Главная Учебники - Разные Лекции (разные) - часть 50
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Задача 19
. Найти производную второго порядка 19.1. x'= -2sin2t= -4sintcost y'= 4sint/cos3
t y''xx
= 4sint
= -1 _
16sin2
tcos5
t 4sintcos5
t 19.2. x'= -t/√(1-t2
) y'= -1/t2
y''xx
= (1-t2
)2
t4
19.3. x'= et
cost-et
sint= et
(cost-sint) y'= et
sint+et
cost= et
(sint+cost) y''xx
= et
(sint+cost)
= sint+cost
e2t
(cost-sint)2
et
(cost-sint)2
19.4. x'= 2shtcht y'= -2sht/ch3
t y''xx
= -2
sht
= -1_
4shtch4
t 2ch4
t 19.5. x'= 1+cost y'= sint y''xx
= sint/(1+cost)2
19.6. x'= -1/t2
y'= -2t/(1+t2
)2
y''xx
= -2t3
_
(1+t2
)2
19.7. x'= 1/2√t y'= 1/√(1-t)3
y''xx
= 4
t _
√(1-t)3
19.8. x'= cost y'= sint/cos2
t y''xx
= sint/cos4
t 19.9. x'= 1/cos2
t y'= -2cos2t/sin2
2t y''xx
= -2cos2tcos4
t
sin2
2t 19.10. x'= 1/2√(t-1) y'= (2-t)/(1-t)3/2
y''xx
= 4(
t
-1)(2-
t
)
= 2
t
-8
(1-t)3/2
√(1-t) 19.11. x'= 1/2√t y'= 1/3
√(t-1)2
y''xx
= 4t/3
√(t-1)2
19.12. x'= -sint/(1+2cost)2
y'= (cost+2)/(1+2cost)2
y''xx
= (
cost+2)(1+2cost)4
= (
cost+2)(1+2cost)2
sin2
t(1+2cost)2
sin2
t 19.13. x'= 3t2
/ 2√(t3
-1) y'= 1/t y''xx
= 2
(
t
3
-1)
3t5
19.14. x'= cht y'= 2tht/ch2
t y''xx
= 2tht/ch4
t 19.15. x'= 1/2√(t-1) y'= -1/2√t3
y''xx
= -
2t
+
2
√t3
19.16. x'= -2cost sint y'= 2sint/cos3
t y''xx
= 2sint
= 1/2cos4
t 4cos4
tsint 19.17. x'= 1/2√(t-3) y'= 1/(t-2) y''xx
= 4(t-3)/(t-2) 19.18. x'= cost y'= -sint/cost y''xx
= -sint/cos3
t 19.19. x'= 1+cost y'= -sint y''xx
= -sint/(1+cost)2
19.20. x'= 1-cost y'= sint y''xx
= sint/(1-cost)2
19.21. x'= -sint y'= cost/sint y''xx
= -cost/sin3
t 19.22. x'= -sint+sint+tcost= tcost y'= cost-cost+tsint= tsint y''xx
= sint/cos2
t 19.23. x'= et
y'= 1/√(1-t2
) y''xx
= et
/√(1-t2
) 19.24. x'= -sint y'= 2sin3(t/2)cos(t/2) y''xx
= -2sin3
(t/2)cos(t/2)/sint= -sin2
(t/2) 19.25. x'= sht y'= 2cht/33
√sht y''xx
= 2cht/33
√sh4
t 19.26. x'= 1/(1+t2
) y'= t y''xx
= t(1+t2
)2
19.27. x'= 2-2cost y'= -4sint y''xx
= -2sint/(1-cost) 19.28. x'= cost-cost+tsint= tsint y'= -sint+sint+tcost= tcost y''xx
= cost/sin2
t 19.29. x'= -2/t3
y'= -2t/(t2
+1)2
y''xx
= -t7
/2(t2
+1)2
19.30. x'= cost-sint y'= 2cos2t y''xx
= 2cos2t/( cost-sint)= 2cost+2sint 19.31. x'= 1/t y'= 1/(1+t2
) y''xx
= t2
/(1+t2
)
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