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LOGARITHMS & EXPONENTIALS
loga x = y ⇐⇒ ay^ = x, loga(xy) = loga x + loga y, loga
x y
= loga x − loga y
loga(xr) = r loga(x), loga(ax) = x, aloga^ x^ = x
(ab)c^ = abc, abac^ = ab+c, a−b^ =
ab
TRIG & HYPERBOLIC IDENTITIES
sin^2 θ + cos^2 θ = 1, tan^2 θ + 1 = sec^2 θ, 1 + cot^2 θ = csc^2 θ
sin^2 θ = 12 (1 − cos 2θ), cos^2 θ = 12 (1 + cos 2θ)
sin 2θ = 2 sin θ cos θ, cos 2θ = cos^2 θ − sin^2 θ, tan 2θ = 2 tan θ 1 − tan^2 θ sin A cos B = 12 [sin(A + B) + sin(A − B)]
cos A cos B = 12 [cos(A + B) + cos(A − B)]
sin A sin B = 12 [cos(A − B) − cos(A + B)]
sin A + sin B = 2 sin 12 (A + B) cos 12 (A − B)
sin A − sin B = 2 cos 12 (A + B) sin 12 (A − B)
cos A + cos B = 2 cos 12 (A + B) cos 12 (A − B)
cos A − cos B = 2 sin 12 (A + B) sin 12 (B − A)
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = tan A ± tan B 1 ∓ tan A tan B cosh^2 x − sinh^2 x = 1, tanh^2 x = 1 − sech 2 x, coth^2 x = 1 + csch 2 x
cosh^2 x = 12 (cosh 2x + 1), sinh^2 x = 12 (cosh 2x − 1)
sinh 2x = 2 sinh x cosh x, cosh 2x = cosh^2 x + sinh^2 x
HYPERBOLIC FUNCTIONS
sinh x = ex^ − e−x 2 , cosh x = ex^ + e−x 2 , tanh x = sinh x cosh x csch x =
sinh x , sech x =
cosh x , coth x = cosh x sinh x
sinh−^1 x = ln(x +
x^2 + 1), cosh−^1 x = ln(x +
x^2 − 1), tanh−^1 x =
ln
1 + x 1 − x
STANDARD ANGLES
θ (rad) 0
π 6
π 4
π 3
π 2 sin θ 0
√^1
cos θ 1
INVERSE TRIGONOMETRIC FUNCTIONS
sin−^1 (sin x) = x, for all x ∈
[
−π 2 , π 2
]
sin(sin−^1 x) = x, for all x ∈ [− 1 , 1]
cos−^1 (cos x) = x, for all x ∈ [0, π]
cos(cos−^1 x) = x, for all x ∈ [− 1 , 1]
tan−^1 (tan x) = x, for all x ∈
−π 2 , π 2
tan(tan−^1 x) = x, for all x
csc−^1 (csc x) = x, for all x ∈
0 , π 2
]
π, 32 π
]
csc(csc−^1 x) = x, for all |x| ≥ 1
sec−^1 (sec x) = x, for all x ∈
[
0 , π 2
[
π, 32 π
sec(sec−^1 x) = x, for all |x| ≥ 1
cot−^1 (cot x) = x, for all x ∈ (0, π)
cot(cot−^1 x) = x, for all x ∈ R
STANDARD DERIVATIVES
d dx (uv) = v du dx
d dx
(u v
v dudx − u dvdx v^2
dy dx
dy du
du dx d dx (sin x) = cos x, d dx (cos x) = − sin x, d dx (tan x) = sec^2 x d dx (sec x) = sec x tan x, d dx (csc x) = − csc x cot x, d dx (cot x) = − csc^2 x d dx
(sin−^1 x) =
1 − x^2
d dx
(cos−^1 x) = −
1 − x^2
d dx
(tan−^1 x) =
1 + x^2 d dx (sec−^1 x) =
x
x^2 − 1
d dx (csc−^1 x) = −
x
x^2 − 1
d dx (cot−^1 x) = −
1 + x^2 d dx (sinh x) = cosh x, d dx (cosh x) = sinh x, d dx (tanh x) = sech 2 x d dx (sech x) = −sech x tanh x, d dx (csch x) = −csch x coth x, d dx (coth x) = −csch 2 x d dx (sinh−^1 x) =
1 + x^2
d dx (cosh−^1 x) =
x^2 − 1
d dx (tanh−^1 x) =
1 − x^2 d dx (sech −^1 x) = −
x
1 − x^2
d dx (csch −^1 x) = −
|x|
x^2 + 1
d dx (coth−^1 x) =
1 − x^2
TAYLOR SERIES
f (x) =
∑^ ∞
n=
f (n)(a) n!
