DMOL.St Thomas More C3: Starters Revise formulae and develop problem solving skills. 123456789...

DMO’L.St Thomas More

C3: StartersRevise formulae and develop

problem solving skills.1 2 3 4 5 6 7 8 9

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Starter 1Solve the equation

for 01sin2cos3 20



for 01sin2cos3 20

01sin)sin21(3 2



for 01sin2cos3 20 01sin)sin21(3 2 02sinsin6 2

0)1sin2)(2sin3(

21sin

32sin or

65

6 ,,55.5,87.3 Back


Starter 2Prove the identity

2244 tansectansec



44 tansec LHS



44 tansec LHS)tan)(sectan(sec 2222

)tantan1)(tan(sec 2222 )tan(sec 22

Back

RHS



tan

cos3cossin3sin


Starter 3

cos3cossin3sin

LHS

cos3cos)sin()cos(2 2

32

3

)cos()cos(2)sin()cos(2

23

23

23

23

)cos()2cos(2)sin()2cos(2


Starter 3

cos3cossin3sin

LHS

cos3cos)sin()cos(2 2

32

3

)cos()cos(2)sin()cos(2

23

23

23

23

RHS tan

)cos()2cos(2)sin()2cos(2

Back


Starter 4Given that andwhere A is acute and B is obtuse, find

43tan A 13

5sin B

)(cos BAec


Starter 4

By Pythagoras

43tan A 13

5sin B

54

53

cossin

AA

1312

125

costan

BB

1312

125

costan

BB

A is acute

B is obtuse


Starter 4

)sin(1)(cosBA

BAec

BABA sincoscossin1

135

54

1312

53

1

6520

6536

1


Starter 4

)sin(1)(cosBA

BAec

BABA sincoscossin1

135

54

1312

53

1

16651

6520

6536

Back


Starter 5Differentiate

xey x sin2

xey 2sin

xxxy

lncossin



xey x sin2 xeu 2 xv sinx

dxdu e22 xdx

dv cos

xexedxdy xx cossin2 22



xey 2sin xu 2sin uey

xdxdu 2cos2 u

dudy e

xexedxdy xu 2cos22cos2. 2sin



xxxy

lncossin

xxu cossin xv ln

xxdxdu 22 sincos xdx

dv 1

2

1

)(lncossin2cosln

xxxxx

dxdy x

xdxdu 2cos



xxxy

lncossin

xxu cossin xv ln


dv 1

221

)(ln2sin2cosln

xxxx

dxdy x

xdxdu 2cos



xxxy

lncossin

xxu cossin xv ln


dv 1

2)(ln22sin2cosln2

xxxxxx

dxdy

xdxdu 2cos

Back



)ln(sec xy

7)3sin1( xy

xexy sin

2cos



1)(cossec xxu uy ln

xxdxdu sin)(cos 2 udu

dy 1

xx

udxdy

2cossin.1

xy ln(sec)

xx

dxdu

2cossin



1)(cossec xxu uy ln

xxdxdu sin)(cos 2 udu

dy 1

xxxx

xx

udxdy tan

coscossin

cossin.1

22

)ln(sec xy

xx

dxdu

2cossin



xu 3sin1 7uy

xdxdu 3cos3 67udu

dy

xxxudxdy 3cos)3sin1(213cos3.7 66

7)3sin1( xy



xu 2cos xev sinxdx

du 2sin2 xdxdv ex sin.cos

x

xx

eexxxe

dxdy

sin2

sinsin ).(cos2cos)2sin2(

xexy sin

2cos



xu 2cos xev sinxdx

du 2sin2 xdxdv ex sin.cos

)cos2cos2sin2(sin xxxedxdy x

xexy sin

2cos

Back


Starter 7Solve the following equations, giving

exact solutions72 xe

5)12ln( 2 x



exact solutions72 xe

7ln2 x7ln2

1x

5)12ln( 2 x5)12ln(2 x

25)12ln( x

25

12 ex

212

5

ex

Back


Starter 8Show that can in be written in

the form Use the iteration starting with to generate Show that 5.5 is a root of the equation to

one decimal place.

0352 xx35 xx

351 nn xx

50 x 654321 ,,,,, xxxxxx

352 xx

35 xx


Starter 8Use the iteration starting with to generate Show that 5.5 is a root of the equation to

one decimal place.

351 nn xx

50 x 654321 ,,,,, xxxxxx

2915.55

1

0

xx

Calculator:5 =5Ans+3====

537.5530.5518.5489.54275.52915.5

5

6

5

4

3

2

1

0

xxxxxxx


Starter 8Show that 5.5 is a root of the equation to

one decimal place. 35)( 2 xxxf

525.0)55.5( f

547.0)45.5( f

Change of sign Root between 5.55 and 5.45Change of sign Root between 5.55 and 5.45Hence, x = 5.5 is a root to 1 decimal place.Back


Starter 9Sketch the graph Hence, or otherwise, solve

52 xy

552 x

y=2x-5

52 xyy=5

x = 0 or 5 Back

when x = 3


Starter 10 By differentiating find the coordinates of

the turning point on the curve State the nature of the turning point (i.e.

maximum or minimum).

xxy ln813

0813 2 x

xdxdyFor turning points

813 3 x3x22

2 816x

xdxyd

0816 22

2

x

xdxydwhen x =

3

Hence, minmum point at (3,-61.99)

Back



exact solutions422 xxx eee

12)35ln( 2 x



exact solutions422 xxx eee

432 xx ee

432 xxTake logs base e

4x



exact solutions12)35ln( 2 x12)35ln(2 x6)35ln( x

e to the power of 635 ex

536

ex Back


Starter 12Complete the table:

ydxdy

x

x

n

a

e

xxxxx

kx

lnsectancossin

aa

e

xxx

xx

nkx

x

x

x

n

ln

tansecsec

sincos

1

2

1

Back


Starter 13Complete the table:

ydxdy

x

x

n

e

xxxx

xx

5

6

3

2

3

6ln7sec4tan

cos

2sin)13(

3ln)3(5

6

7tan7sec74sec4

sincos3

2cos2)13(6

5

6

1

2

2

12

x

x

x

n

e

xxx

xx

xxnx

Back

DMOL.St Thomas More C3: Starters Revise formulae and develop problem solving skills. 123456789...

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Transcript of DMOL.St Thomas More C3: Starters Revise formulae and develop problem solving skills. 123456789...