Download FORMULA SHEET FOR MAT187 Trigonometric Identities. • cos 2(x

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)