3

© 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,

2

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 )

3

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,

4

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.

5

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

6

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

7

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