos( 2 π 10. cos( A − B) = cos A cos B + sin A sin B − θ ) = cosθ 18. sin( 2 11. sin 2θ = 2 sin θ cos θ 12. cos 2θ = cos θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ 2
• 3. Integration Formulas Definition of a Improper Integral b ∫ f ( x) dx is an improper integral if a f becomes infinite at one or more points of the interval of integration, or 1. 2. one or both of the limits of integration is infinite, or 3. both (1) and (2) hold. ∫ a dx = ax + C ∫ csc x dx = ln csc x − cot x + C 1. 12. ∫ sec x d x = tan x + C x n +1 2 13. ∫ x dx = n + C , n ≠ −1 2. n +1 ∫ sec x tan x dx = sec x + C 14. 1 ∫ x dx = ln x + C 3. ∫ csc x dx = − cot x + C 2 15. ∫ e dx = e + C ∫ csc x cot x dx = − csc x + C x x 4. 16. ax ∫ tan x dx = tan x − x + C 2 17. ∫ a dx = x +C 5. ln a  x dx 1 ∫a = Arc tan  + C ∫ ln x dx = x ln x − x + C 18. 6. 2 2 +x a a ∫ sin x dx = − cos x + C  x dx 7. ∫ = Arc sin   + C 19. a ∫ cos x dx = sin x + C a2 − x2 8. x dx 1 1 a ∫ tan x dx = ln sec x + C or − ln cos x + C ∫x 9. = Arc sec + C = Arc cos + C 20. a a a x x2 − a2 ∫ cot x dx = ln sin x + C 10. ∫ sec x dx = ln sec x + tan x + C 11.
• 4. 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 . A function y = f (x ) is continuous at x = a if 1b. i). f(a) exists lim f ( x) exists ii). x→a lim = f (a) iii). 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) . Limits of Rational Functions as x → ±∞ 5. f ( x) = 0 if the degree of f ( x) < the degree of g ( x) lim i). x → ±∞ g ( x) x 2 − 2x =0 Example: lim x → ∞ x3 + 3 f ( x) is infinite if the degrees of f ( x ) > the degree of g ( x ) lim ii). x → ±∞ g ( x ) x3 + 2x =∞ Example: lim x → ∞ x2 − 8 f ( x) is finite if the degree of f ( x ) = the degree of g ( x ) lim iii). x → ±∞ g ( x ) 2 x 2 − 3x + 2 2 =− Example: lim x → ∞ 10 x − 5 x 2 5 6. Average and Instantaneous Rate of Change ( 0 0 ) and (x1, y1 ) are points on the graph of Average Rate of Change: If x , y i). y = f (x) , then the average rate of change of y with respect to x over the interval [x0 , x1 ] is f ( x1 ) − f ( x0 ) = y1 − y 0 = ∆y . x1 − x0 x1 − x0 ∆x Instantaneous Rate of Change: If ( x 0 , y 0 ) is a point on the graph of y = f (x ) , then ii). the instantaneous rate of change of y with respect to x at x 0 is f ′( x 0 ) . f ( x + h) − f ( x ) f ′( x) = lim 7. h h→0
• 5. The Number e as a limit 8. n  1 lim 1 +  = e i). n → +∞ n  1  nn lim 1 +  = e ii). n → 0 1  9. Rolle’s Theorem [] (a, b ) such that f (a) = f (b) , then there If f is continuous on a, b and differentiable on is at least one number c in the open interval (a, b ) such that f ′(c) = 0 . 10. Mean Value Theorem [] (a, b ) , then there is at least one number If f is continuous on a, b and differentiable on c f (b) − f (a) (a, b ) such that = f ′(c) . in 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 To find the maximum and minimum values of a function y = f (x ) , locate 12. the points where f ′(x ) is zero or where f ′(x ) fails to exist. 1. the end points, if any, on the domain of f (x ) . 2. Note: These are the only candidates for the value of x where f (x ) may have a maximum or a minimum. Let f be differentiable for a < x < b and continuous for a a ≤ x ≤ b , 13. [] (a, b ) , If f ′( x ) > 0 for every x in then f is increasing on a, b . 1. is decreasing on [a, b ]. (a, b ) , If f ′( x ) < 0 for every x in then f 2.