12 - 0 or 2 θ= π (x,y) = (1,0) π (-1,0) 7π/6 5π/4...

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Unformatted text preview: 0 or 2 θ= π (x,y) = (1,0) π (-1,0) 7π/6 5π/4 (0,1) π/2 ( 1 / 2, 3 / 2) π/3 π/4 ( 2 / 2, 2 / 2) π/6 ( 3 / 2, 1 / 2) 2π/3 ( -1 / 2, 3 / 2) 3π/4 5π/6 4π/3 3π/2 (0,-1) x = cos θ y = sin θ 5π/3 7π/4 11π/6 ( 1 / 2, - 3 / 2) ( 3 / 2, - 1 / 2) (- 2 / 2, 2 / 2) (- 3 / 2, 1 / 2) (- 3 / 2, - 1 / 2) (- 2 / 2 , - 2 / 2) ( -1 / 2, - 3 / 2) ( 2 / 2 , - 2 / 2) tan θ = sin θ/ cos θ, cot θ = cos θ/ sin θ, sec θ = 1 / cos θ, csc θ = 1 / sin θ, cot θ = 1 / tan θ, sin 2 θ + cos 2 θ = 1 , 1 + tan 2 θ = sec 2 θ, 1 + cot 2 θ = csc 2 θ, sin( A ± B ) = sin A cos B ± cos A sin B, cos( A ± B ) = cos A cos B ∓ sin A sin B. [sin x ] ′ = cos x [cos x ] ′ = − sin x [tan x ] ′ = sec 2 x [sec x ] ′ = sec x tan x [csc x ] ′ = − csc x cot x [cot x ] ′ = − csc 2 x [ln x ] ′ = 1 x [ln | x | ] ′ = 1 x [ e x ] ′ = e x [ fg ] ′ = f ′ g + fg ′ bracketleftbigg f g bracketrightbigg ′ = gf ′ − fg ′ g 2 [ f ( g )] ′ = f ′ ( g ) g ′ integraldisplay b a f ( x ) dx = F ( b ) − F ( a ) , F ′ ( x ) = f ( x ) . integraldisplay f ( g ( x )) g ′ ( x ) dx = integraldisplay f ( u ) du, u = g ( x ) . integraldisplay u dv = uv − integraldisplay v du. L’Hospital’s rule: If f ( x ) g ( x ) → or ± ∞ ∞ . Then lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) . Exponent rule: If f ( x ) g ( x ) → , 1 ∞ , ∞ or 0 ∞ . Then lim x → c f ( x ) g ( x ) = e L where L = lim x → c ln[ f ( x ) g ( x ) ]. integraldisplay ∞ a f ( x ) dx = lim b →∞ integraldisplay b a f ( x ) dx. integraldisplay b −∞ f ( x ) dx = lim a →−∞ integraldisplay b a f ( x ) dx. integraldisplay ∞ −∞ f ( x ) dx = integraldisplay −∞ f ( x ) dx + integraldisplay ∞ f ( x ) dx. integraldisplay b a f ( x ) dx = lim c → a + integraldisplay b c f ( x ) dx (if asymptote at x = a ) . integraldisplay b a f ( x ) dx = lim c → b- integraldisplay c a f ( x ) dx (if asymptote at x = b ) . Absolute value rule: If lim n →∞ | a n | = 0. Then lim n →∞ a n = 0. Squeeze rule: If a n ≤ b n ≤ c n and lim n →∞ a n = L = lim n →∞ c n . Then lim n →∞ b n = L . Geometric (Exponential) sequence rule: If a n = r n . Then lim n →∞ a n = , if − 1 < r < 1 1 , if r = 1 ∞ , if r > 1 dne , if r ≤ − 1 . Monotonic sequence theorem: If a n is bounded and monotonic. Then a n is convergent. Geometric series rule: ∞ summationdisplay k =1 cr k − 1 = braceleftbigg c/ (1 − r ) , if | r | < 1 divergent , if | r | ≥ 1 . p-series rule: ∞ summationdisplay k = q 1 k p converges if p > 1, diverges if p ≤ 1....
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This note was uploaded on 01/27/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas.

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12 - 0 or 2 θ= π (x,y) = (1,0) π (-1,0) 7π/6 5π/4...

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