Formula Booklet AIHL
Prior Learning
A=bh
A=21(bh)
A=21(a+b)h
A=πr2
C=2πr
V=lwh
V=πr2h
V=Ah
A=2πrh
d=√(x2−x1)2+(y2−y1)2
(2x1+x2,2y1+y2)
x=2a−b±√b2−4ac
Sequences Series
un=u1+(n−1)d
Sn=2n(2u1+(n−1)d)=2n(u1+un)
un=u1rn−1
Sn=r−1u1(rn−1)=1−ru1(1−rn),r=1
S∞=1−ru1,∣r∣<1
Financial Mathematics
FV=PV(1+100kr)kn
Exponents & Logs
logab=x⟺ax=b, where a>0, b>0, a=1
loga(xy)=logax+logay
logayx=logax−logay
loga(xm)=mlogax
Approximations Error
ε=∣∣∣∣vEvA−vE∣∣∣∣×100%
Complex Numbers
z=a+bi
Δ=b2−4ac
z=r(cosθ+isinθ)=reiθ=rcisθ
Matrices
detA=∣A∣=ad−bc
A−1=detA1(d−c−ba), ad=bc
Mn=PDnP−1
Coordinates & Lines
y=mx+c
ax+by+d=0
y−y1=m(x−x1)
m=x2−x1y2−y1
Modelling
x=−2ab
f(x)=1+Ce−kxL, L,C,k>0
2&3D Geometry
d=√(x2−x1)2+(y2−y1)2+(z2−z1)2
(2x1+x2,2y1+y2,2z1+z2)
V=31Ah
V=31πr2h
A=πrl
V=34πr3
A=4πr2
l=360∘θ⋅2πr
A=360∘θ⋅πr2
l=rθ
A=21r2θ
Trig IDs
sinAa=sinBb=sinCc
c2=a2+b2−2abcosC
cosC=2aba2+b2−c2
A=21absinC
cos2θ+sin2θ=1
tanθ=cosθsinθ
Matrices
(cos2θsin2θsin2θ−cos2θ)
(k001)
(100k)
(k00k)
(cosθsinθ−sinθcosθ)
(cosθ−sinθsinθcosθ)
Vectors
∣v∣=√v12+v22+v32
r=a+λb
x=x0+λl,y=y0+λm,z=z0+λn
v⋅w=v1w1+v2w2+v3w3
v⋅w=∣v∣∣w∣cosθ
cosθ=∣v∣∣w∣v1w1+v2w2+v3w3
v×w=⎝⎛v2w3−v3w2v3w1−v1w3v1w2−v2w1⎠⎞
∣v×w∣=∣v∣∣w∣sinθ
A=∣v×w∣
Descriptive Statistics
IQR=Q3−Q1
xˉ=ni=1∑kfixi
sn2=n−1ns2
Probability
P(A)=n(U)n(A)
P(A)+P(A′)=1
P(A∪B)=P(A)+P(B)−P(A∩B)
P(A∪B)=P(A)+P(B)
P(A∣B)=P(B)P(A∩B)
P(A∩B)=P(A)P(B)
sn=Tns0
Distributions & Vars
E(X)=∑xP(X=x)
X∼B(n,p)
E(X)=np
Var(X)=np(1−p)
E(aX+b)=aE(X)+b
Var(aX+b)=a2Var(X)
E(a1X1+⋯+anXn)=a1E(X1)+⋯+anE(Xn)
Var(a1X1+⋯+anXn)=a12Var(X1)+⋯+an2Var(Xn)
X∼Po(m)
E(X)=m
Var(X)=m
Differentiation
f(x)=xn⟹f′(x)=nxn−1
f(x)=sinx⟹f′(x)=cosx
f(x)=cosx⟹f′(x)=−sinx
f(x)=tanx⟹f′(x)=cos2x1
f(x)=ex⟹f′(x)=ex
f(x)=lnx⟹f′(x)=x1
dxdy=dudy⋅dxdu
y=uv⟹dxdy=udxdv+vdxdu
y=vu⟹dxdy=v2vdxdu−udxdv
Integration
∫xndx=n+1xn+1+C,n=−1
A=∫abydx
∫abydx≈2h(y0+2(y1+⋯+yn−1)+yn)
∫x1dx=lnx+C
∫sinxdx=−cosx+C
∫cosxdx=sinx+C
∫cos2x1dx=tanx+C
∫exdx=ex+C
A=∫ab∣y∣dx or A=∫ab∣x∣dy
V=π∫aby2dx or V=π∫abx2dy
a=dtdv=dt2d2s
distance =∫t1t2∣v(t)∣dt
displacement =∫t1t2v(t)dt
Differential Equations
yn+1=yn+hf(xn,yn),xn+1=xn+h
xn+1=xn+hf1(xn,yn,tn),yn+1=yn+hf2(xn,yn,tn),tn+1=tn+h
