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Contents
1 krj_bo pbqpNKN p f 1NKO p k 5
NKP _~ f 7NKQ ` k 8
2 ^idb_o^OKN c~ c~ 12OKO m c~ 13OKP m 14OKQ o 15
OKR i~ 16OKS b~ 18OKT f~ 19OKU ` f c~ 22
3 dbljbqovPKN o q~ 24PKO f q~ 27PKP b~~ q~ 28PKQ p~ q~ 29PKR p~ 33PKS o~ 34PKT m~~~ 35PKU o 36PKV q~ 37
PKNM f q~ 38PKNN f q~ f ` 40PKNO q~ f ` 41
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iv
PKNP h 42PKNQ ` n~~~ 43
PKNR q~~ n~~~ 45PKNS d~ n~~~ 46PKNT o~ e~ 47PKNU o~ m 48PKNV ` 50PKOM p ~ ` 53PKON p ~ ` 54PKOO ` 55PKOP o~~ m~~ 56PKOQ m 57PKOR o~ q~ 58PKOS o~ m~ 59PKOT c ~ o~ m~ 61PKOU o~~ o t 62PKOV m~ p 63
PKPM o `~ ` 66PKPN o `~ ` ~ l m~ c~ 68PKPO o `~ ` 69PKPP c ~ o `~ ` 70PKPQ p 72PKPR p~ `~ 72PKPS p~ p 73PKPT p~ p 74
PKPU p~ t 75PKPV b 76PKQM `~ q 78
4 qofdlkljbqovQKN o~~ ~ a j~ ^ 80QKO a ~ d~ q c 81
QKP p q c 86QKQ q c ` ^ 87QKR j f~ c~ 88
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v
QKS o c~ 89QKT m q c 90
QKU o~ q c 90QKV ^ ~ p~ c~ 91QKNM a ^ c~ 92QKNN j ^ c~ 93QKNO e~ ^ c~ 94QKNP e~ ^ q~ f 94QKNQ q~ q b m 95QKNR q~ q b p 97QKNS m q c 98QKNT d~ f q c 99QKNU m~ s~ f q c 102QKNV o~ f q c 103QKOM q b~ 106QKON o~ e c 106
5 j^qof`bp ^ka abqbojfk^kqpRKN a~ 107RKO m a~ 109RKP j~ 110RKQ l~ j~ 111RKR p i~ b~ 114
6 sb`qlop
SKN s `~ 118SKO s ^ 120SKP s p~ 122SKQ p~ s 122SKR p~~ m 123SKS s m 125SKT q m 127
7 ^k^ivqf` dbljbqovTKN l -a~ `~ p 130
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TKO q -a~ `~ p 131TKP p~ i m~ 139
TKQ ` 149TKR b 152TKS e~ 154TKT m~~~ 158TKU q -a~ `~ p 161TKV m~ 165TKNM p~ i p~ 175TKNN n~ p~ 180TKNO p 189
8 afccbobkqf^i `^i`rirpUKN c ~ q d~ 191UKO i c 208UKP a ~ m a~ 209UKQ q~ a~ 211
UKR e l a~ 215UKS ^~ a~ 217UKT a~ 221UKU j~~ c 222UKV a~ l~ 225
9 fkqbdo^i `^i`rirpVKN f f~ 227
VKO f~ o~~ c 228VKP f~ f~~ c 231VKQ f~ q c 237VKR f~ e c 241VKS f~ b~ ~ i~ c 242VKT o c~ 243VKU a f~ 247
VKV f f~ 253VKNM a f~ 257VKNN q f~ 269
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VKNO i f~ 275VKNP p~ f~ 285
10 afccbobkqf^i bnr^qflkpNMKN c l l~ a~ b~ 295NMKO p l l~ a~ b~ 298NMKP p m~~ a~ b~ 302
11 pbofbpNNKN ^ p 304NNKO d p 305NNKP p c p 305NNKQ f p 307NNKR m ` p 307NNKS ` q 308NNKT ^~ p 310NNKU m p 311
NNKV a~ ~ f~ m p 312NNKNM q~ ~ j~~ p 313NNKNN m p b~ p c 314NNKNO _~ p 316NNKNP c p 316
12 mol_^_fifqvNOKN m~ ~ `~ 318
NOKO m~ c~ 319
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q ~ ~ ~K
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1
Chapter 1
Number Sets
1.1 Set Identities
pW ^I _I `r~ W f` W ^ m W _^ b W r W _^
f W _^ a W _y^
1. f^
2. ^^
3. _^ = _^ ~ ^_ .
4. b p^
5. r p{ }_^_^` ==
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CHAPTER 1. NUMBER SETS
2
Figure 1.
6. `~
^__^ =
7. ^~( ) ( ) `_^`_^ =
8. f p
{ }_~^_^` ==
Figure 2.
9. `~^__^ =
10. ^~( ) ( ) `_^`_^ =
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3
11. a( ) ( ) ( )`^_^`_^ = I
( ) ( ) ( )`^_^`_^ = K
12. f^^^ = I^^^ =
13. a~
=^ Iff^ =
14. f^^ = I
^f^ =
15. `
{ }^f^ =
16. ` f ~ rf^^ = I=^^
17. a j~ i~
( ) _^_^ =
I( ) _^_^ =
18. a p{ }^~_^y_` ==
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CHAPTER 1. NUMBER SETS
4
Figure 3.
19. ( )_^y_^y_ =
20. ^_^y_ =
21. =^y^
22. ^_y^ = =_^ .
Figure 4.
23. ( ) ( ) ( )`_y`^`_y^ =
24. ^yf^ =
25. `~~ m
( ){ }_~^I_^` ==
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5
1.2 Sets of Numbers
k~~ W kt W Mk
fW w
m W +w
k~ W wo~~ W n
o~ W o` W `
26. k~~ k` W { }KIPIOINk = K
27. t k
` ~ W { }KIPIOINIMk M= K
28. ft ~ ~ W
{ }KIPIOINkw ==+ I
{ }NIOIPIw = K I
{ } { }KK IPIOINIMINIOIPIwMww == + K
29. o~~ ko~ ~ ~W
== M~w~w~~
~n K
30. f~~ k
k~ ~ ~ ~K
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6
31. o~ kr ~~ ~ ~~ W oK
32. ` k{ }o~o` += I
~~ K
33. `onwk
Figure 5.
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CHAPTER 1. NUMBER SETS
7
1.3 Basic Identities
o~ W ~I I
34. ^ f~M~ =+
35. ^ f
( ) M~~ =+
36. `~ ^~~ +=+
37. ^~ ^( ) ( )~~ ++=++
38. a p~( )~~ +=
39. j~ f~N~ =
40. j~ f
N~
N~ = I M~
41. j~ q MMM~ =
42. `~ j~
~~ =
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CHAPTER 1. NUMBER SETS
8
43. ^~ j~( ) ( )~~ =
44. a i~( ) ~~~ +=+
45. a aN
~~
=
1.4 Complex Numbers
k~~ W f~~ W
` W o~ ~W ~I f~~ ~W I j ~ W I N I O
^ ~ W I N I O
N
= R
= NQ
=+
N O = N S = N OQ =+ P = T = PQ =+
46.
N Q = N U = N Q =
47. ~ +=
48. ` m~
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CHAPTER 1. NUMBER SETS
9
Figure 6.
49. ( ) ( ) ( ) ( )~~ +++=+++
50. ( ) ( ) ( ) ( )~~ +=++
51. ( )( ) ( ) ( )~~~ ++=++
52.
~
~
~OOOO
+
+++
=++
53. `~ ` k
~~|||||||
=+
54. = ~ I =
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CHAPTER 1. NUMBER SETS
10
Figure 7.
55. m~ m~ ` k( )+=+ ~
56. j ~ ^ ~ ` kf ~ + ~ I OO ~ += EFI
~
~~= E~FK
57. m m~ o~
( ) ( )OOONNNON ++= ( ) ( )[ ]ONONON +++=
58. `~ k m~ o~
( ) ( ) ( )[ ]+=+ |||||||||||||||||||||
59. f ~ ` k m~ o~
( ) ( ) ( )[ ]+=
+
N
N
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11
60. n m~ o~( )
( ) ( ) ( )[ ]ONONON
OOO
NNN
O
N
+=+
+
=
61. m ~ ` k
( )[ ] ( ) ( )[ ]+=+=
62. c~ a j
( ) ( ) ( )+=+
63. k o ~ ` k
( )
++
+=+=
O
O I
NIIOINIM = K K
64. b c~
+=
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12
Chapter 2
Algebra
2.1 Factoring Formulas
o~ W ~I I k~~ W
65. ( )( )~~~ OO +=
66. ( )( )OOPP ~~~~ ++=
67. ( )( )OOPP ~~~~ ++=+
68. ( )( ) ( )( )( )OOOOOOQQ ~~~~~~ ++=+=
69.( )( )
QPOOPQRR
~~~~~~ ++++=
70. ( )( )QPOOPQRR ~~~~~~ +++=+
71. f I
( )( )NOOPON ~~~~~~ +++=+ K K
72. f I ( )( )NOOPON ~~~~~~ +++++= K I
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CHAPTER 2. ALGEBRA
13
( )( )NOOPON ~~~~~~ +++=+ K K
2.2 Product Formulas
o~ W ~I I t W I
73. ( ) OOO ~O~~ +=
74. ( ) OOO ~O~~ ++=+
75. ( ) POOPP ~P~P~~ +=
76. ( ) POOPP ~P~P~~ +++=+
77. ( ) QPOOPQQ ~Q~S~Q~~ ++=
78. ( ) QPOOPQQ ~Q~S~Q~~ ++++=+
79. _~ c~
( ) I`~`~`~`~`~ N
NOO
ON
N
M +++++=+ K
( )>>
>`
= ~ ~ K
80. ( ) O~O~O~~ OOOO +++++=++
81. ( ) ++++++=+++++ OOOOOO
~~ KK ( )~~~~O +++++++++++ KKK
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CHAPTER 2. ALGEBRA
14
2.3 Powers
_~ E ~ FW ~I m E~~ FW I
82. ~~~ +=
83.
~~
~ =
84. ( ) ~~ =
85.
~
~=
86. ( ) ~~ =
87. N~ M = I M~
88. N~ N =
89.
~N~ =
90.
~~ =
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CHAPTER 2. ALGEBRA
15
2.4 Roots
_~W ~I m E~~ FW I
MI~ E O = I k F
91. ~~=
92. ~~ =
93.
