1 3 1.1.......................... 3 1............................... 3 1.3....................... 5 1.4.......................... 6 1.5........................ 7 8.1......................... 8.............................. 9.3.............................. 11.4.................................. 1.5............................... 14.6....................... 14.7..................... 15.8.................. 16.9....................... 17 3 18 3.1......................... 18 3........................ 19 3.3......................... 0 4 4.1............................. 4.............................. 4.3................................. 4.4.................................. 1

5 4 5.1.................................. 4 5.................................. 4 5.3.......................... 5

1 1.1 F, F d F P + F P = d P F, F P F F 1: 1. α, 0), α, 0) α > 0, ) + α +, α + d + α) + + α + = d 3

0 < α < d + α) + + α + = d d = + α) + + α + d > 0 d = + α + + + α + α + + α + d = + α + + + α + ) α + ) d + α + ) = + α + ) α + ) d + α + ) = 4 + α + ) α + ) + α + ) d d + α + ) = 4 + α + + α ) + α + α ) + α + ) d d + α + ) = 4 + α + 4α ) + α + ) d d 4 4d + α + ) + 4 + α + = 4 + α + 16α + α + ) d d 4 4d + α + ) = 16α + α + ) d 4d 16α ) + 4d = d 4 4d α + α + ) d 4 d + 4 d + 4 d 4α = 1 + α + ) d 4 d 4α = 1 4 d 4 4 d + d 4α = 1 d 4 4 d 4α 4 4 d + d 4α = 1 d 4α 4 ) d + α + ) 4 + α + d 4α = d 4 + α) + + α + = d 4 d + 4 d 4 d 4α = 1 + = 1 d 4α

d =, d 4α = + = 1 1) 1) + = 1 { > ±, 0) < 0, ± ), 1.3 1) { = X = Y ) X + Y = 1 3) 1) 3) 1 1 1 1 : 5

, ) X 1 1, ) + = 1 ), 0), 0) 0, ) 0, ) 1. + 9 4 = 1-3 3-3: 9 + 4 = 1 1.4 X + Y = 1 { X = cos θ Y = sin θ 0 θ < π) ) + = 1 { = cos θ = sin θ 0 θ < π) 1. θ, ) 6

1.5 + = 1 α β, ) α β α, β) α β + = 1 + α β + α + β = 0 A + C + D + E + F = 0 A, C > 0 4) 4) A + A) D + C + E = D C 4A + E 4C F, A + A) D + C + E 0 C = D, = E A C K = D + E F K < 0 0 4A 4C K = 0 = D, = E K > 0 A C K + D A K A + + E C K C. 9 + 4 + 18 16 11 = 0 = 1 7

9 + 4 + 18 16 11 = 0 9 + 1 + 4 = 36 + 1 + = 1 3 3-1 -3 4: +1 + 3 = 1.1 F, F d F P F P = d P F, F 8

P F F 5:. α, 0), α, 0) α > 0, ) + α +, α + d + α + α + = d 9

0 < d < α + α + α + = d d = + α) + α + d > 0 d = + α + + α + α + + α + d = + α + + α + ) α + ) d + α + ) = + α + ) α + ) d + α + ) = 4 + α + ) α + ) + α + ) d d + α + ) = 4 + α + + α ) + α + α ) + α + ) d d + α + ) = 4 + α + 4α ) + α + ) d d 4 4d + α + ) + 4 + α + = 4 + α + 16α + α + ) d d 4 4d + α + ) = 16α + α + ) d 4d 16α ) + 4d = d 4 4d α + α + ) d 4 d 4 d 4 4α d = 1 + α + ) d 4 4α d = 1 4 d 4 4 d 4α d = 1 d 4 α d 4 ) ) d + α + d ) 4 + α 4 + d = d 4 + α + 4 4 α + = d d 4α d = 1 = 1 d =, 4α d = d 4α d = 1 5) 10

= 1 ± +, 0),.3 5) = 1 ) = 1 = ± 1 + f) = 1 + ) f f f ) = 1 + f ) = 1 + ) 1 ) 3 > 0 f < 0 > 0 f 6: f) = 1 + ) 11

.4 f) = 1 + ) 1 1 + f) = 1 + ) = 1 + ) ) 1 + = 1 + ) ) + 1 + ) + = 1 1 + + 0 ± ) f = f) = = = ± = f) = = f) = = f) 7: f) = 1 + = ± = 1 = ±f) = f) = = = ± = 1 1

