Formula Booklet AAHL
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=√(x1−x2)2+(y1−y2)2
(2x1+x2,2y1+y2)
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
FV=PV(1+100kr)kn
S∞=1−ru1,∣r∣<1
Exponents & Logs
logab=x⟺ax=b, where a,b>0 and a=1
logaxy=logax+logay
logayx=logax−logay
logaxm=mlogax
logax=logbalogbx
Counting
(a+b)n=an+nC1an−1b+⋯+nCran−rbr+bn
nCr=r!(n−r)!n!
nCr=r!(n−r)!n!
nPr=(n−r)!n!
Complex Numbers
z=a+bi
z=r(cosθ+isinθ)=reiθ=rcisθ
[r(cosθ+isinθ)]n=rn(cosnθ+isinnθ)=rneinθ=rncisnθ
Coordinates & Lines
y=mx+c
ax+by+d=0
y−y1=m(x−x1)
m=x2−x1y2−y1
Quadratics
x=−2ab
x=2a−b±√b2−4ac
Δ=b2−4ac
Exponents & Logs
ax=exlna
logaax=x=alogax
Polynomials
sum is −anan−1; product is (−1)nana0
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
Trig IDs
sinAa=sinBb=sinCc
c2=a2+b2−2abcosC
cosC=2aba2+b2−c2
A=21absinC
l=rθ
A=21r2θ
tanθ=cosθsinθ
sin2θ+cos2θ=1
sin2θ=2sinθcosθ
cos2θ=cos2θ−sin2θ=2cos2θ−1=1−2sin2θ
secθ=cosθ1
1+tan2θ=sec2θ
1+cot2θ=cosec2θ
sin(A±B)=sinAcosB±cosAsinB
cos(A±B)=cosAcosB∓sinAsinB
tan(A±B)=1∓tanAtanBtanA±tanB
tan2θ=1−tan2θ2tanθ
Vectors
∣v∣=√v12+v22+v32
v⋅w=v1w1+v2w2+v3w3
v⋅w=∣v∣∣w∣cosθ
cosθ=∣v∣∣w∣v1w1+v2w2+v3w3
r=a+λb
x=x0+λl, y=y0+λm, z=z0+λn
lx−x0=my−y0=nz−z0
v×w=⎝⎛v2w3−v3w2v3w1−v1w3v1w2−v2w1⎠⎞
∣v×w∣=∣v∣∣w∣sinθ
A=∣v×w∣
r=a+λb+μc
r⋅n=a⋅n
ax+by+cz=d
Descriptive Statistics
IQR=Q3−Q1
xˉ=ni=1∑kfixi
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)=0⇒P(A∪B)=P(A)+P(B)
P(A∣B)=P(B)P(A∩B)
P(A∩B)=P(A)P(B)
P(B∣A)=P(B)P(A∣B)+P(B′)P(A∣B′)P(B)P(A∣B)
P(Bi∣A)=P(B1)P(A∣B1)+P(B2)P(A∣B2)+P(B3)P(A∣B3)P(Bi)P(A∣Bi)
Distributions & Vars
E(X)=∑xP(X=x)
E(X)=np
Var(X)=np(1−p)
z=σx−μ
σ2=ni=1∑kfi(xi−μ)2=ni=1∑kfixi2−μ2
σ=⎷ni=1∑kfi(xi−μ)2
E(aX+b)=aE(X)+b
Var(aX+b)=a2Var(X)
E(X)=∫−∞∞xf(x)dx
Var(X)=E(X−μ)2
Var(X)=E(X2)−[E(X)]2
Var(X)=∑(x−μ)2P(X=x)
Var(X)=∑x2P(X=x)−μ2
Var(X)=∫−∞∞(x−μ)2f(x)dx
Var(X)=∫−∞∞x2f(x)dx−μ2
Differentiation
f(x)=xn⟹f′(x)=nxn−1
f(x)=sinx⟹f′(x)=cosx
f(x)=cosx⟹f′(x)=−sinx
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
a=dtdv=dt2d2s
f′(x)=h→0lim(hf(x+h)−f(x))
f(x)=tanx⟹f′(x)=sec2x
f(x)=secx⟹f′(x)=secxtanx
f(x)=cosecx⟹f′(x)=−cosecxcotx
f(x)=cotx⟹f′(x)=−cosec2x
f(x)=ax⟹f′(x)=axlna
f(x)=logax⟹f′(x)=xlna1
f(x)=arcsinx⟹f′(x)=√1−x21
f(x)=arccosx⟹f′(x)=−√1−x21
f(x)=arctanx⟹f′(x)=1+x21
Integration
∫xndx=n+1xn+1+C,n=−1
A=∫abydx
distance =∫t1t2∣v(t)∣dt
displacement =∫t1t2v(t)dt
∫x1dx=lnx+C
∫sinxdx=−cosx+C
∫cosxdx=sinx+C
∫exdx=ex+C
A=∫ab∣y∣dx
∫axdx=lnaax+C
∫a2+x2dx=a1arctan(ax)+C
∫√a2−x2dx=arcsin(ax)+C,∣x∣<a
∫udxdvdx=uv−∫vdxdudx or ∫udv=uv−∫vdu
A=∫ab∣x∣dy
V=π∫aby2dx or V=π∫abx2dy
Differential Equations
yn+1=yn+hf(xn,yn),xn+1=xn+h
μ(x)=e∫P(x)dx
Maclaurin Series
f(x)=f(0)+xf′(0)+2!x2f′′(0)+⋯
ex=1+x+2!x2+⋯
ln(1+x)=x−2x2+3x3−⋯
sinx=x−3!x3+5!x5−⋯
cosx=1−2!x2+4!x4−⋯
arctanx=x−3x3+5x5−⋯
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=√(x1−x2)2+(y1−y2)2
(2x1+x2,2y1+y2)
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
FV=PV(1+100kr)kn
S∞=1−ru1,∣r∣<1
Exponents & Logs
logab=x⟺ax=b, where a,b>0 and a=1
logaxy=logax+logay
logayx=logax−logay
logaxm=mlogax
logax=logbalogbx
Counting
(a+b)n=an+nC1an−1b+⋯+nCran−rbr+bn
nCr=r!(n−r)!n!
nCr=r!(n−r)!n!
nPr=(n−r)!n!
