WorkSHEET 8.1 Advanced exponential functions...Maths Quest Maths C Year 12 for Queensland 2e 1...
Transcript of WorkSHEET 8.1 Advanced exponential functions...Maths Quest Maths C Year 12 for Queensland 2e 1...
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Maths Quest Maths C Year 12 for Queensland 2e 1
WorkSHEET 8.1 Advanced exponential functions
Name: ___________________________ 1 Using de Moivre’s theorem and the binomial
expansion, prove that ( ) 1cos2sincos2cos 222 -=-= qqqq
( )22 sincossincos
qq
qq
iziz+=
+=
Using de Moivre’s theorem, qq 2sin2cos2 iz +=
Writing the binomial expansion of ,2z we have
qqqqqqq
22
222
sincos2cossincos2sincos
-=\
+-= iz
Applying the Pythagorean Identity,
( )
1cos2sincos2cos1cos2cos1cos2cos
cos1sin
2
22
2
22
22
-=
-=\
-=
--=\
-=
qqqq
qqqq
qq
2 Using the multiple angle formulas, prove that ( ) ( ) ( ) ( )2sin cos = sin sinx x x x4 2 6 + 2
( ) ( )( )( )( )( )
xxizz
izz
zzzzi
zzzzi
zzzzi
zzzzi
xx
2sin6sin22
212121
21
212
2cos4sin2
2266
2266
6226
2244
2244
+=
-+
-=
-+-=
--+=
--=
-´-´=
--
--
--
--
--
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Maths Quest Maths C Year 12 for Queensland 2e 2
3 Prove that ( ) ( ) ( )[ ]32cos44cos
81sin 4 +-= xxx
( )
( )( ) ( )
( )
( )
( )32cos44cos81
32cos44cos162
62cos84cos2161sin
62cos84cos262cos244cos264
.4.6.4
16sin
2sin
4
2244
432213441
414
1
+-=
+-=
+-=
+-=+´-=++-+=
+-+-=-
-=
-=
--
-----
-
-
xx
xx
xxx
xxxxzzzz
zzzzzzzzzz
zzx
izzx
4 Express 3
2
2i
ep
in standard form.
31
232
212
32sin2
32cos2
2 32
i
i
i
ei
+-=
´+÷øö
çèæ-´=
+=pp
p
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Maths Quest Maths C Year 12 for Queensland 2e 3
5 If iu -= 3 and ,1 iw += (a) express both u and w in Euler’s form.
(b) express 3
5
uw
in standard form.
(c) find values for m and n such that
8m nu w i= .
iwiu +=-= 1,3 (a) 2=u
u is a complex number in the 4th quadrant of the complex plane
22
6arg
6
=
=\
-=\
-
weu
u
ip
p
w is in the first quadrant of the complex plane
42
4arg
i
ew
w
p
p
-=\
-=
(b)
i
i
e
e
e
e
ewu
i
i
i
i
i
+=
÷ø
öçè
æ+=
=
=
=
-
-
-
122
12222
24
8
2
2
4
43
2
45
25
23
5
3
p
p
p
p
p
(cont.)
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Maths Quest Maths C Year 12 for Queensland 2e 4
5 (cont.)
(c)
23
23
1223
2
642
426
22
2 RHS
.2
.2
2.2
LHS8
nm
i
mninm
mninm
ninmim
nm
nm
e
e
e
ee
wuiwu
+
÷øö
çèæ -
+
÷øö
çèæ -+
-
=\
=
=
=
=
=
=
p
p
p
pp
.integer an for 221
1223 and
622
3 i.e.
kkmnnm
nm
+=-
=+
+=
kmkmk
nmknkn
mnkmn
35.1i.e.632
63662
6324124
62 and24623 i.e.