~
~= I M
94.
~
~
~
== I M K
95. ( ) ~~ =
96. ( ) ~~ =
97. ~~ =
98.
~~ =
99. ~~ =
100. ( ) ~~ =
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16
101.~
~
~
N N
= I M~ K
102.O
~~
O
~~~
OO
+=
103.~
~
~
N
=
m
2.5 Logarithms
m ~ W I I ~I I k~~ W
104. a i~
~= ~ ~= I M~ > I N~ K
105. MN~ =
106. N~~ =
107.
=
N~
N~M~
108. ( ) ~~~ +=
109.
~~~ =
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CHAPTER 2. ALGEBRA
17
110. ( ) ~~ =
111.
N ~
~ =
112. ~
~
~ == I M > I N K
113. ~
N
~ =
114. ~~=
115. i~ _~ NMNM =
116. k~~ i~ = I
KTNUOUNUOUKO
NN
=
+=
117. QPQOVQKM
NM
N ==
118. PMORURKO
N ==
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CHAPTER 2. ALGEBRA
18
2.6 Equations
o~ W ~I I I I I I pW N I O I N I O I P
119. i~ b~ l s~~
M~ =+ I
~
= K
120. n~~ b~
M~O =++ I~O
~Q
O
OIN
= K
121. a~
~Qa
O
=
122. s c~
f MO =++ I
=
=+
ON
ON K
123. M~O =+ I MN= I~
O = K
124. M~O =+ I~
OIN = K
125. ` b~K `~~ c~K
MP
=++ I
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CHAPTER 2. ALGEBRA
19
N += I ( ) ( ) O
P
O
N PIO ++= I
P
OO
P
O
O
+
+= I POO
P
O
O
+
= K
2.7 Inequalities
s~~W I I
o~ W
PON ~II~I~I~
III~
K I I
a~W aI a I a I a
126. f~I f~ k~ ~ d~
f~ f~ k~ d~~ [ ]I~
~ < ( ]I~
~
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20
127. f ~> I ~ < K
128. f ~> I M~ > M~ I ~ +>+ K
130. f ~> I ~ > K
131. f ~> ~ > I ~ +>+ K
132. f ~> ~ > I ~ > K
133. f ~> ~ M > I ~> K
134. f ~> ~ M > I
~> K
135. f ~> ~ M < I ~ < K
136. f ~> ~ M < I
~< K
137. f ~M I ~ < K
138. f ~M K
139. f ~M I M > X ~ ~ ~ ~= K
141. O~
N~ + I M~> X ~ ~ ~ ~ ~ N~= K
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CHAPTER 2. ALGEBRA
21
142.
~~~~~~ ON ON
+++
KK I M~II~I~ ON >K K
143. f M~ >+ ~ M~> I ~
> K
144. f M~ >+ ~ M~ < I ~
< K
145. M~O >++
M~> M~ <
Ma>
N< I O> ON
Ma<
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22
146. ~~ ++
147. f ~< I ~~ I ~< ~ ~> I M~> K
149. f ~O < I ~< I M~ > K
150. f ~O
> I ~> I M~> K
151. f( )( )
M
> I
( ) ( )
( )
>
M
MK
152.( )( )
M
< I
( ) ( )
( )
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CHAPTER 2. ALGEBRA
23
154. p ` f c~f ~I
~ W( )N`^ += K
155. ` ` ff ~ E FI
`^ = K
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24
Chapter 3
Geometry
3.1 Right Triangle
i ~ ~W ~I eW ^W j~W ~ I I
^W I o~ W oo~ W ^~W p
Figure 8.
156. =+ VM
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CHAPTER 3. GEOMETRY
25
157. ==
~
158. ==
159. == ~
~
160. == ~~
161. ==
162. == ~
163. m~~ qOOO ~ =+
164. ~ O = I O = I ~ ~ ~ ~ I -
I K
Figure 9.
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CHAPTER 3. GEOMETRY
26
165. O = I ~ ~K
166.Q
~
OOO
~ = IQ
~
OOO
= I
~ ~ ~ ~ ~ ~ K
Figure 10.
167.O
= I
~ K
168. Oo ==
169.~
~
O
~
++=
+=
170. ~ =
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CHAPTER 3. GEOMETRY
27
171.O
O
~p ==
3.2 Isosceles Triangle
_~W ~iW _~ ~W s ~W ^ ~W mW i^~W p
Figure 11.
172.O
VM =
173.Q
~OOO =
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CHAPTER 3. GEOMETRY
28
174. O~i +=
175. == O
O
~pO
3.3 Equilateral Triangle
p ~ ~~ ~W ~^W o~ W oo~ W mW i^~W p
Figure 12.
176.O
P~ =
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29
177.P
P~
P
Oo ==
178.O
o
S
P~
P
N ===
179. ~Pi =
180. Q
P~
O
~p
O
==
3.4 Scalene Triangle
E^ ~ ~F
p ~ ~W ~I I
pWO
~
++=
^ ~ ~W II ^ ~I I W ~ II
j~ ~I I W ~ II _ ~ II W ~ IIo~ W oo~ W ^~W p
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30
Figure 13.
181. =++ NUM
182. ~ >+ I~ >+ I
~ >+ K
183. ~
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CHAPTER 3. GEOMETRY
31
185. i~ `
+= O~ OOO I
+= ~O~ OOO I
+= ~O~ OOO K
186. i~ p
oO
~=
=
=
I
o ~ K
187.pQ
~
O
~
O
~
O
O
O
O
~o
~
====
=
=
=
188.( )( )( )
~ O
= I
~ N
N
N
N ++= K
189.( )( )
O
=
I
( )
~
O
=
I
( )( )( )~
O~
=
K
190. ( )( )( )~~
O ~ = I
( )( )( )~O
= I
( )( )( )~
O = K
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32
191. == ~ I
== ~ I
== ~ K
192.Q
~
O
OOOO~
+= I
Q
O
~
OOOO
+
= I
Q
O~
OOO
O += K
Figure 15.
193.~
P
O^j = I
P
O_j = I
P
O`j = EcKNRFK
194.( )
( )OO~
~Q
+
= I
( )( )O
O
~
~Q
+
= I
( )( )O
O
~
~Q +
= K
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33
195.O
O
O
~p ~ === I
O
O~
O~p === I
( )( )( )~p = Ee c~FIp = I
oQ
~p = I
= oOp O
I
O~
O~
O~p O
= K
3.5 Square
p ~ ~W ~a~~W o~ W oo~ W mW i^~W p
Figure 16.
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CHAPTER 3. GEOMETRY
34
196. O~ =
197.O
O~
O
o ==
198.O
~ =
199. ~Qi =
200. O~p =
3.6 Rectangle
p ~ ~W ~I a~~W o~ W omW i^~W p
Figure 17.
201. OO ~ +=
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CHAPTER 3. GEOMETRY
35
202.O
o=
203. ( )~Oi +=
204. ~p =
3.7 Parallelogram
p ~ ~~~W ~I a~~W ON I
` ~W I^ ~~W ^W mW i^~W p
Figure 18.
205. =+ NUM
206. ( )OOOOON ~O +=+
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37
210. =+ NUM
211. OOOON ~Q =+
212.~O
~ ON==
213.O
~
~Q
O
ON
===
214. ~Qi =
215. == ~~p O I
ONO
Np = K
3.9 Trapezoid
_~ ~ ~W ~I jW ^W
^~W p
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CHAPTER 3. GEOMETRY
38
Figure 20.
216.O
~ +=
217. O
~p =
+=
3.10 Isosceles Trapezoid
_~ ~ ~W ~I iW
jW ^W a~~W o~ W o^~W p
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CHAPTER 3. GEOMETRY
39
Figure 21.
218.O
~ +=
219. O~ +=
220. ( )OO ~Q
N =
221.( )( )~O~O
~o
O
++
+=
222. O
~p =
+=
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CHAPTER 3. GEOMETRY
40
3.11 Isosceles Trapezoid with
Inscribed Circle
_~ ~ ~W ~I iW jW ^W a~~W o~ W oo~ W mW i^~W p
Figure 22.
223. O~ =+
224. O
~ =
+=
225. OOO +=
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CHAPTER 3. GEOMETRY
41
226.O
~
O
==
227.~
S
~
U
~
O
~
N
O
Q
O
o OO
O
+++
=+=+===
228. ( ) Q~Oi =+=
229. ( ) OiO ~~O~p ===+=+=
3.12 Trapezoid with Inscribed Circle
_~ ~ ~W ~I i~~ W I jW ^W a~~W ON I
^ ~~W o~ W
o~ W omW i^~W p
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CHAPTER 3. GEOMETRY
42
Figure 23.
230. ~ +=+
231.O
O
~
+=
+=
232. ( ) ( )O~Oi +=+=
233. O
O
~p =
+=
+= I
= O
Np ON K
3.13 Kite
p ~ W ~I a~~W ON I
^W II mW i
^~W p
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CHAPTER 3. GEOMETRY
43
Figure 24.
234. =++ PSMO
235. ( )~Oi +=
236.O
p ON=
3.14 Cyclic Quadrilateralp ~ ~~~W ~I I I a~~W ON I
^ ~~W f~ ~W IIIo~ W omW ipW ^~W p
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CHAPTER 3. GEOMETRY
44
Figure 25.
237. =+=+ NUM
238. m q
ON~ =+
239. ~i +++=
240.( )( )( )( )( )( )( )~
~~~
Q
No
+++
= I
O
i = K
241. = O
Np ON I
( )( )( )( )~p = I
O
i = K
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CHAPTER 3. GEOMETRY
45
3.15 Tangential Quadrilateral
p ~ ~~~W ~I I I a~~W ON I
^ ~~W o~ W mW ipW ^~W p
Figure 26.
242. ~ +=+
243. ( ) ( )O~O~i +=+=+++=
244.( ) ( )
O
~~
OOOO
ON += I
Oi = K
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CHAPTER 3. GEOMETRY
46
245. == O
Np ON
3.16 General Quadrilateral
p ~ ~~~W ~I I I a~~W
ON
I
^ ~~W f~ ~W IIImW i^~W p
Figure 27.
246. =+++ PSM
247. ~i +++=
-
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CHAPTER 3. GEOMETRY
47
248. = O
Np ON
3.17 Regular Hexagon
pW ~f~ ~W p~ W o~ W o~ W omW ipW ^~W p
Figure 28.
