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UNIVERSITY OF DURHAM DEPARTMENT OF MATHEMATICAL SCIENCES This formula sheet should be given to all candidates taking Mathematics for Engineers & Scientists (MATH1551/01). TRIGONOMETRIC FUNCTIONS HYPERBOLIC FUNCTIONS cos2 A + sin2 A = 1 1 x e + e−x 2 1 x sinh x = e − e−x 2 p cosh−1 x = ln x ± x2 − 1 p sinh−1 x = ln x + x2 + 1 1 + tan2 A = sec2 A cosh(iA) = cos A sin(A + B) = sin A cos B + cos A sin B cosh x = cos(A + B) = cos A cos B − sin A sin B tan(A + B) = 2 tan A + tan B 1 − tan A tan B 2 1 + cot A = cosec A 1 sin A cos B = (sin(A + B) + sin(A − B)) 2 1 cos A cos B = (cos(A + B) + cos(A − B)) 2 1 sin A sin B = (cos(A − B) − cos(A + B)) 2 1 1 cos C + cos D = 2 cos (C + D) cos (C − D) 2 2 1 1 sin C + sin D = 2 sin (C + D) cos (C − D) 2 2 1 1 cos C − cos D = −2 sin (C + D) sin (C − D) 2 2 1 1 sin C − sin D = 2 cos (C + D) sin (C − D) 2 2 sinh(iA) = i sin A cosh2 A − sinh2 A = 1 cosh(A + B) = cosh A cosh B + sinh A sinh B sinh(A + B) = sinh A cosh B + cosh A sinh B tanh(A + B) = tanh A + tanh B 1 + tanh A tanh B ELEMENTARY RULES FOR DIFFERENTIATION AND INTEGRATION Z Z u 0 u0 v − uv 0 0 0 0 0 (u+v)0 = u0 +v 0 (uv)0 = u0 v+uv 0 = (u(v)) = u (v)v u v dx = uv− uv 0 dx v v2 TAYLOR’S THEOREM 1 (n) f (a)(x − a)n n! |x − a|n+1 |f (n+1) (x)| ≤ M for c ≤ x ≤ b then |f (x) − pn,a (x)| ≤ M. (n + 1)! Taylor approximation: and if f (x) ≈ pn,a (x) = f (a) + f 0 (a)(x − a) + · · · + Date: October 6, 2010. 1 2 DEPARTMENT OF MATHEMATICAL SCIENCES TABLE OF DERIVATIVES y(x) dy/dx TABLE OF INTEGRALS Z f (x) f (x) dx xn nxn−1 xn ln x x−1 x−1 xn+1 (n 6= −1) n+1 ln |x| ex sin x ex cos x ex sin x ex − cos x cos x − sin x cos x sin x tan x 2 sec x tan x − ln cos x cosec x − cosec x cot x cosec x − ln(cosec x + cot x) sec x sec x tan x sec x ln(sec x + tan x) 2 cot x − cosec x cot x ln sin x sinh x cosh x sinh x cosh x cosh x sinh x cosh x sinh x tanh x 1 √ a2 − x2 ln cosh x x sin−1 (a > x) a 1 a2 + x2 1 √ 2 a + x2 1 √ x2 − a2 1 x tan−1 a a x sinh−1 a −1 x (x > a) cosh a 2 tanh x sech x 1 √ 1 − x2 −1 √ 1 − x2 1 1 + x2 1 √ 1 + x2 1 √ x2 − 1 sin−1 x cos−1 x tan−1 x sinh−1 x cosh−1 x CRITICAL POINTS 2 2 2 ∂ f ∂f ∂2f ∂2f = = 0, − > 0, and ∂∂xf2 < 0. Local maximum: ∂f ∂x ∂y ∂x2 ∂y 2 ∂y∂x 2 2 2 ∂f ∂2f ∂2f ∂ f Local minimum: ∂f > 0, and ∂∂xf2 > 0. ∂x = ∂y = 0, ∂x2 ∂y 2 − ∂y∂x 2 2 ∂ f ∂f ∂2f ∂2f < 0. Saddle point: ∂f = = 0 and − 2 2 ∂x ∂y ∂x ∂y ∂y∂x 2 2 2 2 ∂f ∂ f ∂ f ∂ f Inconclusive: ∂f = 0. ∂x = ∂y = 0 and ∂x2 ∂y 2 − ∂y∂x ITERATION METHODS Tj = D−1 (L + U ) A=D−L−U Ax = b Tg = (D − L)−1 U Jacobi’s Method: Dx(k+1) = b + (L + U )x(k) , x(k+1) = D−1 b + D−1 (L + U )x(k) Gauss-Seidel Method: Dx(k+1) = b + Lx(k+1) + U x(k) , x(k+1) = (D − L)−1 b + (D − L)−1 U x(k) SOR Method: x(k+1) = (1 − ω)x(k) + ωD−1 b + Lx(k+1) + U x(k) Optimal Value of ω: ω= 2 q 2 1 + 1 − ρ (Tj )