Скачать .docx Скачать .pdf

Реферат: 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/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 )