249. = NOM
250.O
P~ ==
-
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CHAPTER 3. GEOMETRY
48
251. ~o=
252. ~Si =
253.O
PP~p
O
== I
O
i = K
3.18 Regular Polygon
pW ~k W f~ ~W p~ W o~ W o~ W omW ipW ^~W p
-
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CHAPTER 3. GEOMETRY
49
Figure 29.
254.
= NUMO
O
255. = NUMO O
256.
O
~o
=
257.Q
~o
~O
~
O
O ===
258. ~i =
259.
O
O
op
O = I
Q
~op
OO == I
-
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CHAPTER 3. GEOMETRY
50
O
i = K
3.19 Circle
o~W oa~W `W ~p~ W I q~ W `~ ~W f ~W mW i^~W p
260.O
oO~
=
Figure 30.
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CHAPTER 3. GEOMETRY
51
261. ONON ~~ =
Figure 31.
262. NN =
Figure 32.
263. NO =
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CHAPTER 3. GEOMETRY
52
Figure 33.
264.O
=
Figure 34.
265. oOi ==
266. O
io
Q
op
OO
=
==
-
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CHAPTER 3. GEOMETRY
53
3.20 Sector of a Circle
o~ ~ W o^ W `~ ~ E ~~FW `~ ~ E FW mW i^~W p
Figure 35.
267. o =
268.=
NUM
o
269. oOi +=
270.
===PSM
o
O
o
O
op
OO
-
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CHAPTER 3. GEOMETRY
54
3.21 Segment of a Circle
o~ ~ W o^ W `W ~`~ ~ E ~~FW `~ ~ E FW e W mW i
^~W p
Figure 36.
271. OoOO~ =
272. OO ~oQO
No = I o <
273. ~i +=
-
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CHAPTER 3. GEOMETRY
55
274. ( )[ ] ( )O
o
NUMO
oo~o
O
Np
OO
=
== I
~P
Op K
3.22 Cube
bW ~a~~W o~ W o~ W p~ ~~W psW s
Figure 37.
275. P~ =
276.O~ =
-
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CHAPTER 3. GEOMETRY
56
277.O
P~o=
278. O~Sp =
279. P~s =
3.23 Rectangular Parallelepiped
bW ~I I a~~W p~ ~~W psW s
Figure 38.
280. OOO ~ ++=
281. ( )~~Op ++=
282. ~s=
-
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CHAPTER 3. GEOMETRY
57
3.24 Prism
i~~ W eW i~~ ~~W ip
^~ ~W _p
q~ ~ ~~W psW s
Figure 39.
283. _i pOpp += K
284. i~~ ^~ ~ o m( )~~~~p PONi ++++= K
285. i~~ ^~ ~ l mp i= I
K
-
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CHAPTER 3. GEOMETRY
58
286. ps _=
287. `~~D md ~~ ~K f ~ ~~ ~ ~ ~~~ I ~ ~K
3.25 Regular Tetrahedron
q~ W ~eW ^~ ~W _p
p~ ~~W psW s
Figure 40.
288. ~P
O
=
-
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CHAPTER 3. GEOMETRY
59
289.Q
~Pp
O
_=
290. O~Pp =
291.OS
~p
P
Ns
P
_ == K
3.26 Regular Pyramid
p ~W ~i~~ W eW
p~ W k W p ~W o~ ~W ^~ ~W _p
i~~ ~ ~~W ip
q~ ~ ~~W psW s
-
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CHAPTER 3. GEOMETRY
60
Figure 41.
292.Q
~
OO =
293.
O
~Q
OOO
=
294. ~Q~Q
N~
O
Np OOi ===
295. p _=
296. i_ ppp +=
297. P
Np
P
Ns _ ==
-
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CHAPTER 3. GEOMETRY
61
3.27 Frustum of a Regular Pyramid
_~ ~ W
PON
PON
IIII
~II~I~I~
K
K
eW p~ W ^~ ~W Np I Op
i~~ ~ ~~W ip
m ~W Nm I Omp~ ~W q~ ~ ~~W psW s
Figure 42.
298.
~
~
~
~
~
P
P
O
O
N
N ====== K
-
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CHAPTER 3. GEOMETRY
62
299. O
N
O p
p=
300.( )
O
mmp ONi
+=
301. ONi pppp ++=
302. ( )OONN ppppP
s ++=
303. [ ]ONO
N NP
p
~
~
N
P
ps ++=
++=
3.28 Rectangular Right Wedge
p ~W ~I q W eW i~~ ~ ~~W ip
^~ ~W _pq~ ~ ~~W psW s
-
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CHAPTER 3. GEOMETRY
63
Figure 43.
304. ( ) ( )OOOOi ~Q~O
Np ++++=
305. ~p _=
306. i_ ppp +=
307. ( )~OS
s +=
3.29 Platonic Solids
bW ~o~ W o~ W op~ ~~W p
sW s
-
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CHAPTER 3. GEOMETRY
64
308. c m~ pq ~ ~ ~ ~
~ ~ K
p k s
k b
k c~
p
q~ Q S Q PKOR` U NO S PKOO
l~ S NO U PKOTf~ NO PM OM PKOT
a~ OM PM NO PKOT
Octahedron
Figure 44.
309.S
S~ =
310. O
O~
o=
-
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CHAPTER 3. GEOMETRY
65
311. P~Op O=
312.P
O~s
P
=
Icosahedron
Figure 45.
313.)
NO
RPP~
+=
314. ( )RROQ~o +=
315. P~Rp O=
316.)
NO
RP~Rs
P +=
-
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CHAPTER 3. GEOMETRY
66
Dodecahedron
Figure 46.
317.( )
O
RNNORNM~
+=
318.)
Q
RNP~o
+=
319. ( )RORR~Pp O +=
320.Q
RTNR~s
P +=
3.30 Right Circular Cylinder
o~ ~W oa~ ~W
-
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CHAPTER 3. GEOMETRY
67
eW ei~~ ~ ~~W ip
^~ ~W _pq~ ~ ~~W psW s
Figure 47.
321. oeOp i =
322. ( )
+=+=+= O
eoeoOpOpp _i
323. eoeps O_ ==
-
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CHAPTER 3. GEOMETRY
68
3.31 Right Circular Cylinder with
an Oblique Plane Face
o~ ~W oq ~ ~ W N
q ~ W O
i~~ ~ ~~W ip
^~ ~ ~W _p
q~ ~ ~~W psW s
Figure 48.
324. ( )ONi op +=
325.
O
ONOO
_ O
ooop
++=
-
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CHAPTER 3. GEOMETRY
69
326.
++++=+=O
ONOON_i
O
oooppp
327. ( )ONO
O
os +
=
3.32 Right Circular Coneo~ ~W oa~ ~W eW ep~ W i~~ ~ ~~W ip
^~ ~W _p
q~ ~ ~~W psW s
Figure 49.
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CHAPTER 3. GEOMETRY
70
328. OO oe =
329.O
op i
==
330. O_ op =
331. ( )
+=+=+=O
O
Nooppp _i
332. eoP
Nep
P
Ns O_ ==
3.33 Frustum of a Right Circular Cone
o~ ~W oI eW ep~ W p~ ~W ^~ ~W Np I Op
i~~ ~ ~~Wi
p
q~ ~ ~~W psW s
-
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CHAPTER 3. GEOMETRY
71
Figure 50.
333. ( )OO oe =
334.
o=
335. OO
O
N
O
o
p
p==
336. ( )op i +=
337. ( )oopppp OOiON +++=++=
338. ( )OONN ppppP
s ++=
339. [ ]ONO
N NP
p
o
oN
P
ps ++=
++=
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CHAPTER 3. GEOMETRY
72
3.34 Sphere
o~W oa~W p~ ~~W psW s
Figure 51.
340. OoQp =
341. poP
N
S
Neo
P
Qs PP ===
3.35 Spherical Cap
o~ W oo~ ~W eW ^~ ~ ~W _p
^~ ~ ~W `p q~ ~ ~~W psW s
-
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CHAPTER 3. GEOMETRY
73
Figure 52.
342.O
o
OO +=
343. O_ p =
344. ( )OO
` p +=
345. ( ) ( )OOO`_ oOOppp +=+=+=
346. ( ) ( )OOO PS
oPS
s +
=
=
3.36 Spherical Sector
o~ W oo~ ~ ~ ~W eW
q~ ~ ~~W psW s
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CHAPTER 3. GEOMETRY
74
Figure 53.
347. ( )Oop +=
348. oP
Os O=
kW q ~ ~ ~ ~ K
3.37 Spherical Segment
o~ W oo~ ~W N I O
eW ^~ ~ ~W pp
^~ ~ ~W Np I Op
q~ ~ ~~W p
sW s
-
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CHAPTER 3. GEOMETRY
75
Figure 54.
349. oOp p =
350. ( )OOONONp oOpppp ++=++=
351. ( )OOOON PPS
Ns ++=
3.38 Spherical Wedge
o~W oa~ ~ W a~ ~ ~~W ^~ ~ W ip
q~ ~ ~~W psW s
-
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CHAPTER 3. GEOMETRY
76
Figure 55.
352. oOVM
op O
O
i =
=
353. oOoVM
oop OO
OO +=
+=
354. oP
O
OTM
os P
P
=
=
3.39 Ellipsoid
p-~W ~I I sW s
-
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CHAPTER 3. GEOMETRY
77
Figure 56.
355. ~P
Qs =
Prolate Spheroid
p-~W ~I I E ~> Fp~ ~~W psW s
356.
+=
~~Op I
~
~
OO = K
357. ~PQs O=
-
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CHAPTER 3. GEOMETRY
78
Oblate Spheroid
p-~W ~I I E ~ < Fp~ ~~W psW s
358.
+= ~L ~
~~
Op I
~
OO
= K
359. ~P
Qs O=
3.40 Circular Torus
j~ ~W o
j ~W p~ ~~W psW s
-
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CHAPTER 3. GEOMETRY
79
Picture 57.
360. oQp O=
361. OO oOs =
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80
Chapter 4
Trigonometry
^W I
o~ E~ ~ FW I t W
4.1 Radian and Degree Measures of Angles
362. ?QRDNTRT
NUM
~N
=
363. ~MNTQRPKM~NUM
N
=
364. ~MMMOVNKM~SMNUM
DN
=
365. ~MMMMMRKM~PSMMNUM
?N
=
366. ^EF
M PM QR SM VM NUM OTM PSM
^
E~~F
M
S
Q
P
O
O
P O
-
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CHAPTER 4. TRIGONOMETRY
81
4.2 Definitions and Graphs of Trigonometric
Functions
Figure 58.