-=-=
--=-=\+=+=\
=++=-
There is an infinite solution set given by
.integer for 63and 35.1
kknkm
+=-=
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Maths Quest Maths C Year 12 for Queensland 2e 5
6 Apply Euler’s formula to evaluate
( )sin dxe x x2ò
( )
( )( )
( )( )( )
( )
( )
( ) ( )
( )
( )( )
2
2
2
2
1 2
1 2
1 2
1 2
2
sin 2 d
cos 2 sin 2
Im sin 2
sin 2 d
Im d
Im . d
Im d
Im1 2
1 2Im1 2 1 2
1 2Im
5
Im . 1 25
Im cos 2 sin 2 1 25
x
ix
ix
x
x ix
x ix
i x
i x
i x
i x
xix
x
e x x
e x i x
e x
e x x
e e x
e e x
e x
ei
e ii i
e i
e e i
e x i x i
+
+
+
+
= +
=
\
=
=
=
é ù= ê ú+ë û
é ù-= ´ê ú+ -ë û
é ù-= ê ú
ê úë ûé ù
= -ê úë ûé
= + -
ò
òòòò
( )
( )
( )
( )
cos 2 2 cos 2Im
sin 2 2sin 25
cos 2 2sin 25Im
sin 2 2cos 25
sin 2 d
sin 2 2cos 25
x
x
x
x
x
x i xei x x
e x x
ie x x
e x x
e x x c
ùê úë ûé + ùæ ö
= ê úç ÷+ +è øë ûé ù
+ê úê ú=ê ú
+ -ê úë û
\ =
= - +
ò
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Maths Quest Maths C Year 12 for Queensland 2e 6
7 (a) Sketch the function ( )= cosxy e x- over the domain xp p- £ £ .
(b) Determine ( )xe x
xcoslim -
¥®
(c) Evaluate ( )ò¥
-
0
cos dxxe x
(a) Here, xcos is squeezed between the envelopes .xe-± Graph xey -±= and then squeeze
xy cos= between the envelopes
(b) ( ) 0coslim =- xe x
xe- converges rapidly to zero while xcos oscillates between .1±
Hence ( )xe x cos- oscillates towards 0=y as x increases.
(c) ò¥
-
0
dcos xxe x
formula. parts Apply the
dcosConsider ò -= xxeI x
( )xxexe
xxex
xe
xxx
eI
xx
xx
x
dsinsin
dsinddsin
dsindd
ò
ò
ò
--
--
-
+=
-=
÷øö
çèæ=
Now consider ò - xxe x dsin
*** It is interesting that the book uses Integration by Parts here, instead of converting to Euler form … I would have thought using the new integration technique would be easier and quicker … ??? ***
(cont.)
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Maths Quest Maths C Year 12 for Queensland 2e 7
7 (cont.)
( )
( )
( )xxeI
xxe
xexeI
IxexeI
Ixe
xxexe
xxx
exxe
x
x
xx
xx
x
xx
xx
cossin21
cossin
cossin2
cossin i.e.
cos
dcoscos
dcosdddsin
-=\
-=
-=\
--=
--=
--=
-=
-
-
--
--
-
--
--
ò
òò
.
Hence,
( )
( ) ( )( )
.21
1210
0cos0sin21
cossin21lim
d cos
dcoslimdcos
0
0
00
=
-´-=
þýü--
îíì -=
\
=
-
-
¥®
¥-
-
¥®
¥-
ò
òò
e
nne
xxe
xxexxe
nn
x
nx
nx
*** Not really sure that Limit theory is required here … we are just doing a definite integral … ??? … just doing a standard definite integral process here gets the correct answer … J
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Maths Quest Maths C Year 12 for Queensland 2e 8
8 (a) Find the set of complex numbers where n = 1, 2 and 3 given that
(b) Graph in the complex plane joining
the points together to form a closed figure. What shape is this figure?
(a)
(b) Plot each point in the sequence in the
complex plane. Join them together. The figure formed is an equilateral triangle.
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Maths Quest Maths C Year 12 for Queensland 2e 9
9 Show that
10 Given that represents the displacement of a particle at time t, (a) show that
(b) If show that
(c) By making appropriate use of graphics
calculator functions, find the first positive
value of a such that .
(a)
(b)
(c) Solve
(Hint: Use a ‘solver’ routine in the graphics calculated to show that )