Download UNIVERSITY OF DURHAM This formula sheet should be given to all

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 )