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MATH1051
Calculus & Linear Algebra
University of Queensland
1 Complex Numbers
z = a + ib
Where:
i2
= −1
Modulus
|z| = a2 + b2
Conjugate
If
z = a + ib
then
z = z − ib
Properties of Modulus and Conjugate
|zw| = |z||w|
z
w
|z|
|w|
|z + w| ≤ |z| + |w|
|z − w| ≥ |z| − |w|
z ± w = z ± w
zw = zw
Polar Form
z = rcisθ = r(cos θ + i sin θ)
Where r is the modulus ||z|| and θ the argument arg z.
Polar Form Operations and Properties
rcis(θ) × pcis(φ) = rpcis(θ + φ)
rcis(θ)
pcis(φ)
=
r
p
cis(θ − φ)
arg(zw) = arg(z) + arg(w)
arg
z
w
= arg(z) − arg(w)
Euler’s Formula
eix
= cos x + i sin x
Where e is Euler’s Number and x is real.
De Moivre’s Theorem
Given z = reiθ
:
zn
= rn
einθ
2 Vectors
Length of a vector
|v| = x2 + y2 + z2
Dot Product
v · w = vxwx + vywy + vzwz = |v||w|cosθ
Angle Between Vectors
cos θ =
a · b
a b
Cross Product
v × w =
ˆi ˆj ˆk
vx vy vz
wx wy wz
Area of a Triangle
Formed by vectors v and u is:
A =
|v × u|
2
Scalar projection v in direction of w
vw =
v · w
|w|
Vector projection v in direction of w
vw =
v · w
|w|2
w
Triangle Inequality
|v + w| ≤ |v| + |w|
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Scalar Triple
[u, v, w] = u · (v × w)
v × w
v
u
w
V = [u, v, w]
Volume of a parallelpiped
3 Matrices
Matrix Multiplication
AB =
a b c
x y z


α ρ
β σ
γ τ


=
aα + bβ + cγ aρ + bσ + cτ
xα + yβ + zγ xρ + yσ + zτ
Determinants
2x2 Matrix
det
a b
c d
=
a b
c d
= ad − cb
3x3 Matrix
a b c
d e f
g h i
= a
e f
h i
− b
d f
g i
+ c
d e
g h
4 Lines in 3D
Parametric Equation of a Line



x(t) = a + pt
y(t) = b + qt
z(t) = c + rt



Symmetric Form of Line Equation
x − a
p
=
y − b
q
=
z − c
r
Vector Equation of a Line
r(t) = d + tv
5 Planes in 3D
Cartesian Equation of a Plane
ax + by + cz = d
Parametric Equation of a Plane



