PDF Level 3 Calculus (90635) 2009

Level 3 Calculus, 2009 Level 3 CAS Calculus, 2009

Refer to this booklet to answer the questions in your Question and Answer booklets.

Check that this booklet has pages 2–4 in the correct order and that none of these pages is blank.

YOU MAY KEEP THIS BOOKLET AT THE END OF THE EXAMINATION.

L 3 – C A L C F

© New Zealand Qualifications Authority, 2009

All rights reserved. No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.

2.00 pm Thursday 26 November 2009

FORMULAE AND TABLES BOOKLET for 90635, 90636, 90638, 90639,

90833, 90834 and 90835

993203

3

CALCULUS – USEFUL FORMULAE2

ALGEBRA Quadratics

If ax² +bx+c=0 then x= −b± b² −4ac

2a Logarithms

y=log_bx⇔x=b^y log_b

( )

xy ^=log_b^x+log_b ^y

log_b x y

⎛

⎝⎜

⎞

⎠⎟ =log_bx−log_b y

log_b

( )

x^{n =nlog}bx

log log

b loga

a

x x

= b

Binomial Theorem (a+b)ⁿ =

0

⎛n

⎝⎜ ⎞

⎠⎟aⁿ+

1

⎛n

⎝⎜ ⎞

⎠⎟aⁿ⁻¹b¹+

2

⎛n

⎝⎜ ⎞

⎠⎟aⁿ⁻²b²+ ... +

r

⎛n

⎝⎜ ⎞

⎠⎟a^n−rb^r+ ... +

n

⎛n

⎝⎜ ⎞

⎠⎟bⁿ

r

⎛n

⎝⎜ ⎞

⎠⎟ = ⁿC

r = n!

(n−r)!r!

Some values of

r

⎛n

⎝⎜ ⎞

⎠⎟= ⁿC

r = n!

(n−r)!r!

are given in the table below.

n r 0 1 2 3 4 5 6 7 8 9 10

0 1

1 1 1

2 1 2 1

3 1 3 3 1

4 1 4 6 4 1

5 1 5 10 10 5 1

6 1 6 15 20 15 6 1

7 1 7 21 35 35 21 7 1

8 1 8 28 56 70 56 28 8 1

9 1 9 36 84 126 126 84 36 9 1

10 1 10 45 120 210 252 210 120 45 10 1 11 1 11 55 165 330 462 462 330 165 55 11 12 1 12 66 220 495 792 924 792 495 220 66

Complex Numbers z x y= +i

^{=r cis}θ

=r(cosθ+isin )θ z = −x iy

^{=rcis (−}θ₎ =r(cosθ−isin )θ

r = z = zz = (x²+y²)

θ = arg z where cosθ₌ ^x

r

and sinθ = y r

De Moivre’s Theorem If n is any integer then

(rcisθ₎ⁿ ₌rⁿ cis (nθ₎

COORDINATE GEOMETRY Straight Line

Equation y y− ₁=m x x( − ₁) 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 a

= c

Hyperbola x²

a²

− y²

b² =1 or (asecθ_,btanθ₎ asymptotes y b

ax

= ±

Foci (c,0) (–c,0) where b² = c² – a² Eccentricity: e a

= c

3

CALCULUS Differentiation

y f x= ( ) ^dy

dx = f′(x)

lnx

e^ax sinx cosx tanx secx cosec x

cotx

1 x ae^ax cosx

−sinx sec²x sec tanx x

−cosec x cotx

−cosec²x Integration

f x( )

∫

^{f(x) dx}

xⁿ

1 x

′ f (x)

f(x) x

nⁿ⁺ c +¹ +

1 ln x c+

ln ( )f x +c

First Principles

f '(x)=lim

h→0

f(x+h)− f(x) h

Parametric Function dy

dx =dy dt

.dt dx

d²y dx² = d

dt dy dx

⎛

⎝⎜

⎞

⎠⎟.dt dx

Product Rule

(f.g)′= f.g′+g.f′ or if y=uv then dy dx =udv

dx+vdu dx

Quotient Rule

f g

⎛

⎝⎜

⎞

⎠⎟

′

= g.f′− f.g′ g²

or if y= u v

then dy dx=

vdu dx−udv

dx v²

Composite Function or Chain Rule f g

( )

( )

^{′ = f′}

^{( )}

^{g .g′}

or if and then d d

d d y f u u g x y d

x y u

= ( ) = ( ) = . u ddx Volume of Revolution

y = f(x) between x=a and x=b rotated about the x-axis

Volume = π

a

∫

b ^y2dx

NUMERICAL METHODS Trapezium Rule

f(x) dx

a

∫

b ^≈1₂^{h y⎡⎣ 0+ yn+2(y1+ y2+...+yn−1)⎤⎦}

where h b a and

n y_r f x_r

= −

= ( ) Simpson’s Rule

f(x) dx≈1

3h y⎡⎣ ₀+ y_n+4(y₁+ y₃+...+ y_n−1)+2(y₂+y₄+...+ y_n−2)⎤⎦

a

∫

b

where h b a and is even.

n y_r f x_r n

= −

, = ( )

4

L 3 – C A L C F

TRIGONOMETRY cosec θ

= 1θ sin sec θ

= 1θ cos cot θ

= 1θ tan

cot θ θ

=cosθ sin Sine Rule

a A

b B

c sin =sin =sinC Cosine Rule

c² =a²+b²−2abcosC Identities

cos sin

tan sec

cot cos

2 2

2 2

2 2

1 1

1

θ θ

θ θ

θ θ

+ =

+ = + = ec General Solutions

If sin = sin then If cos

θ α θ α

θ

=nπ+ −( )1 ⁿ

== cos then If tan = tan then

α θ α

θ α

=2nπ± θθ =n +α n

π where is any integer

Compound Angles

sin(A±B)=sinAcosB±cosAsinB cos(A±B)=cosAcosBsinAsinB tan(A±B)= tanA±tanB

1tanAtanB Double Angles

sin sin cos

tan tan

tan

cos cos

2 2

2 2

1 2

2

A A A

A A

A A

=

= −

= ^{22 2}

2 2 1

1 2

A A

A

−

= −

= −

sin cos

s

iin²A

Products 2

2

sin cos sin( ) sin( )

cos sin sin(

A B A B A B

A B A

= + + −

= +BB A B

A B A B A B

) sin( )

cos cos cos( ) cos( )

sin

− −

= + + −

2

2 AAsinB=cos(A B− ) cos(− A B+ ) Sums

sin sin sin cos

sin sin cos

C D C D C D

C D C D

+ = + −

− = +

2 2 2

2 2 ssin

cos cos cos cos

cos cos

C D

C D C D C D

C D

−

+ = + −

− =

2

2 2 2

−−2 + −

2 2

sinC DsinC D

MEASUREMENT Triangle

Area= 1

2ab Csin Trapezium

Area= 1 + 2(a b h) Sector

Area Arc length

=

= 1 2r²

r θ

θ

Cylinder Volume π

Curved surface area π

=

= r h

rh

2

2 Cone

Volume π

Curved surface area π where

=

= 1

3 r h²

rl ll= slant height

Sphere Volume π Surface area π

=

= 4 3

4

3 2

r r

1 1 1

2 3

2 6–

3–

4–

4–