Theme 3 PROPERTIES AND SOCIAL-ECONOMICAL RELATIONS

1) Calculate a determinant .

2) Calculate a minor of an element of a matrix .

3) Calculate a determinant .

4) Find a matrix , if .

5) Calculate product of matrices .

6) Find an inverse matrix , if a matrix .

7) Calculate a determinant .

8) Matrices and are given. Find product of matrices .

9) Matrices and are given. Find the sum of matrices .

10) Find a matrix , if a matrix .

11) Find a matrix , if a matrix .

12) Determinant of a matrix is the number, equal to:

13) At transposing of a square matrix its determinant:

14) If elements of two lines are proportional, a determinant:

15) Solve the matrix equation .

16) Find product of matrices .

17) Find , if a matrix .

18) Determinant of the third order is equal:

19) Calculate product of matrices , if and are given.

20) If - a minor of a determinant of an element , then algebraic complement of this element is equal:

21) If in a determinant to change places of a lines, then:

22) If any line in a determinant to add with other line:

23) If in a determinant two columns are proportional, then:

24) To what value the determinant of an unitary matrix is equal?

25) Matrix is given. Find an inverse matrix .

26) Calculate a determinant .

27) Calculate a minor of a determinant .

28) Calculate a determinant .

29) Calculate algebraic complement of a determinant .

30) Find product of matrices .

31) Find an inverse matrix , if .

32) Find an inverse matrix , if .

33) Find a matrix , if a matrix .

34) Calculate product of matrices .

35) Find an inverse matrix , if .

36) Matrix is given. Find the transposed matrix .

37) Calculate a determinant .

38) Calculate a determinant .

39) Ñalculate a minor of an element of a matrix .

40) Minor is:

41) Calculate product of matrices: .

42) Matrices and are given. Find product of a matrices .

43) Matrices and . Find a difference of matrices .

44) How to add two matrixes?

45) How to subtract one matrix from another matrix?

46) Matrices and are given. Find .

47) Matrices and are given. Find .

48) Calculate .

49) Calculate a determinant .

50) Calculate a determinant .

Theme 2. Systems of the linear equations

51) Solve system of the equations . Find .

52) Solve system of the equations . Find .

53) Solve system of the equations . Find .

54) Solve system of the equations . Find .

55) If the determinant of a matrix of system of the linear equations from unknown variables is different from zero, then system of the equations :

56) Solve system of the equations by Kramer's formulas. Find .

57) Find symbolical record of Kramer’s formulas, defining the decision of system if the determinant of a matrix of system is not equal to zero.

58) System of the linear equations is called compatible, if it:

59) Solve system of the equations by Kramer's rule . Find .

60) Solve system of the equations .

61) Solve system of the equations .

62) How it is possible to find by means of Kramer's formula?

63) How it is possible to find by means of Kramer's formula?

64) How it is possible to find by means of Kramer's formula?

65) What from the specified formulas below is Kramer's formula?

66) Solve system of the equations , using Kramer's rule.

67) Solve system of the equations , using Kramer's rule.

68) Solve system of the linear equations .

69) Solve system of the linear equations .

70) Solve system of the linear equations .

Theme 3. Elements of vector algebra

71) Points and are given. Find coordinates of a vector .

72) Find length of a vector .

73) Vectors and are given. Find coordinates of a vector .

74) Find scalar product of vectors and .

75) Find an angle between vectors and .

76) At what value vectors and are collinear?

77) At what value vectors and are orthogonal?

78) , , an angle between and is equal . Find scalar product .

79) Points and are given. Find coordinates of a vector .

80) Find length of a vector .

81) Vectors and are given. Find coordinates of a vector .

82) Find scalar product of vectors and .

83) Find an angle between vectors and .

84) At what value vectors and are collinear?

85) At what value vectors and are orthogonal?

86) Points and are given. Find coordinates of a vector .

87) Find length of a vector .

88) Vectors and are given. Find coordinates of a vector .

89) Find scalar product of vectors and .

90) Find an angle between vectors and .

91) At what value vectors and are collinear?

92) At what value vectors and are orthogonal?