367.
=
368.
=
369.
~ =
370.
=
-
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CHAPTER 4. TRIGONOMETRY
82
371.
=
372.
=
373. p c= I NN K
Figure 59.
374. ` c
= I NN K
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CHAPTER 4. TRIGONOMETRY
83
Figure 60.
375. q~ c
~= I ( )O
NO + I K~
Figure 61.
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CHAPTER 4. TRIGONOMETRY
84
376. `~ c= I I K
Figure 62.
377. p~ c
= I ( )O
NO + K
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CHAPTER 4. TRIGONOMETRY
85
Figure 63.
378. `~ c= I K
Figure 64.
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CHAPTER 4. TRIGONOMETRY
86
4.3. Signs of Trigonometric Functions
379.n~~
p
`
q~
`
p
`
f H H H H H Hff H H
fff H Hfs H H
380.
Figure 65.
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-
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CHAPTER 4. TRIGONOMETRY
88
382.
~ ~
NRNO
Q
OS
Q
OS+ PO PO+
NUNM
Q
NR
Q
RONM+R
ROR ROR+
PS
R
Q
RONM
Q
NR +
NR
RONM
+
RONM
NR
+
RQNM
P
Q
NR+
Q
RONMRONM
NR
+NR
RONM
+
TOR
O
Q
RONM+
Q
NR ROR+ R
ROR
TRNO
R
Q
OS+
Q
OS PO+ PO
4.5 Most Important Formulas
383. N OO =+
384. N~ OO =
385. N OO =
386.
=
~
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CHAPTER 4. TRIGONOMETRY
89
387.
=
388. N~ =
389.
=
N
390. = N
4.6 Reduction Formulas
391. ~ + ~
VM + + + + ~
+VM + ~
NUM + ~
+NUM + ~ +
OTM + + ~
+OTM + ~
PSM + ~
+PSM + + + ~ +
-
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CHAPTER 4. TRIGONOMETRY
90
4.7 Periodicity of Trigonometric Functions
392. ( ) = O I O PSM K
393. ( ) = O I O PSM K
394. ( ) = ~~ I NUM K
395. ( ) = I NUM K
4.8 Relations between TrigonometricFunctions
396. ( ) NQO
OONO
NN OO
===
O~N
O~O
O+
=
397. ( ) NO
OONO
NN OO
=+==
O~N
O~N
O
O
+
=
398.
=+
==
=O
ON
ON
ON
~ O
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CHAPTER 4. TRIGONOMETRY
91
O~N
O~O
ON
ON
O+
=+
=
399.
=+
==
=ON
O
O
ONN
O
O~O
O~N
ON
ONO
=
+=
400.
O~N
O~N
~N
N
O
O
O
+
=+=
=
401.
O~O
O~N
N
N
O
O
+
=+=
=
4.9 Addition and Subtraction Formulas
402. ( ) +=+
403. ( ) =
404. ( ) =+
405. ( ) +=
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CHAPTER 4. TRIGONOMETRY
92
406. ( )+
=+~~N
~~~
407. ( )+
=~~N
~~~
408. ( )+
=+~~
~~N
409. ( )+=
~~
~~N
4.10 Double Angle Formulas
410. = OO
411. NOONO OOOO ===
412.
=
=~
O
~N
~OO~
O
413.O
~
O
NO
O =
=
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CHAPTER 4. TRIGONOMETRY
93
4.11 Multiple Angle Formulas
414. == POP PQPP
415. = UQQ P
416. += RP NSOMRR
417. == OPP PPQP
418. NUUQ OQ +=
419. += ROMNSR PR
420.
= OP
~PN
~~PP~
421.+
=QO
P
~~SN
~Q~QQ~
422.
+
+=
QO
PR
~R~NMN
~R~NM~R~
423.NP
PP
O
P
=
424.+
=P
QO
~Q~Q
~~SNQ
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CHAPTER 4. TRIGONOMETRY
94
425.+
+=
~R~NM~
~R~NMNR
PR
QO
4.12 Half Angle Formulas
426.O
N
O
=
427.O
N
O
+=
428. =
=+
=+
=
N
N
N
N
O~
429. +=+
=
=+
=
N
N
N
N
O
4.13 Half Angle Tangent Identities
430.
O~N
O~O
O+
=
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CHAPTER 4. TRIGONOMETRY
95
431.
O~N
O~N
O
O
+
=
432.
O~N
O~O
~O
=
433.
O~O
O~N
O
=
4.14 Transforming of TrigonometricExpressions to Product
434.O
O
O +
=+
435.O
O
O +=
436.O
O
O +
=+
437.O
O
O +
=
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CHAPTER 4. TRIGONOMETRY
96
438.( )
+
=+
~~
439.( )
=
~~
440.( )
+
=+
441.( )
=
442.
+
=
=+Q
OQ
O
443.
+=
=
QO
QO
444.( )
=+
~
445.( )
+=
~
446.O
ON O
=+
447.O
ON O
=
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CHAPTER 4. TRIGONOMETRY
97
448.
=+OQ
ON O
449.
=OQ
ON O
4.15 Transforming of TrigonometricExpressions to Sum
450.( ) ( )
O
+=
451.( ) ( )
O
++=
452.( ) ( )
O
++=
453.++
=
~~~~
454.++
=~~
455.++
=~
~~
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CHAPTER 4. TRIGONOMETRY
98
4.16 Powers of Trigonometric Functions
456.O
ON O
=
457.Q
PP P
=
458.U
POQQ Q +=
459.NS
RPRNM R
+=
460.
PO
SQSONRNM S
+=
461.O
ON O
+=
462.Q
PP P
+=
463.U
POQQ Q ++=
464.NS
RPRNM R
++=
465. PO
SQSONRNM
S +++
=
-
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CHAPTER 4. TRIGONOMETRY
99
4.17 Graphs of Inverse Trigonometric
Functions
466. f p c
~= I NN IO
~O
K
Figure 66.
467. f ` c~= I NN I ~M K
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CHAPTER 4. TRIGONOMETRY
100
Figure 67.
468. f q~ c
~~= I IO
~~O
-
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CHAPTER 4. TRIGONOMETRY
101
469. f `~ c~= I I
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CHAPTER 4. TRIGONOMETRY
102
471. f `~ c
( ] [ ) KOIMMIO~IINNII~
=
Figure 71.
4.18 Principal Values of InverseTrigonometric Functions
472.
M O
N
O
O
O
P
N~ M PM QR SM VM
~ VM SM QR PM M
O
N
O
O
O
P N
~PM
QR SM
VM
~ NOM
NPR NRM NUM
-
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CHAPTER 4. TRIGONOMETRY
103
473.M
P
PN P
P
P N P
~~ M PM QR SM PM QR
SM
~ VM SM QR PM NOM NPR
NRM
4.19 Relations between InverseTrigonometric Functions
474. ( ) ~~ =
475. ~
O
~
=
476. ON~~ = I NM K
477. ON~~ = I MN K
478.ON
~~~
= I N
O < K
479.N
~~O
= I NM < K
480.
= N
~~O
I MN
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CHAPTER 4. TRIGONOMETRY
104
482. ~O
~
=
483. ON~~ = I NM K
484. ON~~ = I MN K
485.N
~~~O
= I NM < K
486.N
~~~O
+= I MN
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CHAPTER 4. TRIGONOMETRY
105
493.N
~~O
~~
= I M> K
494.N
~~O
~~
= I M< K
495.N
~~~ = I M> K
496. = N~~~ I M< K
497. ( ) ~~ =
498. ~~O
~
=
499.ON
N~~
+= I M> K
500.ON
N~~
+= I M< K
501.ON
~~+
=
502.N
~~~ = I M> K
503.N
~~~ += I M< K
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CHAPTER 4. TRIGONOMETRY
106
4.20 Trigonometric Equations
t W
504. ~ = I ( ) ~~N +=
505. ~ = I O~~ +=
506. ~~ = I ~~~ +=
507. ~ = I ~~ +=
4.21 Relations to Hyperbolic Functions
f~~ W
508. ( ) =
509. ( ) ~~ =
510. ( ) =
511. ( ) =
512. ( ) =
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107
Chapter 5
Matrices and Determinants
j~W ^I _I `
b ~ ~W ~ I I ~ I I a~ ~ ~W ^j ~ ~ W j
`~ ~ ~ W `
q~ ~ ~W q^ I ^
^ ~ ~W ^~
q~ ~ ~W ^
f ~ ~W N^ o~ W o~ ~~W
k~~ W I
5.1 Determinants
513. p l a~
NOON
OO
NN~~
~
~^ ==
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CHAPTER 5. MATRICES AND DETERMINANTS
108
514. q l a~
++== POONNPPNOPNOPPOONN
PPPOPN
OPOOON
NPNONN
~~~~~~~~~
~~~
~~~
~~~
^
PNOONPPPONNOPOOPNN ~~~~~~~~~
515. p~ o E^ oF
Figure 72.