x=Aeλ1tp1+Beλ2tp2
Prior Learning
A=bh
A=21(bh)
A=21(a+b)h
A=πr2
C=2πr
V=lwh
V=πr2h
V=Ah
A=2πrh
d=√(x2−x1)2+(y2−y1)2
(2x1+x2,2y1+y2)
x=2a−b±√b2−4ac
Sequences Series
un=u1+(n−1)d
Sn=2n(2u1+(n−1)d)=2n(u1+un)
un=u1rn−1
Sn=r−1u1(rn−1)=1−ru1(1−rn),r=1
S∞=1−ru1,∣r∣<1
Financial Mathematics
FV=PV(1+100kr)kn
Exponents & Logs
logab=x⟺ax=b, where a>0, b>0, a=1
loga(xy)=logax+logay
logayx=logax−logay
loga(xm)=mlogax
Approximations Error
ε=∣∣∣∣vEvA−vE∣∣∣∣×100%
Complex Numbers
z=a+bi
Δ=b2−4ac
z=r(cosθ+isinθ)=reiθ=rcisθ
Matrices
detA=∣A∣=ad−bc
A−1=detA1(d−c−ba), ad=bc
Mn=PDnP−1
Coordinates & Lines
y=mx+c
ax+by+d=0
y−y1=m(x−x1)
m=x2−x1y2−y1
Modelling
x=−2ab
f(x)=1+Ce−kxL, L,C,k>0
2&3D Geometry
d=√(x2−x1)2+(y2−y1)2+(z2−z1)2
(2x1+x2,2y1+y2,2z1+z2)
V=31Ah
V=31πr2h
A=πrl
V=34πr3
A=4πr2
l=360∘θ⋅2πr
A=360∘θ⋅πr2
l=rθ
A=21r2θ
Trig IDs
sinAa=sinBb=sinCc
c2=a2+b2−2abcosC
cosC=2aba2+b2−c2
A=21absinC
cos2θ+sin2θ=1
tanθ=cosθsinθ
Matrices
(cos2θsin2θsin2θ−cos2θ)
(k001)
(100k)
(k00k)
(cosθsinθ−sinθcosθ)
(cosθ−sinθsinθcosθ)
Vectors
∣v∣=√v12+v22+v32
r=a+λb
x=x0+λl,y=y0+λm,z=z0+λn
v⋅w=v1w1+v2w2+v3w3
v⋅w=∣v∣∣w∣cosθ
cosθ=∣v∣∣w∣v1w1+v2w2+v3w3
v×w=⎝⎛v2w3−v3w2v3w1−v1w3v1w2−v2w1⎠⎞
∣v×w∣=∣v∣∣w∣sinθ
A=∣v×w∣
Descriptive Statistics
IQR=Q3−Q1
xˉ=ni=1∑kfixi
sn2=n−1ns2
Probability
P(A)=n(U)n(A)
P(A)+P(A′)=1
P(A∪B)=P(A)+P(B)−P(A∩B)
P(A∪B)=P(A)+P(B)
P(A∣B)=P(B)P(A∩B)
P(A∩B)=P(A)P(B)
sn=Tns0
Distributions & Vars
E(X)=∑xP(X=x)
X∼B(n,p)
E(X)=np
Var(X)=np(1−p)
E(aX+b)=aE(X)+b
Var(aX+b)=a2Var(X)
E(a1X1+⋯+anXn)=a1E(X1)+⋯+anE(Xn)
Var(a1X1+⋯+anXn)=a12Var(X1)+⋯+an2Var(Xn)
X∼Po(m)
E(X)=m
Var(X)=m
Differentiation
f(x)=xn⟹f′(x)=nxn−1
f(x)=sinx⟹f′(x)=cosx
f(x)=cosx⟹f′(x)=−sinx
f(x)=tanx⟹f′(x)=cos2x1
f(x)=ex⟹f′(x)=ex
f(x)=lnx⟹f′(x)=x1
dxdy=dudy⋅dxdu
y=uv⟹dxdy=udxdv+vdxdu
y=vu⟹dxdy=v2vdxdu−udxdv
Integration
∫xndx=n+1xn+1+C,n=−1
A=∫abydx
∫abydx≈2h(y0+2(y1+⋯+yn−1)+yn)
∫x1dx=lnx+C
∫sinxdx=−cosx+C
∫cosxdx=sinx+C
∫cos2x1dx=tanx+C
∫exdx=ex+C
A=∫ab∣y∣dx or A=∫ab∣x∣dy
V=π∫aby2dx or V=π∫abx2dy
a=dtdv=dt2d2s
distance =∫t1t2∣v(t)∣dt
displacement =∫t1t2v(t)dt
Differential Equations
yn+1=yn+hf(xn,yn),xn+1=xn+h
xn+1=xn+hf1(xn,yn,tn),yn+1=yn+hf2(xn,yn,tn),tn+1=tn+h
x=Aeλ1tp1+Beλ2tp2