(x − a)n^ = f (a) + f ′(a)(x − a) + f ′′(a) 2!
(x − a)^2 + f ′′′(a) 3!
(x − a)^3 +...
Pn(x) = f (a) + f ′(a)(x − a) + f ′′(a) 2! (x − a)^2 + f ′′′(a) 3! (x − a)^3 +... + f (n)(a) n! (x − a)n
f (x) = Pn(x) + Rn(x), Rn(x) = f (n+1)(c) (n + 1)! (x − a)n+1, c ∈ (a, x)
NUMERICAL INTEGRATION
T =
h 2
(y 0 + 2y 1 + 2y 2 +... + 2yn− 1 + yn), | ET |≤ (b − a) 12
h^2 M
S =
h 3
(y 0 + 4y 1 + 2y 2 + 4y 3 +... + 2yn− 2 + 4yn− 1 + yn), | ES |≤ (b − a) 180 h^4 M
NUMERICAL ROOT FINDING ALGORITHMS
Bisection method ([a,b] contains root): error ≤ b − a 2 n Newton’s Method: xn+1 = xn − f (xn) f ′(xn)
STANDARD INTEGRALS
u dv = uv −
v du,
∫ (^) b
a
f (u(x)) du dx dx =
∫ (^) u(b)
u(a)
f (u) du ∫ f ′(u) f (u) du = ln |f (u)| + C,
(f (u))n^ f ′(u) du =
(f (u))n+ n + 1
+ C
sin u du = − cos u + C,
cos u du = sin u + C,
sec^2 u du = tan u + C ∫ tan u du = ln | sec u| + C,
tanh u du = ln(cosh u) + C ∫ cot u du = ln | sin u| + C,
coth u du = ln | sinh u| + C ∫ sec u du = ln | sec u + tan u| + C,
sech u du = tan−^1 | sinh u| + C ∫ csc u du = ln | csc u − cot u| + C,
csch u du = ln | tanh u 2 | + C ∫ du a^2 − u^2
2 a ln
∣∣^ u^ +^ a u − a
∣∣ + C,
du u^2 − a^2
2 a ln
∣∣^ u^ −^ a u + a
∣∣ + C
∫ (^) du √ a^2 − u^2
= sin−^1
(u a
+ C,
du a^2 + u^2
a
tan−^1
(u a
+ C
∫ (^) du
u
u^2 − a^2
a sec−^1
(u a
+ C
∫ (^) du √ u^2 + a^2
= sinh−^1
(u a
u^2 + a^2 | + C ∫ (^) du √ u^2 − a^2
= cosh−^1
(u a
u^2 − a^2 | + C ∫ (^) du
u
a^2 − u^2
a sech −^1
(u a
+ C = −
a ln
∣∣^ a^ +^
a^2 − u^2 u
∣∣ + C
LENGTHS OF CURVES AND SURFACES OF
REVOLUTION
L =
∫ (^) b
a
dy dx
dx =
∫ (^) d
c
dx dy
dy =
ds
rotation around x-axis: S =
∫ (^) b
a
2 πy
dy dx
dx =
∫ (^) d
c
2 πy
dx dy
dy
rotation around y-axis: S =
∫ (^) d
c
2 πx
dx dy
dy =
∫ (^) b
a
2 πx
dy dx
dx
