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FORMULA SHEET FOR MAT187 Trigonometric Identities. • cos2 (x) = 1 + cos(2x) 2 • sin2 (x) = 1 − cos(2x) 2 • sin(x) = cos π 2 −x Applications of integration. Z bq 2 1 + f 0 (x) dx • Arc length for y = f (x) a b Z • Area of a surface of revolution y = f (x) revolved around x−axis q 2 2πf (x) 1 + f 0 (x) dx a x −x sinh(x) = e − e 2 −x x cosh(x) = e + e 2 • Hyperbolic Functions Numerical Integration. Z • Midpoint Rule a b n X xi−1 + xi f (x) dx ≈ ∆x f 2 i=1 2 b−a ∆x max f 00 (x) 24 x∈[a,b] Z b ∆x f (x) dx ≈ f (a) + 2f (x1 ) + 2f (x2 ) + · · · + 2f (xn−1 ) + f (b) 2 a |EM | 6 • Trapezoid Rule 2 b−a ∆x max f 00 (x) 12 x∈[a,b] Z b ∆x f (x) dx ≈ f (a) + 4f (x1 ) + 2f (x2 ) + · · · + 4f (xn−1 ) + f (b) 3 a |ET | 6 • Simpson’s Rule (n even) |ES | 6 4 b−a ∆x max f 0000 (x) 180 x∈[a,b] Sequences. • Important limit • Geometric Sum lim n→∞ n X k=0 1+ a n = ea n xk = 1 − xn+1 1−x Power Series. f (x) = pn (x) + Rn (x) n X f (k) (a) (x − a)k pn (x) = k! k=0 f (n+1) (c) (x − a)n+1 R (x) = n (n + 1)! • Taylor Theorem ∞ X 1 = xk 1−x • Geometric Series k=0 p • Binomial Series (1 + x) = ∞ X p(p − 1) · · · (p − k + 1) k! k=0 • Sine Series sin(x) = xk ∞ X (−1)k 2k+1 x (2k + 1)! k=0 • Cosine Series cos(x) = ∞ X (−1)k k=0 (2k)! x2k ∞ X 1 k ln(1 − x) = − x k • Logarithmic Series k=1 Vector-Valued Functions. 1 2 • Area of a polar function r = f (θ) β Z 2 f (θ) dθ α Z • Length of a parametric curve ~r(t) b ~r 0 (t) dt a Z • Length of a polar curve r = f (θ) βq 2 2 f (θ) + f 0 (θ) dθ α • Unit Tangent vector ~r 0 (t) T~ (t) = 0 ~r (t) • Principal Unit Normal vector ~0 ~ (t) = T (t) N T~ 0 (t) • Binormal vector • Curvature • Torsion κ = dT~ ds = τ =− ~ dB ~ ·N ds 0 T~ (t) 0 ~r (t) = ~ ~ B(t) = T~ × N ~v × ~a = 3 ~v − ~ 0 (t) · N ~ (t) B ~r0 (t)