Tuanaki, Kaupae 3, 2017 · cosecc osec cot cotc osec () d 1 (1 ) 1 ln () ln d d d d.d d d d d d d...
Transcript of Tuanaki, Kaupae 3, 2017 · cosecc osec cot cotc osec () d 1 (1 ) 1 ln () ln d d d d.d d d d d d d...
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© Mana Tohu Mātauranga o Aotearoa, 2017. Pūmau te mana.Kia kaua rawa he wāhi o tēnei tuhinga e tāruatia ki te kore te whakaaetanga a te Mana Tohu Mātauranga o Aotearoa.
Tuanaki, Kaupae 3, 2017
9.30 i te ata Rāpare 23 Whiringa-ā-rangi 2017
TE PUKAPUKA O NGĀ TIKANGA TĀTAI ME NGĀ TŪTOHI
mō 91577M, 91578M me 91579M
Tirohia tēnei pukapuka hei whakatutuki i ngā tūmahi o ō Pukapuka Tūmahi, Tuhinga hoki.
Tirohia mēnā e tika ana te raupapatanga o ngā whārangi 2 – 7 kei roto i tēnei pukapuka, ka mutu, kāore tētahi o aua whārangi i te takoto kau.
KA TAEA TĒNEI PUKAPUKA TE PUPURI HEI TE MUTUNGA O TE WHAKAMĀTAUTAU.
L 3 – C A L C M F993203
TE PĀNGARAU – ĒTAHI TURE WHAI HUA
TE ĀHUAHANGA TAUNGA
Te Rārangi Torotika
Whārite y y m x x− = −1 1( )
Te Porohita( ) ( )x a y b r− + − =2 2 2
ko te (a,b) te pū, ko te r te pūtoro
Te Unahi
Arotahi (a,0) x = – a
Te Pororapa
Arotahi (c,0) (–c,0) ina b2 = a2 – c2
Pūkētanga: e ac=
Te Pūwerewere
rārangi pātata y ba
x= ±
Arotahi (c,0) (–c,0) ina b2 = c2 – a2
Pūkētanga: e ac=
Rārangi whakarite
rānei
rānei
x2
a2y2
b2= 1, (asecθ ,b tanθ )
x2
a2+
y2
b2= 1, (acosθ ,bsinθ )
rānei
y2 = 4ax , (at2 ,2at)
∫
= = ʹ
−
−
−
++
≠ −
+
ʹ+
=
=⎛
⎝⎜⎞
⎠⎟
+
y f x dydx
f x
xxa
x xx x
x xx x xx x x
x x
f x f x x
x xn
c
n
xx c
f xf x
f x c
yx
yttx
yx t
yx
tx
( ) ( )
ln 1
e esin coscos sin
tan secsec sec tan
cosec cosec cot
cot cosec
( ) ( )d
1
( 1)
1 ln
( )( )
ln ( )
dd
dd
. dd
dd
dd
dd
. dd
ax ax
nn
2
2
1
2
2
TE TUANAKIKimi Pārōnaki
Ngā Tikanga Pāwhaitua
Te Pānga Tawhā
ax2 + bx + c = 0
x = b ± b2 4ac2a
y = logb x x = by
logb xy( ) = logb x + logb y
logbxy
= logb x logb y
logb xn( ) = n logb x
logb x =loga xloga b
z = x + iy= r cis= r(cos + isin )
z = x iy= r cis ( )= r(cos isin )
r = z = zz = (x2 + y2 )
= arg z
cos =xr
sin =yr
(r cis )n = rn cis (n )
TE TAURANGINgā Whārite Pūrua
Mēnā
kāti
Ngā Taupū Kōaro
Ngā Tau Matatini
ina
ā,
Te Ture a De MoivreMēnā he tau tōpū a n, kāti,
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MATHEMATICS – USEFUL FORMULAE
COORDINATE GEOMETRYStraight LineEquation y − y1 = m(x − x1)
Circle
(x − a)2 + ( y − b)2 = r2
has a centre (a,b) and radius r
Parabola
y2 = 4ax or (at2,2at)Focus (a,0) Directrix x = − a
Ellipse
x2
a2+
y2
b2= 1 or (acosθ ,bsinθ)
Foci (c,0) (–c,0) where b2 = a2 – c2
Eccentricity: e = ca
Hyperbola
x2
a2−
y2
b2= 1 or (asecθ ,b tanθ)
asymptotes y = ± ba
x
Foci (c,0) ( − c,0) where b2 = c2 – a2
Eccentricity: e = ca
∫
= = ′
−
−
−
++
≠ −
+
′ +
=
=
+
y f x dydx
f x
xx
ax xx x
x xx x xx x x
x x
f x f x x
x xn
c
n
xx c
f xf x
f x c
yx
yt
tx
yx t
yx
tx
CALCULUSDifferentiation