x(u, v) = a + pu + lv
y(u, v) = b + qu + mv
z(u, v) = c + ru + nv



Vector Equation of a Plane
n · (r − d) = 0
6 Logarithms
Basic Properties
loga 1 = 0
loga a = 1
loga xy = loga x + loga y
loga
x
y
= loga x − loga y
loga xn
= n loga x
Change of Base
ax
= bx logb a
loga x =
logb x
logb a
7 Inverse Trigonometric Functions
−1 −0.5 0.5 1
−π
−
π
2
π
2
π
y = arcsin x
−1 −0.5 0.5 1
π
2
π
3π
2
2π
y = arccos x
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−3 −2 −1 1 2 3
−
π
2
π
2
y = arctan x
8 Function Composition
(g ◦ f)(x) = g(f(x))
Important Limits
lim
x→0
sin x
x
= 1
lim
x→0
1 − cos x
x2
=
1
2
lim
x→0
ex
− 1
x
= 1
9 Differentiation
Chain Rule
dy
dx
=
dy
du
du
dx
Or if f(x) = g(h(x)) then:
f (x) = g (h(x))h (x)
Differentiating Inverse Functions
Use x = f(f−1
(x)) = f(y) and:
dy
dx
=
1
dx
dy
Derivatives of Inverse Trig Functions
d[arcsin x]
dx
=
1
√
1 − x2
for − 1 < x < 1
d[arccos x]
dx
=
−1
√
1 − x2
for − 1 < x < 1
d[arctan x]
dx
=
1
1 + x2
L’Hopital’s Rule
If
lim
x→a
f(x)
g(x)
=
0
0
OR
±∞
±∞
Then use:
lim
x→a
f(x)
g(x)
= lim
x→a
f (x)
g (x)
10 Integration
General Riemann Sum Formula (Area Under a
Curve)
Over interval [a, b] with n rectangles:
A =
n
k=1
f(xk)∆x
Where: ∆x =
b − a
n
Integral as Riemann Summ
b
a
f(x) dx = lim
n→∞
n
k=1
f(xk)∆x
Where: ∆x =
b − a
n
Fundamental Theorem of Calculus
Given f is continuous over [a, b]
1. If F(x) =
x
a
f(u) du then F (x) = f(x)
2. If G(x) is any antiderivative of f(x) then:
b
a
f(x) dx = G(b) − G(a)
Trapezoidal Rule
Tn =
∆x
2
{f(x0) + 2f(x1) + ... + 2f(xn−1 + f(xn)}
Where: ∆x =
b − a
n
Integration By Substitution
I = f(x) dx = f(x(u))
dx
du
du
Integration By Parts
u dv = uv − v du
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Partial Fractions: Important Rules
Irreducible Quadratic Q(x) in Denominator:
Factor on RHS is:
Ax + B
Q(x)
Multiple Root (x + a)n
in Denominator:
Include on RHS:
A1
x + a
+
A2
(x + a)2
+
A3
(x + a)3
+ ... +
An
(x + a)n
Volumes of Revolution
Rotating Around x-Axis
V = π
b
a
[f(x)]2
dx
11 Series and Sequences
Arithmetic
an = an−1 + d = a + nd
Sn = (n + 1)a +
n(n + 1)d
2
Geometric
an = san−1 = a0sn
Partial sums for geometric series
Sn =



(n + 1)a0 : s = 1
(1 − sn+1
)a0
1 − s
: s = 1
Newton’s Method for Finding Roots of Polyno-
mials
xn+1 = xn −
f(xn)
f (xn)
Power Series
In x around the point x = a:
a0 + a1(x − a) + a2(x − a)2
+ ... + an(x − a)n
+ ...
= Σ∞
n=0an(x − a)n
Maclaurin Series
f(x) = a0 + a1x + a2x2
+ ... + anxn
+ ...
with:
an =
1
n!
dn
f
dxn
x=0
Taylor Series
If f(x) is infinitely differentiable at x=0, then:
f(x) = a0 + a1(x − 1) + a2(x − a)2
+ ... + an(x − a)n
+ ...
with:
an =
1
n!
dn
f
dxn
x=a
12 Convergence Tests
For
∞
n=1
an
Necessary and Sufficient Conditions
Necessary:
If convergent, it is necessary that
an → 0
Sufficient:
an → 0
is not sufficient (by itself) to prove convergence
Ratio Test
L = lim
n→∞
an+1
an
L < 1 Absolute convergence
L > 1 does not converge
L = 1 or if no limit exists, then inconclusive
Comparison Test
If C = Σ∞
n=0cn is a convergent series and D = Σ∞
n=0dn a
divergent one then:
Convergence: When 0 < an < cn ∀n > m
Divergence: When 0 < dn < an ∀n > m
Alternating Series
If |an| → ∞:
Convergence: When |an+1| < |an| ∀n > m
Divergence: Otherwise
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Alternating Series Estimation Theorem
Given an = s then:
|s − an| ≤ an+1
Absolute Convergence
If Σ∞
n=0|an| converges then Σ∞
n=0an converges absolutely,
otherwise we can’t say anything.
P-series
∞
n=1
1
np
,
For positive, real p. Cases:
p = 1 Divergent (harmonic series)
p > 1 Convergent
p < 1 Divergent
13 Taylor Series
Useful Functions
ex
= 1 + x +
x2
2!
+
x3
3!
+ ... = Σ∞
n=0
xn
n!
cos x = 1 −
x2
2!
+
x4
4!
−
x6
6!
+ ... = Σ∞
n=0(−1)n x2n
2n!
sin x = x −
x3
3!
+
x5
5!
−
x7
7!
+ ... = Σ∞
n=0(−1)n+1 x2n+1
(2n + 1)!
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