93) Vector, located on one straight line or parallel straight lines are called:

94) If vector have identical directions and length they are called:

95) Points and are given. Find coordinates of a vector .

96) Find length of a vector .

97) Vectors and are given. Find coordinates of a vector .

98) Find scalar product of vectors and .

99) Find an angle between vectors and .

100) At what value vectors and are collinear to each other?

101) At what value vectors and are orthogonal to each other?

102) An angle between vectors and is found by the formula:

103) Scalar product of vectors and is equal:

104) Scalar product of vectors and is equal to zero if:

105) Points and are given. Find length of a vector .

106) Find , if , , .

107) Vectors are called equipollent, if they:

108) Find scalar product of vectors , if , , .

109) Find , if , , .

110) Find , if , , .

Theme 4. The equation of a line on a plane and in space

111) Points and are given. Find coordinates of a midpoint .

112) In what point the straight line crosses an axis ?

113) Find angular coefficient of a straight line, parallel straight line .

114) Write down the equation of the straight line, which is passing through points and .

115) Find the equation of a straight line on a plane in a general view.

116) Find the equation of a straight line with angular coefficient.

117) Points and are given. Find coordinates of a midpoint .

118) In what point the straight line crosses an axis ?

119) Find angular coefficient of a straight line, parallel straight line .

120) Write down the equation of the straight line, which is passing through points and .

121) Points and are given. Find coordinates of a midpoint .

122) In what point the straight line crosses an axis ?

123) Find angular coefficient of a straight line, parallel straight line .

124) Write down the equation of the straight line which is passing through points and .

125) Find the equation of the straight line, which is passing through points and .

126) Find the equation of an axis .

127) In what point the straight line crosses an axis ?

128) Find angular coefficient of the straight line, parallel given straight line .

129) Write down the equation of the straight line, which is passing through points and .

130) Calculate distance from a point up to a straight line .

131) Find the equation of a straight line on a plane in segments.

132) Find coordinates of a midpoint, connecting points and .

133) Two straight lines and are parallel, if:

134) Write the equation of a straight line with angular coefficient and passing through a point (2;-1).

135) Write the equation of the straight line, which is passing through points and .

136) Find a point of crossing of straight lines and .

137) Find the equation of a straight line in segments for a straight line .

138) Points and are given. Find coordinates of a midpoint .

139) Find distance between points and .

140) Construct the equation of the straight line, which is passing through a point in parallel to the straight line .

141) Find a point of crossing of straight lines and .

142) Construct the equation of the straight line, which is passing through a point with angular coefficient 4.

143) Find a point of crossing of straight lines and .

144) Find the equation of the straight line, which is passing through points and .

145) Find a condition of parallelism of straight lines and .

146) Find a condition of perpendicularity of straight lines and .

147) Equation of a straight line with angular coefficient is given by:

148) Find the equation of the straight line, which is passing through points and .

149) Find the equation of the straight line, which is passing through a point , in parallel a vector .

150) If - an angle between straight lines and , what of following parameters expression defines?

151) Find distance from a point up to a straight line .

152) Find an angle between straight lines and .

153) Find angular coefficient of a straight line, perpendicular straight line .

154) Find the equation of the straight line, which is passing through a point and a parallel straight line .

155) Construct the equation of the straight line, which is passing through a point in parallel axes .

Theme 5. Function. Limits and a continuity

156) Find a limit .

157) Find a limit .

158) Find a limit .

159) Find a limit .

160) Find a limit .

161) Find a limit .

162) Find a limit .

163) Find a limit .

164) Ñalculate a limit by means of Lhopital rule.

165) Ñalculate a limit by means of Lhopital rule.

166) Ñalculate a limit by means of Lhopital rule.

167) Find a limit .

168) Find a limit .

169) Find a limit .

170) Find a limit .

171) Find the first remarkable limit.

172) Find the second remarkable limit.

173) Find a limit .

174) Ñalculate a limit by means of Lhopital rule.

175) Find a domain of function .

176) Find a domain of function .

177) Find a limit .

178) Find a limit .

179) Find a limit .

180) Find a limit .

181) Find a limit .

182) Find a limit .

183) Find a limit .

184) Ñalculate a limit by means of Lhopital rule.