8: ) = 1 = ± q 1 + ` ` ` = 1 = 1, > 0 = 1 + 13

= 1 ) + ) = 1 = 1 + 0 = 0 ) = 1.5 ) ) = 1 ± = 0 1 0 = 1 1 ) ± 1 ) 1 1 = 1 1 =, > 0 = 1 = = ±.6 θ ) ) 1 c d 0 ) ) cos θ 0 θ sin θ 1 14

cos )) ) θ + π θ sin sin θ ) θ + π = cos θ ) ) ) ) 1 cos θ = = c c d 0 sin θ ) ) ) ) 0 sin θ = = d c d 1 cos θ c ) cos θ = d sin θ ) sin θ cos θ, ) θ, ) ) ) ) cos θ sin θ = sin θ cos θ F, ) = 0 θ, ) θ cosθ) sinθ), sinθ)+ cosθ)) = cos θ + sin θ, sin θ + cos θ) F, ) = 0 θ F cos θ + sin θ, sin θ + cos θ) = 0.7 = c c > 0 = c θ cos θ + sin θ) sin θ + cos θ) = c cos θ sin θ) + cos θ sin θ) + cos θ sin θ) = c sin θ + cos θ) + sin θ = c 6) cos θ = 0 θ = π sin θ = 4 sin π = 1 6) ) = c = 1 c c 15

.5 = c c c = 1 π 4 9: = c c c = 1.8 { cosh t = e t +e t sinh t = et e t cosh, sinh cosh t sinh t = 1 { = cosh t = sinh t 16

= 1 = 1 = 1 { = cosh t = sinh t.9 = 1 α β, ) α β α, β) α β = 1 α + β + α β = 0 A + C + D + E + F = 0 A > 0 > C 7) 4) A + A) D + C + E = D C 4A + E 4C F K = D + E F K = 0 4A 4C = D, = E A C K > 0 K + D A K A + E C K C = 1 17

K < 0 K + D A K A + + E C K C 3. 4 = 10 = 1 4 = 10 + 1 = 8 + 1 + = 1-1 10: +1 = 1 3 3.1 l F P l 18

P F P l F l P _ F 11: 3. = p 0, p), ) = p + p, ) 0, p) + p + p = + p + p = + p + p p = 4p = = 1 4p 19

3.3 ) 0 m mg ) m d dt Newton ) 0 = mg m d dt = 0 m d dt = mg m d dt = 0 = 0 + v 1 t d dt = g = 0 + v t 1 gt 0, 0 ) t = 0 ) 0, 0 ) v 1 v 0, 0, v 1, v v 1 v ) t = 0 t 0, ) = 0 + v 1 t, 0 + v t 1 gt ) { { } = 0 + v 1 t, ) t 0 = 0 + v t 1 gt v 1 < 0 = 0 + v 1 t = 0 + v 1g t v g t = 0 v 1 = 0 + v 1g t v g = 0 + v g 1 g g g 0 v v 1 g 0 t 0 t 0, 0

0 + v 1v g, 0 + v g ) 0 v 1 = 0 = 0 = 0 + v t 1 gt = 0 + v 1g t 0 t v g g = 0 = 0 + v 1g t v g g t 0 = 0 + v g 1 g t v g t 0 v 0 0 + v g v < 0 0 v 0 0, 0 + v ) g v 0 0, 0 ) v 1 > 0 = 0 + v 1 t = 0 + v 1g t v g t = 0 v 1 = 0 + v 1g t v g = 0 + v g 1 g g g 0 v v 1 g t 0 t 0, 0 0 + v 1v g, 0 + v g ) 0 prol) 000 1

4 4.1, ) = 0 A + B + C + D + E + F = 0 A, B, C) 0, 0, 0) 8) 4., 8) 4.3 4.4 1 : 1 1 : e

α, 0) α > 0), ), ) α + e = α + e = α + e 1) + α α = 0 { + α α = 0 e = 1 ) e 1) ) + α e 1 e α = 0 e 1 ) e 1 + = 1 0 < e < 1 ) + α e 1 eα 1e = 1 α + α + α e 1 eα e 1 eα 1e eα e 1 e = 1 ) = 1 e > 1 ) 0 < e < 1 e = 1 e > 1 e e 10 e e 1 e 0.5 e 0.1 1: e 1) + α α = 0 0 < e < 1 ) + α e 1 eα + 1e eα 1e = 1 3

eα 1e + eα 1e = 1 + = 1 e e = { = eα 1e = eα 1e { = eα 1e = 1 e 1. = 1 e, 3. 5 5.1 P OP r OP θ P r, θ) r, θ) r, θ) r, θ), ) { = r cos θ = r sin θ 5. rθ rθ, = f) f r = fθ) f 4

4. r = θ 0 θ π) Π 13: 5.3 = α α > 0) e, ) + α = r cos θ + α, ) r 1 : e r = e r cos θ + α r = er cos θ + α) r = eα 1 e cos θ Newton Newton Kepler 5