Integration
Parametric Function
( ) ( )
ln 1
e esin coscos sin
tan secsec sec tan
cosec cosec cot
cot cosec
( ) ( )d
1
( 1)
1 ln
( )( )
ln ( )
dd
dd
. dd
dd
dd
dd
. dd
ax ax
nn
2
2
1
2
2
ALGEBRAQuadratics
If ax2 + bx + c = 0
then x = −b ± b2 − 4ac2a
Logarithms
y = logb x⇔ x = by
logb xy( ) = logb x + logb y
logbxy
⎛⎝⎜
⎞⎠⎟
= logb x − logb y
logb xn( ) = n logb x
logb x =loga xloga b
Complex numbersz = x + iy= r cisθ= r(cosθ + isinθ )
z = x − iy= r cis (−θ )= r(cosθ − isinθ )
r = z = zz = (x2 + y2 )
θ = arg z
where cosθ = xr
and sinθ = yr
De Moivre’s TheoremIf n is any integer, then
(r cisθ )n = rn cis (nθ )
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1 whakarea
( f .g) = f .g + g. f y = uv dydx
= u dvdx
+ v dudx
fg
=g. f f .g
g2 y = uv
dydx
=v du
dxu dv
dxv2
f g( )( ) = f g( ).g
y = f (u) u = g(x) dydx
=dydu
.dudx
f (x) dxa
b 12h y0 + yn + 2( y1 + y2 + ...+ yn 1)
h = b an
yr = f (xr )
f (x) dx 13h y0 + yn + 4( y1 + y3 + ...+ yn 1)+ 2( y2 + y4 + ...+ yn 2 )
a
b
h = b an
, yr = f (xr )
Te Ture mō te Otinga Whakarau1
mēnā rānei kāti
Te Ture mō te Otinga Wehe
Te Ture Pānga Hiato, te Ture Mekameka rānei
NGĀ TIKANGA TAUTe Ture Taparara
ina ā,
Te Ture a Simpson
ina , ā, he taurua te n.
mēnā rānei kāti
mēnā rānei kātiā,
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Product Rule
( f .g ′) = f . ′g + g. ′f or if y = uv then dydx
= u dvdx
+ v dudx
Quotient Rule
fg
⎛⎝⎜
⎞⎠⎟
′= g. ′f − f . ′g
g2 or if y = u
v then
dydx
=v
dudx
− u dvdx
v2
Composite Function or Chain Rule
f g( )( )′ = ′f g( ). ′g
or if y = f (u) and u = g(x) then dydx
= dydu
.dudx
NUMERICAL METHODSTrapezium Rule
f (x) dxa
b
∫ ≈ 12h y0 + yn + 2( y1 + y2 + ...+ yn−1)⎡⎣ ⎤⎦
where h = b− an
and yr = f (xr )
Simpson’s Rule
f (x) dx ≈ 13h y0 + yn + 4( y1 + y3 + ...+ yn−1)+ 2( y2 + y4 + ...+ yn−2 )⎡⎣ ⎤⎦
a
b
∫where h = b− a
n, yr = f (xr ) and n is even.
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1
1
1
2 3
2
� 6 –
� 3 –
� 4 –
� 4 –
cosec θ =1
sinθ
sec θ =1
cosθ
cot θ =1
tanθ
cot θ =cosθsinθ
asin A
=b
sin B=
csinC
c2 = a2 + b2 2ab cosC
sin(A ± B) = sin Acos B ± cos Asin Bcos( A ± B) = cos Acos B sin Asin B
tan( A ± B) =tan A ± tan B
1 tan A tan B
sin2A = 2sin Acos A
tan 2A =2 tan A
1 tan2 A
cos2A = cos2 A sin2 A
= 2cos2 A 1
= 1 2sin2 A
TE PĀKOKI
Te Ture Aho
Te Ture Whenu
Ngā Whārite ka Pono Ahakoa ngā Uara Ka Whakaurua Atu
Ngā Otinga Whānui
Ngā Koki Hiato
Ngā Koki Rearua
±
±
cos2 θ + sin2 θ = 1
tan2 θ + 1 = sec2 θ
cot2 θ + 1 = cosec2 θ
Mēnā sin θ = sin α kāti θ = n� + (−1)n αMēnā cos θ = cos α kāti θ = 2n� ± αMēnā tan θ = tan α kāti θ = n� + αko te n, he tau tōpū ahakoa
Ngā Otinga Whakarau
22
sin cos sin( ) sin( )cos sin sin(
A B A B A BA B A
= + + −= + BB A B
A B A B A B) sin( )
cos cos cos( ) cos( )sin
− −= + + −2
2 AA B A B A Bsin cos( ) cos( )= − − +
Ngā Otinga Tāpiri
sin sin sin cos
