Ride the Storm: Navigating Through Unstable Periods / Katerina Rudko (Belka G...
MATH1131C Past Paper Summary | University of New South Wales
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MATH1131
Mathematics 1A
University of New South Wales
1 Trigonometry
Pythagorean Identity
cos2
x + sin2
x = 1
Angle Sum Formulae
sin(θ + φ) = sin θ cos φ + sin φ cos θ
cos(θ + φ) = cos θ cos φ − sin θ sin φ
tan(θ + φ) =
tan θ + tan φ
1 − tan θ tan φ
Double Angle Formula
sin 2θ = 2 sin θ cos θ =
2 tan θ
1 + tan2
θ
cos 2θ = cos2
θ − sin2
θ
= 2 cos2
θ − 1
= 1 − 2 sin2
θ
tan 2θ =
2 tan θ
1 − tan2
θ
2 Function Composition
(g ◦ f)(x) = g(f(x))
3 Differentiation
Scalar Multiple
d[kf(x)]
dx
= kf (x)
Power Function
d[xn
]
dx
= nxn−1
Addition Rule
d[u(x) + v(x)]
dx
= u (x) + v (x)
Product Rule
d[u(x)v(x)]
dx
= u(x)v (x) + u (x)v(x)
Quotient Rule
If f(x) =
u(x)
v(x)
then:
f (x) =
v(x)u (x) − u(x)v (x)
[v(x)]2
Chain Rule
If f(x) = u[v(x)] then:
f (x) = v (x)u (x)
That is:
df
dx
=
df
du
du
dx
Exponential Function
d[ex
]
dx
= ex
Logarithmic Functions
d[ln x]
dx
=
1
x
for x > 0
Trig Functions
d[sin x]
dx
= cos x
d[cos x]
dx
= − sin x
d[tan x]
dx
= sec2
x
Hyperbolic Functions
d[sinh x]
dx
= cosh x
d[cosh x]
dx
= − sinh x
d[tanh x]
dx
=
1
cosh2
x
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
1
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Derivative Tests
Stationary Points
f (x) = 0
Concavity
• f (x) > 0: Concave up - local minimum
• f (x) < 0: Concave down - local maximum
Point of Inflection
f (x) = 0
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)
4 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
−3 −2 −1 1 2 3
−
π
2
π
2
y = arctan x
5 Integration
Left Riemann Sum Formula
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)
Integration By Substitution
I = f(x) dx = f(x(u))
dx
du
du
Integration By Parts
u dv = uv − v du
2
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Standard Integrals
xn
dx =
xn+1
n + 1
+ C (n = −1)
cos x dx = sin x + C
sin x dx = − cos x + C
sec2
x dx = − tan x + C
ex
dx = ex
+ C
1
x
dx = loge(x) + C
cosh x dx = sinh x + C
sinh x dx = cosh x + C
1
√
1 − x2
dx = arcsin x + C
1
1 + x2
dx = arctan x + C
6 Log and Exp Functions
Properties of Logarithms
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
Logarithmic Differentiation
f (x) = f(x)[ln f(x)]
7 Hyperbolic Trigonometric Func-
tions
y=0.5
cosh(x) =
ex
+ e−x
2
sinh(x) =
ex
− e−x
2
tanh x =
sinh x)
cosh x
Properties of Hyperbolic Functions
cosh2
x − sinh2
x = 1
cosh(x + y) = cosh x cosh y + sinh x sinh y
sinh(x − y) = sinh x cosh y − sinh y cosh x
tanh(x + y) =
tanh x + tanh y
1 + tanh x tanh y
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