Complex Numbers
z=a+bi
z=r(cosθ+isinθ)=reiθ=rcisθ
[r(cosθ+isinθ)]n=rn(cosnθ+isinnθ)=rneinθ=rncisnθ
Coordinates & Lines
y=mx+c
ax+by+d=0
y−y1=m(x−x1)
m=x2−x1y2−y1
Quadratics
x=−2ab
x=2a−b±√b2−4ac
Δ=b2−4ac
Exponents & Logs
ax=exlna
logaax=x=alogax
Polynomials
sum is −anan−1; product is (−1)nana0
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
Trig IDs
sinAa=sinBb=sinCc
c2=a2+b2−2abcosC
cosC=2aba2+b2−c2
A=21absinC
l=rθ
A=21r2θ
tanθ=cosθsinθ
sin2θ+cos2θ=1
sin2θ=2sinθcosθ
cos2θ=cos2θ−sin2θ=2cos2θ−1=1−2sin2θ
secθ=cosθ1
1+tan2θ=sec2θ
1+cot2θ=cosec2θ
sin(A±B)=sinAcosB±cosAsinB
cos(A±B)=cosAcosB∓sinAsinB
tan(A±B)=1∓tanAtanBtanA±tanB
tan2θ=1−tan2θ2tanθ
Vectors
∣v∣=√v12+v22+v32
v⋅w=v1w1+v2w2+v3w3
v⋅w=∣v∣∣w∣cosθ
cosθ=∣v∣∣w∣v1w1+v2w2+v3w3
r=a+λb
x=x0+λl, y=y0+λm, z=z0+λn
lx−x0=my−y0=nz−z0
v×w=⎝⎛v2w3−v3w2v3w1−v1w3v1w2−v2w1⎠⎞
∣v×w∣=∣v∣∣w∣sinθ
A=∣v×w∣
r=a+λb+μc
r⋅n=a⋅n
ax+by+cz=d
Descriptive Statistics
IQR=Q3−Q1
xˉ=ni=1∑kfixi
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)=0⇒P(A∪B)=P(A)+P(B)
P(A∣B)=P(B)P(A∩B)
P(A∩B)=P(A)P(B)
P(B∣A)=P(B)P(A∣B)+P(B′)P(A∣B′)P(B)P(A∣B)
P(Bi∣A)=P(B1)P(A∣B1)+P(B2)P(A∣B2)+P(B3)P(A∣B3)P(Bi)P(A∣Bi)
Distributions & Vars
E(X)=∑xP(X=x)
E(X)=np
Var(X)=np(1−p)
z=σx−μ
σ2=ni=1∑kfi(xi−μ)2=ni=1∑kfixi2−μ2
σ=⎷ni=1∑kfi(xi−μ)2
E(aX+b)=aE(X)+b
Var(aX+b)=a2Var(X)
E(X)=∫−∞∞xf(x)dx
Var(X)=E(X−μ)2
Var(X)=E(X2)−[E(X)]2
Var(X)=∑(x−μ)2P(X=x)
Var(X)=∑x2P(X=x)−μ2
Var(X)=∫−∞∞(x−μ)2f(x)dx
Var(X)=∫−∞∞x2f(x)dx−μ2
Differentiation
f(x)=xn⟹f′(x)=nxn−1
f(x)=sinx⟹f′(x)=cosx
f(x)=cosx⟹f′(x)=−sinx
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
a=dtdv=dt2d2s
f′(x)=h→0lim(hf(x+h)−f(x))
f(x)=tanx⟹f′(x)=sec2x
f(x)=secx⟹f′(x)=secxtanx
f(x)=cosecx⟹f′(x)=−cosecxcotx
f(x)=cotx⟹f′(x)=−cosec2x
f(x)=ax⟹f′(x)=axlna
f(x)=logax⟹f′(x)=xlna1
f(x)=arcsinx⟹f′(x)=√1−x21
f(x)=arccosx⟹f′(x)=−√1−x21
f(x)=arctanx⟹f′(x)=1+x21
Integration
∫xndx=n+1xn+1+C,n=−1
A=∫abydx
distance =∫t1t2∣v(t)∣dt
displacement =∫t1t2v(t)dt
∫x1dx=lnx+C
∫sinxdx=−cosx+C
∫cosxdx=sinx+C
∫exdx=ex+C
A=∫ab∣y∣dx
∫axdx=lnaax+C
∫a2+x2dx=a1arctan(ax)+C
∫√a2−x2dx=arcsin(ax)+C,∣x∣<a
∫udxdvdx=uv−∫vdxdudx or ∫udv=uv−∫vdu
A=∫ab∣x∣dy
V=π∫aby2dx or V=π∫abx2dy
Differential Equations
yn+1=yn+hf(xn,yn),xn+1=xn+h
μ(x)=e∫P(x)dx
Maclaurin Series
f(x)=f(0)+xf′(0)+2!x2f′′(0)+⋯
ex=1+x+2!x2+⋯
ln(1+x)=x−2x2+3x3−⋯
sinx=x−3!x3+5!x5−⋯
cosx=1−2!x2+4!x4−⋯
arctanx=x−3x3+5x5−⋯