MATHEMATICS SPECIALIST
ATAR COURSE
FORMULA SHEET
2017
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2017/1752(v3)
Mathematics Specialist Formula Sheet 2017
MATHEMATICS SPECIALIST
2
Formula SHEET
Index
Differentiation and integration
3
Applications of calculus
Functions
Statistical inference
4
Mensuration
Vectors in 3D
5
Complex numbers
6
Trigonometry
7
See next page
Formula SHEET
3
MATHEMATICS SPECIALIST
Differentiation and integration
x n+1
+ c,
n +1
d n
(x ) = nx n–1
dx
∫ x ndx =
d ax
(e ) = ae ax
dx
∫e
d
1
(1n x) =
dx
x
∫ x dx = ln x
f ′(x)
d
(1n f (x)) =
f (x)
dx
∫
d
(sin f (x)) = f ′(x) cos ( f (x))
dx
∫ sin (ax) dx = – a
d
(cos f (x)) = – f ′(x) sin ( f (x))
dx
∫ cos (ax) dx = a
f ′(x)
d
(tan f (x)) = f ′(x) sec2 ( f (x)) =
cos2 f (x)
dx
∫ sec (ax) dx = a
1
dx = a e ax + c
ax
1
1
1
1
2
Quotient rule
Chain rule
Fundamental theorem
cos (ax) + c
sin (ax) + c
tan (ax) + c
If y = f (x) g(x)
then
or
d
dv du
(uv) = u
+
v
dx
dx dx
then
y' = f '(x) g(x) + f (x) g'(x)
u
v
If y =
+c
f′(x)
dx = 1n f (x) + c
f(x)
If y = uv
Product rule
n = −1
If y =
then
or
f (x)
g(x)
then
dv
du
–u
d u v
dx
dx
=
dx v
2
v
y' =
If y = f (u) and u = g(x)
If y = f (g(x))
then
or
dy = dy × du
dx du dx
d
dx
∫
x
a
f '(x) g (x) – f (x) g'(x)
(g(x))2
then
y' = f '(g(x)) g'(x)
f(t) dt = f (x)
See next page
and
b
∫ f'(x) d x = f (b) − f (a)
a
MATHEMATICS SPECIALIST
4
Formula SHEET
Applications of calculus
Growth and decay
dP = kP
dt
Exponential equation
dP = rP(k – P)
dt
Logistic equation
P=
Volumes of solids of revolution
∫
V = �∫
V=�
About the x-axis
About the y-axis
P = P0ekt
kP0
P0 + (k – P0 )e –rkt
b
[ f(x)]2dx
a
d
c
[ f(y)]2dy
Simple harmonic motion
If
d 2x
= – k 2x
dt 2
then
x = A sin (kt + α)
x = A cos (kt + β)
or
where A is the amplitude, α and β are phase angles, v is the velocity and x is the displacement
v 2 = k 2 (A 2 – x 2 )
Period: T =
2�
k
Frequency: f =
δy ≈
Incremental formula
dv
dt
Acceleration
v
or
1
T
dy
× δx
dx
dv
dx
or
d 1 v2
dx 2
Functions
Quadratic function
2
– √ b −4ac
If f(x) = ax 2 + bx + c and f(x) = 0, then x = −b +
2a
Absolute value function
x, for x ≥ 0
x = –x, for x < 0
Statistical inference
Confidence interval for the mean of the
population:
X –z s ≤ μ ≤ X +z s
√n
√n
Sample size:
n=
(
z×s
d
See next page
)
2
Formula SHEET
5
MATHEMATICS SPECIALIST
Mensuration
Parallelogram
A = bh
Triangle
A=
1
1
bh or A = ab sin C
2
2
Trapezium
A=
1
(a + b)h
2
Circle
A = πr 2 and C = 2πr = πd
Prism
V = Ah, where A is the area of the cross section
Pyramid
V=
Cylinder
V = πr 2h
Cone
V=
1 πr 2h
3
S = πrs + πr 2, where s is the slant height
Sphere
V=
4 πr 3
3
S = 4πr 2
1
Ah, where A is the area of the cross section
3
S = 2πrh + 2πr 2
Vectors in 3D
Magnitude
Dot product
Cross product
Equation of a line
Equation of a plane
(a1, a2, a3) = √ a12 + a22 + a32
a·b = a b cosθ = a1b1 + a2b2 + a3b3
a1
a2 b3 – a3 b2
b1
a × b = a2 × b2 = a3 b1 – a1 b3
a3
a1 b2 – a2 b1
b3
One point and direction
r = a + λu
Two points A and B
r = a + λ(b − a)
or
r = a + λu1 + µu2
r• n = a• n
r–d =r
Equation of a sphere
Cartesian equation of a line
or
(x – a) 2 + ( y – b) 2 + (z – c) 2 = r 2
x – a1
z – a3
y – a2
=
=
u1
u3
u2
Cartesian equation of a plane
ax + by + cz = d
Parametric equation of a line
x = a1 + λu1 ... ... (1)
y = a2 + λu2 ... ... (2)
z = a3 + λu3 ... ... (3)
See next page
MATHEMATICS SPECIALIST
6
Formula SHEET
Complex numbers
Cartesian form
z = a + bi
z = a – bi
Arg(z) = θ,
Mod (z) = z = √a2 + b2 = r
tan θ =
b
,
a
−π < θ ≤ π
z1 z2 = z1 z2
z1
z1
z2 = z2
arg (z1 z2) = arg (z1) + arg (z2)
z
arg z1 = arg (z1) − arg (z2)
2
z z = |z|2
z−1 =
1 z
=
z | z|2
z1 + z2 = z 1 + z 2
z1 z2 = z 1 z2
z = a + bi = r(cos θ + i sin θ) = r cis θ
z = r cis (– θ)
z1 z2 = r1r2 cis (θ1 + θ2)
z 1 r1
z2 = r2 cis (θ1 – θ2)
Polar form
cis(θ1 + θ2) = cis θ1 cis θ2
cis(−θ) =
1
cis θ
De Moivres theorem
(cis θ)n = cos nθ + i sin nθ
z n = z n cis (nθ)
1
q
z =r
1
q
cos θ + 2πk + isin θ + 2πk ,
q
q
for k an integer
See next page
Formula SHEET
7
MATHEMATICS SPECIALIST
Trigonometry
a
b
c
sin A = sin B = sin C
Length of arc = rθ
a2 = b 2 + c 2 − 2bc cosA
Area of segment =
cosA =
b2 + c2 – a2
2bc
Area of sector =
1 2
r (θ − sinθ)
2
1 r2 θ
2
Identities
cos2 x + sin2 x = 1
cos (x +
– y) = cos x cos y
–
+
1 + tan2 x = sec2 x
cos 2x = cos 2 x – sin 2 x
= 2 cos 2 x – 1
sin x sin y
sin (x +
– y) = sin x cos y +
– cos x sin y
sin 2x = 2sin x cos x
tan x –+ tany
tan (x +
– tan x tan y
– y) = 1 +
cos A cos B =
sin A sin B =
= 1 – 2 sin2 x
tan 2x =
1
(cos(A – B) + cos (A + B))
2
1
(cos(A – B) – cos (A + B))
2
2 tan x
1 – tan 2 x
sin A cos B =
1
(sin(A + B) + sin(A – B))
2
cos A sin B =
1
(sin(A + B) – sin(A – B))
2
Note: Any additional formulas identified by the examination panel as necessary will be
included in the body of the particular question.
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