185) Ñalculate a limit by means of Lhopital rule.

186) Ñalculate a limit by means of Lhopital rule.

187) Find a limit .

188) Find a limit .

189) Find a limit .

190) Find a limit .

191) Find a limit .

192) Find a limit .

193) Find a limit .

194) Find a limit .

195) Find a limit .

196) Find a limit .

197) Find a limit .

198) Find a limit .

199) Find a limit .

200) Find a limit .

Theme 6. Derivative and differential of function

201) Derivative of function is equal:

202) Find the formula of a derivative of quotient of two functions and .

203) Find the formula of derivative of exponential .

204) Find the formula of derivative of logarithmic function .

205) Find the formula of derivative of function .

206) Find a derivative of function .

207) Find a derivative of function .

208) Function is given. Ñalculate value .

209) Find a derivative of function .

210) Function is given. Calculate value .

211) Find a derivative of function .

212) Find a derivative of function .

213) Function is given. Ñalculate value .

214) Function is given. Find .

215) Function is given. Find .

216) Function is given. Find .

217) Function is given. Find .

218) Function is given. Find .

219) Find the formula of a derivative of product of two functions and .

220) Find the formula of derivative of function .

221) Function is given. Calculate value .

222) Find a derivative of function .

223) Find a derivative of function .

224) Find differential of function .

225) Find a derivative of function .

226) Function is given. Find a derivative .

227) Derivative of function is equal:

228) Derivative of function is equal:

229) Derivative of function is equal:

230) Derivative of function is equal:

231) Derivative of function is equal:

232) Derivative of function is equal:

233) Derivative of function is equal:

234) Derivative of function is equal:

235) Derivative of function is equal:

236) Find a derivative of function .

237) Find a derivative of function .

238) Find a derivative of function .

239) Find a derivative of function .

240) Find a derivative of function .

241) Find a derivative of function .

242) Find a derivative of function .

243) Find a derivative of function .

244) Find a derivative of function .

245) Find a derivative of function .

Theme 7. Investigation of function by means of a derivative

246) Find an interval of decrease of function .

247) Find an interval of increase of function .

248) Find an interval of increase of function .

249) Find an interval of convexity of function .

250) Find an interval of concavity of function .

251) Find an interval of convexity of function .

252) Find an interval of concavity of function .

253) Find a point of a maximum of function .

254) Find a point of a minimum of function .

255) Investigate function on an extremum.

256) If the derivative of differentiate function is positive inside of an interval , then function on an interval :

257) Necessary condition of an extremum of function in a point :

258) If at transition through a critical point the derivative of function changes the sign from plus on a minus, then the point is a point:

259) Find interval of increase of function .

260) Find a point of a minimum of function .

261) Find a point of a maximum of function .

262) Find interval of decrease of function .

263) Find points of inflection of function .

264) Find points of inflection of a curve .

265) Find an interval of convexity of the graph of function .

266) To what of following equality even function satisfies:

267) To what of following equality odd function satisfies:

Theme 8. Functions of several variables

268) Function is given. Find a partial derivative .

269) Function is given. Find a partial derivative .

270) Function is given. Find a partial derivative .

271) Function is given. Find a partial derivative .

272) Function is given. Find a partial derivative .

273) Function is given. Find .

274) Function is given. Find .

275) Function is given. Find .

276) Find a critical point of function .

277) Find a critical point of function .

278) Function is given. Find a partial derivative .

279) Find the formula of total differential of function .

280) Function is given. Find a partial derivative .

281) Find a critical point of function .

282) Find a critical point of function .

283) Find a domain of function .

284) Find a domain of function .

285) Function is given. Find a partial derivative .

286) Function is given. Find a partial derivative .

287) Find total differential of function .

288) Function is given. Find a partial derivative .

Theme 9. Antiderivative and indefinite integral

289) Find integral .

290) Find integral .

291) Find integral .

292) If and are differentiable functions, then is equal:

293) Find integral .

294) Find integral .

295) Function is called antiderivative of function on an interval , if function is differentiable on and following equality is executed:

296) Integral is equal:

297) Integral is equal:

298) Integral is equal:

299) Find expedient substitution for a finding of integral .