516. k- l a~
ON
ON
OOOOON
NNNONN
~~~~
~~~~
~~~~
~~~~
^
KK
KKKKKK
KK
KKKKKK
KK
KK
=
517. jq j ~ ~ ~ -
~ ^ ( )N - ~ ~ ^ - ~ - K
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CHAPTER 5. MATRICES AND DETERMINANTS
109
518. `~
( )
jN` +=
519. i~~ b~ - l a~i~~ ~ -
=
=
N`~^ I IIOIN K= K
i~~ ~ -
==
N`~^ I IIOIN K= K
5.2 Properties of Determinants
520. q ~ ~ ~ ~ ~ ~~ ~ K
OO
NN
ON
ON
~
~
~~=
521. f E F ~ ~I ~ ~K
NN
OO
OO
NN
~~
~~ =
522. f E F ~ ~I ~ ~ K
M~~
~~
OO
NN =
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CHAPTER 5. MATRICES AND DETERMINANTS
110
523. f ~ E F ~ ~ ~I ~ ~
~K
OO
NN
OO
NN
~
~
~
~=
524. f ~ E F ~ ~ E~F ~ ~ E FI ~ ~
~K
OO
NN
OOO
NNN
~
~
~
~=
+
+
5.3 Matrices
525. a^ ~ ^ ~ ~~ ~~ E-
F ~ K
[ ]
==
ON
OOOON
NNONN
~~~
~~~
~~~
~^
K
MMM
K
K
526. p~ ~ ~ ~ K
527. ^ ~ ~ ~ ~~ = I KK
~ ~ ~~K
528. ^ ~ ~ ~ - ~~ = K
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CHAPTER 5. MATRICES AND DETERMINANTS
111
529. a~~ ~ ~ ~ ~ ~ ~ ~~K
530. r ~ ~ ~~ ~ ~ ~~ ~ ~ K q ~ fK
531. ^ ~ ~ ~ K
5.4 Operations with Matrices
532. q ~ ^ ~ _ ~ ~ I ~ I ~ ~ ~ ~ ~~K
533. q ~ ^ ~ _ ~ ~ E ~F I ~ I ~ ~ ~ K f
[ ]
==
ON
OOOON
NNONN
~~~
~~~
~~~
~^
K
MMM
K
K
I
[ ]
==
ON
OOOON
NNONN
_
K
MMM
K
K
I
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CHAPTER 5. MATRICES AND DETERMINANTS
112
+++
+++
+++
=+
OONN
OOOOOOONON
NNNONONNNN
~~~
~~~
~~~
_^
K
MMM
K
K
K
534. f ~ ~~I ~ ~^ = ~ ~I
[ ]
==
ON
OOOON
NNONN
~~~
~~~
~~~
~^
K
MMM
K
K
K
535. j~ q j~q ~ ~ ~
K
f
[ ]
==
ON
OOOON
NNONN
~~~
~~~
~~~
~^
K
MMM
K
K
I
[ ]
==
ON
OOOON
NNONN
_
K
MMM
K
K
I
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CHAPTER 5. MATRICES AND DETERMINANTS
113
==
ON
OOOON
NNONN
`^_
K
MMM
K
K
I
=
=+++=
NOONN ~~~~ K
E IIOIN K= X IIOIN K= FK
q
[ ]
==
OPOOON
NPNONN
~~~
~~~~^ I [ ]
==
P
O
N
_ I
=
=
POPOOONON
PNPONONNN
P
O
N
OPOOON
NPNONN
~~~
~~~
~~~
~~~^_ K
536. q~ ~ j~f ~ ~ ~ ~ ~I
~ ~ ~ ~ ~Kf ^ ~ ~I ~ q^
^
K
537. q ~ ^ ~ f^^ q = K
538. f ~ ^_ I
( )
qqq
^_^_ = K
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CHAPTER 5. MATRICES AND DETERMINANTS
114
539. ^ j~f ^ ~ ~ ~I ~I ^~ I
~ ~ ~ ` ^W
[ ]q`^~ = K
540. q~ ~ j~f ^ ~ ~ ~I ~I ^ I ~ ~~W
OONN ~~~^ +++= K K
541. f ~ j~f ^ ~ ~ ~ ~ ~ ~
^ I N^
^
^~^ N= K
542. f ~ ^_ I ( ) NNN ^_^_ = K
543. f ^ ~ ~ ~I u ~ ~
u^u = I ~ ~ ~~ ~
Mf^ = K
5.5 Systems of Linear Equations
s~~W I I I N I KIO
o~ W KI~I~II~I~I~ NONNNPON
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CHAPTER 5. MATRICES AND DETERMINANTS
115
a~W aI a I a I a
j~W ^I _I u
544.
=+
=+
OOO
NNN
~
~I
a
a = I
a
a
= E`~ FI
NOON
OO
NN ~~~
~a == I
NOON
OO
NN
a == I
NOON
OO
NN
~~~
~a == K
545. f Ma I ~ ~ W
a
a = I
a
a
= K
f Ma= ~ Ma E Ma FI ~
Kf Maaa
=== I ~ ~
K
546.
=++
=++
=++
PPPP
OOOO
NNNN
~
~
~
I
a
a
= I a
a
= I aa
= E`~
FI
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CHAPTER 5. MATRICES AND DETERMINANTS
116
PPP
OOO
NNN
~
~
~
a= I
PPP
OOO
NNN
a = I
PPP
OOO
NNN
~
~
~
a = I
PPP
OOO
NNN
~
~
~
a = K
547. f Ma I ~ ~ W
a
a = I
a
a
= Ia
a = K
f Ma= ~ Ma E Ma Ma FI
~ Kf Maaaa ==== I ~
~ K
548. j~ c ~ p i~ b~ rq ~ ~
=+++
=+++
=+++
OONN
OOOOONON
NNONONNN
~~~
~~~
~~~
K
KKKKKKKKKKKK
K
K
~ ~
=
O
N
O
N
ON
OOOON
NNONN
~~~
~~~
~~~
MM
K
MMM
K
K
I
KK_u^ = I
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CHAPTER 5. MATRICES AND DETERMINANTS
117
=
ON
OOOON
NNONN
~~~
~~~
~~~
^
K
MMM
K
K
I
=
O
N
uM
I
=
O
N
_M
K
549. p ~ p i~ b~ _^u N= I
N
^
^K
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118
Chapter 6
Vectors
sW r
I r
I r
I r
I
^ _ I
s W r I r I
r W r
I r
I r
k W Mr
`~
rW NNN wIvIu
`~ r
W OOO wIvIu
p~~W I
a W I I ^ W
6.1 Vector Coordinates
550. r s
( )MIMIN=r
I
( )MINIM=r
I
( )NIMIM=r
I
N === rrr
K
551. ( ) ( ) ( ) ^_ MNMNMNrrrr
++==
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CHAPTER 6. VECTORS
119
Figure 73.
552. ( ) ( ) ( )OMNO
MN
O
MN ^_ ++==r
553. f ^_ r=
I _^ r=
K
Figure 74.
554. = u r
I
= v r I= w
rK
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CHAPTER 6. VECTORS
120
Figure 75.
555. f ( ) ( )NNNN wIvIuwIvIu rr
= I
Nuu = I Nvv= I Nww = K
6.2 Vector Addition
556. rrr+=
Figure 76.
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CHAPTER 6. VECTORS
121
Figure 77.
557. PON r
Krrrr
++++=
Figure 78.
558. `~ i~
rrrr+=+
559. ^~ i~( ) ( ) rrrrrr ++=++
560. ( )ONONON wwIvvIuu +++=+rr
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CHAPTER 6. VECTORS
122
6.3 Vector Subtraction
561. rrr=
rrr=+ K
Figure 79.
Figure 80.
562. ( ) rrrr
+=
563. ( )MIMIMM == rrr
564. MM=r
565. ( )ONONON wwIvvIuu =rr
I
6.4 Scaling Vectors
566. rr=
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CHAPTER 6. VECTORS
123
Figure 81.
567. rr=
568. ( )wIvIu =r
569. = rr
570. ( ) rrr+=+
571. ( ) ( ) ( ) rrr
==
572. ( ) rrrr
+=+
6.5 Scalar Product
573. p~~ m s r
~r
= rrrr
I
~ r
~ r
K
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CHAPTER 6. VECTORS
124
Figure 82.
574. p~~ m `~ cf ( )NNN wIvIu =r I ( )OOO wIvIu=r I
ONONON wwvvuu ++=rr
K
575. ^ _ q sf ( )NNN wIvIu =
rI ( )OOO wIvIu=r
I
O
O
O
O
O
O
O
N
O
N
O
N
ONONON
wvuwvu
wwvvuu
++++
++= K
576. `~ m
rrrr=
577. ^~ m( ) ( )
rrrr=
578. a m( )
rrrrrrr+=+
579. M =rr
r
Ir
~ ~ EO
= FK
580. M >rr
OM
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CHAPTER 6. VECTORS
125
581. M
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CHAPTER 6. VECTORS
126
Figure 83.
588.
OOO
NNN
wvu
wvu
rrr
rrr==
589.
==OO
NN
OO
NN
OO
NN
vu
vuI
wu
wuI
wv
wv
rrr
590. == p rrrr
EcKUPF
591. ^ _ q s EcKUPF
rr
rr
=
592. k~ m( )
rrrr=
593. ^~ m( ) ( )
rrrr=
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CHAPTER 6. VECTORS
127
594. a m( )
rrrrrrr+=+
595. Mrrr
= r
~ r
~ ~~ E M= FK
596. Mrrrrrrr
===
597. rrr
= I rrr
= I rrr
=
6.7 Triple Product
598. p~~ q m[ ] ( ) ( ) ( )
rrrrrrrrrrrr===
599. [ ] [ ] [ ] [ ] [ ] [ ] rrrrrrrrrrrrrrrrrr =====
600. ( ) [ ] rrrrrr=
601. p~~ q m `~ c
( )PPP
OOO
NNN
wvuwvu
wvu
=
rrr
I
( )NNN wIvIu =
rI ( )OOO wIvIu=r
I ( )PPP wIvIu=r
K
602. s m~~( )s rrr=
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CHAPTER 6. VECTORS
128
Figure 84.
603. s m~
( )S
Ns
rrr=
Figure 85.
604. f ( ) M = rrr
I r
I r
I ~ r
~ ~
I rrr+= ~~ ~ K
605. f ( ) M rrr
I r
I r
I ~ r
~ ~
K
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CHAPTER 6. VECTORS
129
606. s q m( ) ( ) ( )
rrrrrrrrr=
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130
Chapter 7
Analytic Geometry
7.1 One-Dimensional Coordinate System
m ~W M I N I O I M I N I O
o~ W a~ W
607. a~ _ q m
ONNO ^_ ===
Figure 86.
608. a ~ i p o~
++
=N
ONM I
`_
^`= I N K
Figure 87.
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CHAPTER 7. ANALYTIC GEOMETRY
131
609. j ~ i p
O
ON
M
+
= I N= K
7.2 Two-Dimensional Coordinate System
m ~WM
IN
IO
IM
IN
IO
m~ ~W Io~ W m ~ W ~I I Ia~ W ^~W p
610. a~ _ q m( ) ( )ONO
O
NO ^_ +==
Figure 88.
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CHAPTER 7. ANALYTIC GEOMETRY
132
611. a ~ i p o~
+
+
= N
ON
M I +
+
= N
ON
M I
`_
^`= I N K
Figure 89.
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CHAPTER 7. ANALYTIC GEOMETRY
133
Figure 90.
612. j ~ i p
O
ONM
+= I
O
ONM
+= I N= K
613. ` Ef j~F ~ q~
P
PONM
++= I
P
PONM
++= I
( )NN I^ I ( )OO I_ I ~ ( )PP I` ~ ~ ^_` K
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CHAPTER 7. ANALYTIC GEOMETRY
134
Figure 91.
