Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

398 Answers
Form a table of values:
Constructing the points (x, y) obtained, we get the desired curve (see Appendix
VI, Fig. 7). (Here, the parameter t cannot be laid off geometrically!)
142. See Appendix VI, Fig. 19. 143. See Appendix VI, Fig. 27. 144. See
Appendix VI, Fig. 29. 145. See Appendix VI, Fig. 22 150. See Appendix VI,
Fig. 28. 151. Hint. Solving the equation for y, we get y= 1^25 x 2 . It is
now easy to construct the desired curve from the points. 153. See Appendix
VI, Fig. 21. 156. See Appendix VI, Fig. 27. It is sufficient to construct
the points (x, y) corresponding to the abscissas x = 0,
-^
, a. 157. Hint.
we get the
Solving the equation for x, we have * 10 \ogy y ( * Whence ]
.
points (x, y) of the sought-for curve, assigning to the ordinate y arbitrary
values (*/>0) and calculating the abscissa x from the formula ( * } Bear in
mind that log y -* oo as y -+ 0. 159. Hint. Passing to polar coordinates
r YX* -f- y* and tancp=~ ,
we will have r = e? (see Appendix VI, Fig 32)
160. Hint. Passing to polar coordinates x = rcosq, and // = rsincp, we will
haver= 3sin cp cos cp
8
3
cos >
(
8
q> + sin 3
(p
162.
d) x = 0.4
b) *,=_;
Appendix KK VI, Fig. 32) 161. F = ft / 32 + l, 8C
i
= 0.6* (10 *); =15 when x = 163. ^ =
r=" 1M. .),,-'
:2.9, y,
_1^ '
2
e) x=1.50; f) x = 0.86.
^=-2; * 2 = -2, = -
2
.1 f / 2
=i= 2.5; d) ^^=
166. n > -~
V e
165. a) *! = 2, ^ = 5; * = 2 5, = f/ 2 2;
; ^ = 2,^ = 3;^ = 3, = y4 2; c) x,=--2,
3.6, y,25s 3.1; A-
2 ^=2.7, yz ^ 2 9;
e) Xl =., yi
Sit
**= ^ T =
a) n ^ 4; b) /i > 10; c) n ^ 32. 167. /i >
1=JV. a) A^=9; b) W = 99; c) A^ = 168. 6 = -
b) 0002; c) 0.0002. 169. a) logAr< N when 0N when \x\>X(N). 170. a) 0; b) 1; c) 2; d) ~ .
oU
171. I. 172. 1. 173. ~~. 174. 1. 175. 3. 176. 1. 177. J .
Use the formula ! 2 + 2 2
. . . -f +i 2 =
-g
( + l) (2/i+ 1).
178. j.
Hint.
179. 0. 180.0. 181. 1.
182. 0. 183. oo. 184. 0. 185. 72. 186. 2. 187. 2. 188. oo. 189. 0. 190. 1. 191. 0.
192. oo. 193. 2. 194. oo. 195. ~ 196.
3a a
197. 3* 2 . 198. -1. 199. ^-