300) Find integral .

301) Find integral .

302) Formula of integration in parts for indefinite integral is given by:

303) Find integral .

304) Find integral .

305) Find integral .

306) Find integral .

307) Find integral .

308) Find integral .

309) Find integral .

310) Integral is equal:

311) Find integral .

312) Find definition of indefinite integral.

313) Integral is equal:

314) Integral is equal:

315) Integral is equal:

316) Integral is equal:

317) Integral is equal:

318) Integral is equal:

319) Integral is equal:

320) Integral is equal:

321) Find integral .

322) Find integral .

323) Find integral .

324) Find integral .

325) Find integral .

326) Find integral .

Theme 10. Definite integral and its applications

327) Calculate integral .

328) Calculate integral .

329) Find geometrical meaning of the definite integral .

330) Find the area of the figure, limited by a parabola and an axis using definite integral.

331) Calculate integral .

332) Calculate integral .

333) Calculate the area of the figure, limited by lines , , .

334) Calculate the area of the figure, limited by lines , .

335) Calculate the area of the figure, limited by lines , , .

336) Calculate improper integral or establish its divergence.

337) Calculate improper integral or establish its divergence.

338) Calculate improper integral or establish its divergence.

339) Calculate improper integral or establish its divergence.

340) Calculate improper integral or establish its divergence.

341) Let function is integrated on a segment , then function also is integrated on this segment and following equality is executed:

342) Let functions and are integrated on a segment , then function also is integrated on the given segment and following equality is executed:

343) What of following equality is correct?

344) Formula of Newton – Leibniz is given by:

345) Formula of integration in parts for definite integral is given by:

346) Calculate integral .

347) Find value of integral .

348) Calculate integral .

349) Integral is equal:

350) Calculate integral .

351) Find value of integral .

352) Calculate integral .

Theme 11. Differential equations and series

353) Define type of the differential equation in a general view .

354) Define type of the differential equation .

355) Define type of the differential equation .

356) Define type of the differential equation .

357) Define type of the differential equation .

358) Differential equation is called homogeneous, if:

359) What is order of the differential equation?

360) Linear differential equation of the first order is the equation of a kind:

Theme 3 PROPERTIES AND SOCIAL-ECONOMICAL RELATIONS

1. Property in economical and legal sense

2. Classification of the property

3. Macroeconomical structure of the property

1. Property in economical and legal sense

The property is rather difficult appearance: economical theory analyze the economical content of this appearance, and jurisprudence - legal.

In economical sense the property is a human relations on appropriation and economic usage of the material and non-material benefits. The property in legal understanding demonstrates how real property relations are arranged and fixed in precepts of law and laws which are established by the state in order for all citizens.

Property as economical system. In each economical relation of the property there are two sides: the subject (owner) and object (property). Appropriation relations are spread primarily to the such property on which economic activities directly depend. It includes production factors (both material benefits, and results of intellectual transactions).

Studying of the property allows to give an answer to three principal social and economic problems:

1. Who (what subjects of activities) has an economical power - appropriate the factors and results of production ?

2. What kind of economical relations give an ability on the best usage of the property?

3. Who will take incomes of economical activities?

So, the system of economical relations of the property includes following elements: A) appropriation of factors and results of production; B) economical usage of the material and other means; C) property income (Figure 1).

Appropriation - economical rela-tions between people which estab-lishes their relation to things as to their own. This relation consist the base of productive process. Because any material benefit production means an appro-priation by people of the material of nature and energy in a purpose of consumption of their needs.

Estrangement relation is opposite to appropriation. It has appeared, if any part of a society occupied all factors of productions, and other part remains without any livelihood. Or when the goods created by one people, appropriates by other.

Owners of factors of productions are not always engaged in creative activity. They give the chance to other persons to use their property (for example, land, house, the equipment) in the economic purposes under certain conditions. In this case between the owner and entrepreneur appeared relations of economic usage of property. Last has an opportunity to temporarily own and use object by another's property. It can be, rent - hire by one person (or by enterprise) at other person (or enterprise) the property in temporary using on certain period and for a certain price.

The property economically justifies itself if brings an income to its o