614. f Ef ^ _F ~ q~
~
~
PONM ++
++= I ~
~
PONM ++
++= I
_`~= I `^ = I ^_ = K
Figure 92.
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CHAPTER 7. ANALYTIC GEOMETRY
135
615. ` Ef p m~_F ~ q~
N
N
N
O
N
N
N
PP
OO
NN
POP
OP
OOO
OO
NON
ON
M
+
++
= I
N
N
N
O
N
N
N
PP
OO
NN
OP
OPP
OO
OOO
ON
ONN
M
+
++
=
Figure 93.
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CHAPTER 7. ANALYTIC GEOMETRY
136
616. l Ef ^F ~ q~
N
N
N
N
N
N
PP
OO
NN
OPONP
OONPO
ONPON
M
+
+
+
= I
N
N
N
N
N
N
PP
OO
NN
PONOP
ONPOO
NPOON
M
+
+
+
=
Figure 94.
617. ^~ ~ q~
( ) ( )NPNP
NONO
PP
OO
NN
O
N
N
N
N
O
Np
==
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CHAPTER 7. ANALYTIC GEOMETRY
137
618. ^~ ~ n~~~
( ) ( )( ) ( )( )[ ++++= POPOONON O
N
p( )( ) ( )( )NQNQQPQP ++++
Figure 95.
kW f ~ SNTI SNU EHF EF ~ ~ ~ ~~K
619. a~ _ q m m~ `~
( )NOONOO
ON O^_ +==
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CHAPTER 7. ANALYTIC GEOMETRY
138
Figure 96.
620. ` o~~ `~ m~ `~= I = K
Figure 97.
621. ` m~ `~ o~~ `~
OO += I
~ = K
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CHAPTER 7. ANALYTIC GEOMETRY
139
7.3 Straight Line in Plane
m ~W uI vI I M I N I M I N I N~ I O~ I
o~ W I ~I I I I ^I _I `I N^ I O^ I
^W I ^ W k~ W
r
m W r
I ~r
I r
622. d~ b~ ~ p~ iM`_^ =++
623. k~ s ~ p~ iq ( )_I^
r ~ M`_^ =++ K
Figure 98.
624. b b~ ~ p~ i Ep-f cF += K
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CHAPTER 7. ANALYTIC GEOMETRY
140
q ~ = ~ K
Figure 99.
625. d~ ~ i
NO
NO
~
==
Figure 100.
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CHAPTER 7. ANALYTIC GEOMETRY
141
626. b~ ~ i d ~ m ~ d~( )MM += I
~I ( )MM Im ~ K
Figure 101.
627. b~ ~ i q~ m~ q q m
NO
N
NO
N
=
MNN
N
OO
NN = K
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CHAPTER 7. ANALYTIC GEOMETRY
142
Figure 102.
628. f c
N
~
=+
Figure 103.
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CHAPTER 7. ANALYTIC GEOMETRY
143
629. k~ cM =+
Figure 104.
630. m a c
v
u
NN =
I
( )vIu ~ ( )NNN Im K
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CHAPTER 7. ANALYTIC GEOMETRY
144
Figure 105.
631. s~ i~=
632. e~ i=
633. s b~ ~ p~ i
~rrr
+= I
l ~Iu ~ ~~ I~r
~ ^ I
r
~ I ~~ I ~ ~~I
= lur
~ u K
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CHAPTER 7. ANALYTIC GEOMETRY
145
Figure 106.
634. p~ i m~~ c
+=
+=
OO
NN
~
~
I( )I ~ ~ ~ I( )ON ~I~ ~ ~ ~ I( )ON I ~ ~ ~ ~~ I ~ ~~K
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CHAPTER 7. ANALYTIC GEOMETRY
146
Figure 107.
635. a~ c ~ m q ~ iq ~ ( )I~m
M`_^ =++
OO _^
`_^~
+++= K
Figure 108.
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CHAPTER 7. ANALYTIC GEOMETRY
147
636. m~~ iq NN += ~ OO += ~ ~~
ON = Kq M`_^ NNN =++ ~ M`_^ OOO =++ ~~~
O
N
O
N
_
_
^
^= K
Figure 109.
637. m~ iq
NN
+= ~OO
+= ~ ~
N
O
N = I ~I N ON = K
q M`_^ NNN =++ ~ M`_^ OOO =++ ~~
M__^^ ONON =+ K
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CHAPTER 7. ANALYTIC GEOMETRY
148
Figure 110.
638. ^ _ q i
ON
NO
N
~
+
= I
OO
OO
ON
ON
ONON
_^_^
__^^
++
+= K
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CHAPTER 7. ANALYTIC GEOMETRY
149
Figure 111.
639. f q if M`_^ NNN =++ ~ M`_^ OOO =++ -I ~ ~
NOON
NOONM
_^_^
_`_`
+
= INOON
NOONM
_^_^
`^`^
+
= K
7.4 Circle
o~W o
` W ( )I~m ~W I I N I N I o~ W ^I _I `I aI bI cI
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CHAPTER 7. ANALYTIC GEOMETRY
150
640. b~ ~ ` ` ~ l Ep~~cF
OOO o =+
Figure 112.
641. b~ ~ ` ` ~ ^ m ( )I~
( ) ( ) OOO o~ =+
Figure 113.
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CHAPTER 7. ANALYTIC GEOMETRY
151
642. q m c
M
N
N
N
N
PPOP
OP
OOOO
OO
NNON
ON
OO
=
+
+
+
+
Figure 114.643. m~~ c
=
=
o
o
I OM K
644. d~ c
Mcba^^ OO =++++ E^ I ^cQba OO >+ FKq ~ ~ ( )I~ I
^O
a~ = I
^O
b = K
q ~
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CHAPTER 7. ANALYTIC GEOMETRY
152
^O
^cQbao
OO += K
7.5 Ellipse
p~ ~W ~
p ~W cW ( )MIcN I ( )MIcO a~ W ObW o~ W ^I _I `I aI bI cI mW i^~W p
645. b~ ~ b Ep~~ cF
N
~
O
O
O
O
=+
Figure 115.
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CHAPTER 7. ANALYTIC GEOMETRY
153
646. ~O ON =+ I NI O ~ ~ ~ ( )Im K
Figure 116.
647. OOO ~ +=
648. b
N~
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CHAPTER 7. ANALYTIC GEOMETRY
154
651. d~ c
Mcba`_^ OO =+++++ I
M^`Q_O K
653. `
( )~bQi = I b ~ K
654. ^~ c~ `
( ) ~~RKNi += I
( )OO ~Oi += K
655. ~p =
7.6 Hyperbola
q~ ~W ~`~ ~W cW ( )MIcN I ( )MIcO a~ W ObW ^W I o~ W ^I _I `I aI bI cI I
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CHAPTER 7. ANALYTIC GEOMETRY
155
656. b~ ~ e~ Ep~~ cF
N
~
O
O
O
O
=
Figure 117.
657. ~O ON = I N I O ~ ~ ~ ( )Im
~ K
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CHAPTER 7. ANALYTIC GEOMETRY
156
Figure 118.
658. b~ ^
~
=
659. OOO ~ +=
660. b
N~
>=
661. b~ a
~
~
O
==
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CHAPTER 7. ANALYTIC GEOMETRY
157
662. m~~ b~ o _~ ~ e~
=
=
~
I OM K
663. d~ c
Mcba`_^ OO =+++++ I
M^`Q_O > K
664. d~ c ^ m~~ `~ ^
Mcba`^ OO =++++ I M^` < K
665. ^ c
Q
O
= I
= I
Q
O
= K
f ~ I ~ ~ ~ M= ~M= K
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CHAPTER 7. ANALYTIC GEOMETRY
158
Figure 119.
7.7 Parabola
c~ ~~W cW c
sW ( )MM Ijo~ W ^I _I `I aI bI cI I ~I I
666. b~ ~ m~~~ Ep~~ cF
OO =
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160
O
= I
`~
O
IMc I
`~ ( )MIMj K
Figure 121.
669. d~ cI ^ m~~ -~Mcba^O =+++ E^I b FI
~ O ++= I~O
N = K
b~
O
M= I
`~
-
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161
+
O
Ic MM I
`~
~O
M = I
~Q
~Q~
O
MOMM
=++= K
Figure 122.
7.8 Three-Dimensional Coordinate System
m ~W M I M I M I N I N I N I
o~ W a~ W
^~W psW s
-
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CHAPTER 7. ANALYTIC GEOMETRY
162
670. a~ _ q m
( ) ( ) ( )ONOO
NO
O
NO ^_ ++==
Figure 123.
671. a ~ i p o~
+
+=
N
ONM I +
+=
N
ONM I +
+=
N
ONM I
`_^`= I N K
-
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163
Figure 124.
Figure 125.
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CHAPTER 7. ANALYTIC GEOMETRY
164
672. j ~ i p
O
ON
M
+
= I O
ON
M
+
= I O
ON
M
+
= I N= K
673. ^~ ~ q~q ~~ ~ ~ ( )NNNN IIm I
( )OOOO IIm I ~ ( )PPPP IIm O
PP
OO
NN
O
PP
OO
NN
O
PP
OO
NN
NN
N
NN
N
NN
N
O
N
p ++= K
674. s ~ q~q ~ ~ ( )NNNN IIm I
( )OOOO IIm I ( )PPPP IIm I ~ ( )QQQQ IIm
N
N
N
N
S
Ns
QQQ
PPP
OOO
NNN
= I
QPQPQP
QOQOQO
QNQNQN
S
Ns
= K
kW t EHF EF ~ ~ ~ K
-
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165
Figure 126.
7.9 Plane
m ~W I I I M I M I M I N I N I N I
o~ W ^I _I `I aIN
^ IO
^ I ~I I IN
~ IO
~ I I I I k~ W r I N
r I Or
a W I I a~ ~W
675. d~ b~ ~ m~Ma`_^ =+++
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166
676. k~ s ~ m~q ( )`I_I^
r ~ ~
Ma`_^ =+++ K
Figure 127.
677. m~~ `~ b~ ~ m~Ma`_^ =+++
f M^ = I ~ ~~ -~Kf M_ = I ~ ~~ -~Kf M` = I ~ ~~ -~Kf Ma = I ~ K
f M_^ == I ~ ~~ -~Kf M`_ == I ~ ~~ -~K
f M`^ == I ~ ~~ -~K
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167
678. m a c( ) ( ) ( ) M`_^ MMM =++ I
( )MMM IIm ~I ~ - ( )`I_I^ ~ ~K
Figure 128.