sin sin cos
C D C D C D
C D C D
+ = + −
− = +
22 2
22
ssin
cos cos cos cos
cos cos
C D
C D C D C D
C D
−
+ = + −
− =
2
22 2
−− + −22 2
sin sinC D C D
TE INE
Te Tapatoru
Te Taparara
Te Pewanga
Te Rango
Te Koeko
Te Poi
Horahanga =
Horahanga =
Horahanga =
Te roa o te pewa = rθ
Rōrahi = πr 2hHorahanga mata kōpiko = 2πrh
Rōrahi =
Rōrahi =
r 2θ
Horahanga mata kōpiko = πrl ina ko te l te teitei o te tītaha
πr 2h
Horahanga mata = 4πr 2
πr 343
13
12
12
12
(a + b)h
absinC
cosec θ =1
sinθ
sec θ =1
cosθ
cot θ =1
tanθ
cot θ =cosθsinθ
asin A
=b
sin B=
csinC
c2 = a2 + b2 2ab cosC
sin(A ± B) = sin Acos B ± cos Asin Bcos( A ± B) = cos Acos B sin Asin B
tan( A ± B) =tan A ± tan B
1 tan A tan B
sin2A = 2sin Acos A
tan 2A =2 tan A
1 tan2 A
cos2A = cos2 A sin2 A
= 2cos2 A 1
= 1 2sin2 A
TE PĀKOKI
Te Ture Aho
Te Ture Whenu
Ngā Whārite ka Pono Ahakoa ngā Uara Ka Whakaurua Atu
Ngā Otinga Whānui
Ngā Koki Hiato
Ngā Koki Rearua
±
±
cos2 θ + sin2 θ = 1
tan2 θ + 1 = sec2 θ
cot2 θ + 1 = cosec2 θ
Mēnā sin θ = sin α kāti θ = n� + (−1)n αMēnā cos θ = cos α kāti θ = 2n� ± αMēnā tan θ = tan α kāti θ = n� + αko te n, he tau tōpū ahakoa
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1
1
1
2 3
2
� 6 –
� 3 –
� 4 –
� 4 –
TRIGONOMETRY
cosec! = 1sin!
sec! = 1cos!
cot! = 1tan!
cot! = cos!sin!
Sine Rulea
sin A= b
sinB= c
sinC
Cosine Rule
c2 = a2 + b2 ! 2ab cosC
Identities
cos2! + sin2! = 1
tan2! +1= sec2!
cot2! +1= cosec2!
General Solutions
If sin ! = sin " then ! = n! + ("1)n" If cos ! = cos " then ! = 2n! ±"If tan ! = tan " then ! = n! +"where n is any integer
Compound Anglessin(A± B) = sin AcosB ± cos AsinBcos(A± B) = cos AcosB ! sin AsinB
tan(A± B) = tan A± tanB1! tan A tanB
Double Anglessin2A = 2sin Acos A
tan2A = 2 tan A1! tan2 A
cos2A = cos2A! sin2A
= 2cos2A!1
= 1! 2sin2A
Products2sin Acos B = sin( A+ B) + sin( A − B)2cos Asin B = sin( A+ B) − sin( A − B)2cos Acos B = cos( A+ B) + cos( A − B)2sin Asin B = cos( A − B) − cos( A+ B)
Sums
sinC + sin D = 2sinC + D
2cos
C − D2
sinC − sin D = 2cosC + D
2sin
C − D2
cosC + cos D = 2cosC + D
2cos
C − D2
cosC − cos D = −2sinC + D
2sin
C − D2
MEASUREMENTTriangle
Area =12
absinC
Trapezium
Area =12
(a + b)h
Sector
Area =12
r2θ
Arc length = rθ
Cylinder
Volume = !r2hCurved surface area = 2!rh
Cone
Volume =13!r2h
Curved surface area = !rl where l = slant height
Sphere
Volume =43!r3
Surface area = 4!r2
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Level 3 Calculus, 2017
9.30 a.m. Thursday 23 November 2017
FORMULAE AND TABLES BOOKLETfor 91577, 91578 and 91579
Refer to this booklet to answer the questions in your Question and Answer booklets.
Check that this booklet has pages 2 – 7 in the correct order and that none of these pages is blank.
YOU MAY KEEP THIS BOOKLET AT THE END OF THE EXAMINATION.
L3
–C
AL
CM
F
English translation of the wording on the front cover