Решения к Сборнику заданий по высшей математике Кузнецова Л.А. - 2. Дифференцирование. Зад.19
Задача 19. Найти производную второго порядка
от функции, заданной параметрически.19.1.
x'= -2sin2t= -4sintcost
y'= 4sint/cos3t
y''xx= 4sint = -1 _
16sin2tcos5t 4sintcos5t
19.2.
x'= -t/√(1-t2)
y'= -1/t2
y''xx= (1-t2)2 t4
19.3.
x'= etcost-etsint= et(cost-sint)
y'= etsint+etcost= et(sint+cost)
y''xx= et(sint+cost) = sint+cost
e2t(cost-sint)2 et(cost-sint)2
19.4.
x'= 2shtcht
y'= -2sht/ch3t
y''xx= -2sht = -1_
4shtch4t 2ch4t
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= 4t _
√(1-t)3
19.8.
x'= cost
y'= sint/cos2t
y''xx= sint/cos4t
19.9.
x'= 1/cos2t
y'= -2cos2t/sin22t
y''xx= -2cos2tcos4t
sin22t
19.10.
x'= 1/2√(t-1)
y'= (2-t)/(1-t)3/2
y''xx= 4(t-1)(2-t) = 2t-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
sin2t(1+2cost)2 sin2t
19.13.
x'= 3t2 / 2√(t3-1)
y'= 1/t
y''xx= 2(t3-1)
3t5
19.14.
x'= cht
y'= 2tht/ch2t
y''xx= 2tht/ch4t
19.15.
x'= 1/2√(t-1)
y'= -1/2√t3
y''xx= -2t+2
√t3
19.16.
x'= -2cost sint
y'= 2sint/cos3t
y''xx= 2sint = 1/2cos4t
4cos4tsint
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/cos3t
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/sin3t
19.22.
x'= -sint+sint+tcost= tcost
y'= cost-cost+tsint= tsint
y''xx= sint/cos2t
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√sh4t
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/sin2t
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)