679. f c
N
~ =++
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CHAPTER 7. ANALYTIC GEOMETRY
168
Figure 129.
680. q m c
M
POPOPO
PNPNPN
PPP
=
I
M
N
N
NN
PPP
OOO
NNN = K
-
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169
Figure 130.
681. k~ cM =++ I
~ ~ ~ I ~ I I ~ ~ ~ ~K
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CHAPTER 7. ANALYTIC GEOMETRY
170
Figure 131.
682. m~~ c
++=
++=
++=
~~
ONN
ONN
ONN
I
( )II ~ ~ ~ I ( )NNN IIm ~I ( )NNN II~ ~ ( )OOO II~ ~ ~~ ~K
-
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CHAPTER 7. ANALYTIC GEOMETRY
171
Figure 132.
683. a~ ^ _ q m~f ~ ~
Ma`_^ NNNN =+++ IMa`_^ OOOO =+++ I
~ ~
OO
OO
OO
ON
ON
ON
ONONON
ON
ON
`_^`_^
``__^^
++++
++=
= rr
rr
K
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172
Figure 133.
684. m~~ m~q ~ Ma`_^ NNNN =+++ ~
Ma`_^ OOOO =+++ ~ ~~
O
N
O
N
O
N
`
`
_
_
^
^== K
685. m~ m~q ~ Ma`_^ NNNN =+++ ~
Ma`_^ OOOO =+++ ~ ~ M``__^^ ONONON =++ K
686. b~ ~ m~ q ( )NNN IIm ~ m~~ q s ( )NNN II~ ~ ( )OOO II~ EcKNPOF
-
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173
M
~
~
OOO
NNN
NNN
=
687. b~ ~ m~ q ( )NNNN IIm ~ ( )OOOO IIm I~ m~~ q s ( )II~
M
~
NONONO
NNN
=
Figure 134.
688. a~ c ~ m q ~ m~q ~ ( )NNNN IIm ~
Ma`_^ =+++
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CHAPTER 7. ANALYTIC GEOMETRY
174
OOO
NNN
`_^
a`_^
++
+++= K
Figure 135.
689. f q m~f ~ Ma`_^ NNNN =+++ ~
Ma`_^ OOOO =+++ I ~
+=
+=
+=
~
N
N
N
I
~
NNN =
=
I
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175
OO
NN
`_
`_~= I
OO
NN
^`
^` = I
OO
NN
_^
_^ = I
OOO
OO
NN
OO
NN
N~
_a
_a
`a
`a
++
= I
OOO
OO
NN
OO
NN
N~
`a
`a~
^a
^a
++
= I
OOO
OO
NN
OO
NN
N~
^a
^a
_a
_a~
++
= K
7.10 Straight Line in Space
m ~W I I I N I N I N I
a W I I o~ W ^I _I `I aI ~I I I N~ I O~ I I
a ~ W r
I Nr
I Or
k~ ~ ~W r
^ W
690. m a c b~ ~ i
~
NNN =
=
I
( )NNNN IIm I ~ ( )II~ K
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176
Figure 136.
691. q m c
NO
N
NO
N
NO
N
=
=
Figure 137.
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CHAPTER 7. ANALYTIC GEOMETRY
177
692. m~~ c
+=+=
+=
N
N
N
I
( )NNNN IIm ~ I I I ~
I ~~ ~ ~ K
Figure 138.
693. ^ _ q p~ i
OO
OO
OO
ON
ON
ON
ONONON
ON
ON
~~
~~
++++
++=
= rrrr
-
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178
Figure 139.
694. m~~ i
q ~ ~~ ON
rrI
O
N
O
N
O
N
~
~== K
695. m~ i
q ~ ~~ M ON =
rrI
M~~ ONONON =++ K
696. f q i
q N
N
N
N
N
N
~
=
=
~
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179
O
O
O
O
O
O
~
=
=
M
~
~
OOO
NNN
NONONO
=
K
697. m~~ i ~ m~
q ~
~
NNN =
=
~ ~
Ma`_^ =+++ ~ ~~ M =
rrI
M`_^~ =++ K
Figure 140.
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CHAPTER 7. ANALYTIC GEOMETRY
180
698. m~ i ~ m~
q ~
~
NNN
=
=
~ ~Ma`_^ =+++ ~ ~
rr
I
`_
~
^== K
Figure 141.
7.11 Quadric Surfaces
m ~ ~ ~W I I o~ W ^I _I `I ~I I I PON II I
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CHAPTER 7. ANALYTIC GEOMETRY
181
699. d~ n~~ b~
MaoOnOmOeOdOcO`_^ OOO =+++++++++
700. `~~ n~ p~
`~ o~EF o~EbF q p~N P Q M< p~ o~ bO P Q M> p~ f~~ bP P Q M> a e N pQ P Q M< a e O pR P P a o~ n~ `S P P p~ f~~ n~ `T O Q M< p~ b m~~U O Q M> a e m~~V O P p~ o~ b
NM O P p~ f~~ b `NN O P a e `NO O O a o~ f m~
NP O O p~ f~~ f m~NQ N P m~~ `NR N O o~ m~~ m~NS N O f~~ m~~ m~NT N N ` m~
e
=
`cd
c_e
de^
I
=
aonm
o`cdnc_e
mne^
b I ( )b= I
PON II ~ ~I
M
`cd
c_e
de^
=
K
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CHAPTER 7. ANALYTIC GEOMETRY
182
701. o~ b E`~ NF
N
~
O
O
O
O
O
O
=++
Figure 142.
702. f~~ b E`~ OF
N
~
O
O
O
O
O
O
=++
703. e N p E`~ PF
N
~
O
O
O
O
O
O
=+
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CHAPTER 7. ANALYTIC GEOMETRY
183
Figure 143.
704. e O p E`~ QF
N
~
O
O
O
O
O
O
=+
Figure 144.
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CHAPTER 7. ANALYTIC GEOMETRY
184
705. o~ n~ ` E`~ RF
M
~
O
O
O
O
O
O
=+
Figure 145.
706. f~~ n~ ` E`~ SF
M
~
O
O
O
O
O
O
=++
707. b m~~ E`~ TF
M
~
O
O
O
O
=+
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CHAPTER 7. ANALYTIC GEOMETRY
185
Figure 146.
708. e m~~ E`~ UF
M
~
O
O
O
O
=
Figure 147.
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CHAPTER 7. ANALYTIC GEOMETRY
186
709. o~ b ` E`~ VF
N
~
O
O
O
O
=+
Figure 148.
710. f~~ b ` E`~ NMF
N
~
O
O
O
O
=+
711. e ` E`~ NNFN
~
O
O
O
O
=
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CHAPTER 7. ANALYTIC GEOMETRY
187
Figure 149.
712. o~ f m~ E`~ NOF
M
~
O
O
O
O
=
713.f~~ f m~ E`~ NPF
M
~
O
O
O
O
=+
714. m~~ ` E`~ NQF
M~
O
O
=
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CHAPTER 7. ANALYTIC GEOMETRY
188
Figure 150.
715. o~ m~~ m~ E`~ NRF
N~
O
O
=
716. f~~ m~~ m~ E`~ NSF
N
~
O
O
=
717. ` m~ E`~ NTF
MO =
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CHAPTER 7. ANALYTIC GEOMETRY
189
7.12 Sphere
o~ ~ W om ~W I I I N I N I N I
` ~ W ( )II~ o~ W ^I aI bI cI j
718. b~ ~ p ` ~ l Ep~~cF
OOOO o =++
Figure 151.719. b~ ~ ` ` ~ ^ m ( )II~
( ) ( ) ( ) OOOO o~ =++
720. a~ c
( )( ) ( )( ) ( )( ) M ONONON =++ I
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CHAPTER 7. ANALYTIC GEOMETRY
190
( )NNNN IIm I ( )OOOO IIm ~ ~ ~K
721. c m c
M
N
N
N
N
N
QQQ
O
Q
O
Q
O
Q
PPPOP
OP
OP
OOOOO
OO
OO
NNNON
ON
ON
OOO
=
++
++
++
++
++
722. d~ c
Mjcba^^^ OOO =++++++ E^ FKq ~ ~ ( )II~ I
^O
a~ = I
^O
b = I
^O
c = K
q ~
^O
j^Qcbao
OOOO ++= K
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191
Chapter 8
Differential Calculus
cW I I I I
^ E ~~FW o~ W ~I I I k~~ W ^W f W N
8.1 Functions and Their Graphs
723. b c( ) ( ) =
724. l c( ) ( ) =
725. m c( ) ( )q =+
726. f c
( )= ~ I ( )= ( ) N= K
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CHAPTER 8. DIFFERENTIAL CALCULUS
192
Figure 152.
727. ` c( )= I ( ) = I ( )( )= ~ K
728. i~ c~ += I o I = ~~ I
-K
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CHAPTER 8. DIFFERENTIAL CALCULUS
193
Figure 153.
729. n~~ cO= I o K
Figure 154.
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CHAPTER 8. DIFFERENTIAL CALCULUS
194
730. ~ O ++= I o K
Figure 155.
731. ` cP= I o K
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CHAPTER 8. DIFFERENTIAL CALCULUS
195
Figure 156.
732. ~ OP +++= I o K
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CHAPTER 8. DIFFERENTIAL CALCULUS
196
Figure 157.
733. m c= I k K
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CHAPTER 8. DIFFERENTIAL CALCULUS
197
Figure 158.
Figure 159.
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CHAPTER 8. DIFFERENTIAL CALCULUS
198
734. p~ o c
= I [ ) IM K
Figure 160.
735. b~ c~= I M~> I N~ I= ~= I KSTNUOUNUOUQKO =
Figure 161.
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CHAPTER 8. DIFFERENTIAL CALCULUS
199
736. i~ c ~= I ( ) IM I M~> I N~ I
= ~= I M> K
Figure 162.
737. e p c
= IO
= I o K
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CHAPTER 8. DIFFERENTIAL CALCULUS
200
Figure 163.
738. e ` c= I
O
+= I o K
Figure 164.
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CHAPTER 8. DIFFERENTIAL CALCULUS
201
739. e q~ c
~= I
~
+
=== I o K
Figure 165.
740. e `~ c
= I
+
=== I o I M K
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CHAPTER 8. DIFFERENTIAL CALCULUS
202
Figure 166.
741. e p~ c
= I
O
N
+=== I o K
Figure 167.
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CHAPTER 8. DIFFERENTIAL CALCULUS
203
742. e `~ c
= I
O
N
=== I o I M K
Figure 168.
743. f e p c~= I o K
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CHAPTER 8. DIFFERENTIAL CALCULUS
204
Figure 169.
744. f e ` c~= I [ ) IN K
Figure 170.
745. f e q~ c~~= I ( )NIN K
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CHAPTER 8. DIFFERENTIAL CALCULUS
205
Figure 171.
746. f e `~ c~= I ( ) ( ) INNI K
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CHAPTER 8. DIFFERENTIAL CALCULUS
206
Figure 172.
747. f e p~ c~= I ( ]NIM K
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CHAPTER 8. DIFFERENTIAL CALCULUS
207
Figure 173.
748. f e `~ c
~= I o I M K
Figure 174.
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CHAPTER 8. DIFFERENTIAL CALCULUS
208
8.2 Limits of Functions
cW ( ) I ( ) ^W o~ ~W ~I
749. ( ) ( )[ ] ( ) ( )~~~
+=+
750. ( ) ( )[ ] ( ) ( )~~~
=
751. ( ) ( )[ ] ( ) ( )~~~
=
752.( )
( )
( )
( )
~
~
~
= I ( ) M
~
K
753. ( )[ ] ( )~~
=
754. ( )( ) ( )~~
=
755.( ) ( )~~
= I ( ) ~ ~
= K
756. N
M
=
757. N~
M
=
758. N
N
M=
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CHAPTER 8. DIFFERENTIAL CALCULUS
209
759. N~
N
M=
760.( )
NN
M
=+
761.
NN
=
+
762.
N =
+
763. N~ M
=
8.3 Definition and Properties of the Derivative
cW I I I I f ~~W o~ ~W ^W
764. ( ) ( ) ( )
MM=
=
+=
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CHAPTER 8. DIFFERENTIAL CALCULUS
210
Figure 175.
765. = ~
766.( )
+=
+
767.( )
=
768.( )
=
769. m o( )
+=
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CHAPTER 8. DIFFERENTIAL CALCULUS
211
770. n o
O
=
771. `~ o( )( )= I ( ) = I
= K
772. a~ f c
N
= I
( ) ( ) K
773. o~ o
O
N
=
774. i~ a~( )= I ( ) = I
( ) ( )[ ]
= K
8.4 Table of Derivatives
f ~~W o~ ~W `I ~I I k~~ W
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CHAPTER 8. DIFFERENTIAL CALCULUS
212
775. ( ) M`
=
776. ( ) N
=
777. ( ) ~~
=+
778. ( ) ~~ O +=++
779. ( ) N
=
780. ( ) N
+
=
781.O
N
N
=
782. ( )O
N
=
783. ( ) N
N
=
784. ( ) N
=
785. ( )~
N
~ = I M~> I N~ K
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CHAPTER 8. DIFFERENTIAL CALCULUS
213
786. ( ) ~~~
= I M~> I N~ K
787. ( )
=
788. ( )
=
789. ( ) =
790. ( )
N~
OO
==
791. ( )
N
OO
==
792. ( ) ~
=
793. ( )
=
794. ( )ON
N~
=
795. ( )ON
N~
=
796. ( ) ONN~~
+=
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CHAPTER 8. DIFFERENTIAL CALCULUS
214
797. ( )ON
N~
+=
798. ( )N
N~
O
=
799. ( )N
N~
O
=
800. ( )
=
801. ( )
=
802. ( ) N~ OO ==
803. ( )
N
OO
==
804. ( ) ~
=
805. ( )
=
806. ( )N
N~
O +
=
807. ( )N
N~
O
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
215
808. ( )ON
N~~
= I N< K
809. ( )N
N~
O
= I N> K
810. ( )
N +=
8.5 Higher Order Derivatives
cW I I I f ~~W k~~ W
811. p ~
( )O
O
=
=
==
812. e-l ~
( ) ( ) ( )( ) === N
813. ( )( ) ( ) ( ) +=+
814. ( )( ) ( ) ( ) =
815. i c~
( ) O ++=
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CHAPTER 8. DIFFERENTIAL CALCULUS
216
( ) PP +++=
( )( ) ( ) ( ) ( ) ( ) ( )ON ON N ++++= K
816. ( )( )( )
>
>
=
817. ( )( ) > =
818. ( )( ) ( ) ( )
~
>NN
N
~
=
819. ( )( ) ( ) ( )
N >NN
=
820. ( )( )
~~~
=
821. ( )( ) =
822. ( )( ) ~~~ =
823. ( )( ) += O
824. ( )( )
+=
O
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CHAPTER 8. DIFFERENTIAL CALCULUS
217
8.6 Applications of Derivative
cW I I m ~ W sW ^~W f ~~W qW k~~ W
825. s ~ ^~( ) = ~ ~ ~
~ ~ ~ I( ) == ~~ I
( ) === ~~ ~~
K
826. q~ i( )( )MMM =
-
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CHAPTER 8. DIFFERENTIAL CALCULUS
218
Figure 176.
827. k~ i
( )( )M
M
M
N
= Ec NTSF
828. f~ ~ a~ cK
f ( ) M M > I EF ~ ~ MK Ec NTTI N< IO< FI
f ( ) M M
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CHAPTER 8. DIFFERENTIAL CALCULUS
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Figure 177.
829. i~ ~^ EF ~ ~ ~ ~~ N ~
~ ~ N ~( ) ( ) N ~ ~ EcKNTTFK
^ EF ~ ~ ~ ~ O ~
~ ~ O ~
( ) ( ) O ~ ~ EcKNTTFK
830. `~ m^ ~ EF ~ M ~
( )M ~ K
831. c a~ q i~ b~Kf EF ~ E ( ) M > F ~ ~( ]NI~ ~ EF ~ E ( ) M
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220
832. f EF ~ E ( ) M F ~ ~ [ )IO I EF ~ ~ ~ ~ O KEcKNTTFK
833. p a~ q i~ b~Kf ( ) M N = ~ ( ) M N I EF ~ ~ ~
~ O K EcKNTTF
834. `~KEF ~ ~ ~ M ~ ( ) ~ ~ M EcKNTTI P< FKEF ~ ~ ~ M ~ ( ) ~ ~ MK EcKNTTI P< FK
835. p a~ q `~Kf ( ) M M > I EF ~ ~ ~ M Kf ( ) M M
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8.7 Differential
cW I I f ~~W a~ ~ W ( ) I ( ) o~ ~W `a~ ( )= W a~ W p~ ~ W p~ ~ W
838. =
839. ( ) ( ) ( ) +=+
Figure 178.
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CHAPTER 8. DIFFERENTIAL CALCULUS
222
840. p~ `~ ( ) ( ) +=
841. ( ) +=+
842. ( ) =
843. ( ) `` =
844. ( ) +=
845.O
=
8.8 Multivariable Functions
c ~~W ( )I I ( )I I ( )I I ( )I^W I I p~ ~ I I I W I I K
846. c l m~~ a~q ~~ ~
=
E~
=
FI
q ~~ ~
=
E~
=
FK
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CHAPTER 8. DIFFERENTIAL CALCULUS
223
847. p l m~~ a~
O
O
=
=
I
O
O
=
=
I
O
=
=
I
O
=
=
K
f ~ ~ I
OO
=
K
848. `~ o
f ( ) ( )( )II = E ~ ~~ FI ( )( )
I
=
I ( )( )
I
=
K
f ( ) ( ) ( )( )I = I ( )
+
= K
f ( ) ( )( )III= I
+
=
I
+
=
K
849. p~ `~
+
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CHAPTER 8. DIFFERENTIAL CALCULUS
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850. i~ j~~ ~ j~( )I ~ ~ ~ ~~ ( )MM I ( ) ( )MM II
~ ( )I ( )MM I K
( )I ~ ~ ~ ~ ( )MM I ( ) ( )MM II ~ ( )I ( )MM I K
851. p~~ m
M
=
=
K
i~ ~~ ~ ~ ~ ~ ~~ K
852. p~ m^ ~~ ~ ~ ~ ~ ~
853. p a~ q p~~ m
i ( )MM I ~ ~~ E M
=
=
FK
( ) ( )
( ) ( )MMMM
MMMM
II
IIa = K
f Ma > I ( ) MI MM > I ( )MM I ~ ~ ~Kf Ma > I ( ) MI MM < I ( )MM I ~ ~ ~~Kf Ma < I ( )MM I ~ ~ Kf Ma = I ~K
854. q~ m~q ~ ~ ~ ~ ( )I= ~ ( )
MMM
II
( )( ) ( )( )MMMMMMM II += K
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225
855. k~ p~q ~ ~ ~ ( )I= ~
( )MMM II
( ) ( ) N
I
I
M
MM
M
MM
M
=
=
K
8.9 Differential Operators
r ~ ~ ~W r
I r
I r
p~~ E~~ FW ( )II I ( )ON III K d~ ~ ~~ W ~ I
a~ ~W
s E FW ( )oInImcr
a ~ W cr
I cr
` ~ W cr
I cr
i~~~ ~W O
856. d~ ~ p~~ c
==
I
I
~ I
==ON
II
I
~ K K
857. a~ a~
+
+
=
I
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CHAPTER 8. DIFFERENTIAL CALCULUS
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( ) IIr
I N OOO =++ K
858. a ~ s c
o
n
mcc
+
+
== rr
859. ` ~ s c
onm
cc
==
rrr
rr
m
n
o
m
n
o rrr
+
+
=
860. i~~~ l~
O
O
O
O
O
OO
+
+
=
861. ( ) ( ) Mcc = rr
862. ( ) ( ) M~ =
863. ( ) ( ) ~ O==
864. ( ) ( ) ( ) cccc~c OO rrrrr
==
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227
Chapter 9
Integral Calculus
cW I I I
f ~~W I I f ~ ~ W ( ) I ( ) I a~ ~ W ( ) I ( ) I ( )c I o~ ~W `I ~I I I I k~~ W I I I
9.1 Indefinite Integral
865. ( ) ( ) `c += ( ) ( )c = K
866. ( )( ) ( ) =
867. ( ) ( ) =
868. ( ) ( )[ ] ( ) ( ) +=+
869. ( ) ( )[ ] (