Задача 71676 Найти производную второго порядка...
Условие
Найти производную второго порядка функции, заданную параметрически:
x = 2tcost
y = 2tsint
Решение
{ y = 2t*sin t
Производная 1 порядка:
{ dx/dt = 2cos t + 2t(-sin t) = 2cos t - 2t*sin t
{ dy/dt = 2sin t + 2t*cos t
[m]\frac{dy}{dx} = \frac{dy}{dt} : \frac{dx}{dt} = \frac{2sin(t) + 2t \cdot cos(t)}{2cos(t) - 2t \cdot sin (t)} = \frac{sin(t) + t \cdot cos(t)}{cos(t) - t \cdot sin (t)}[/m]
Производная 2 порядка:
[m]\frac{d^2y}{dx^2} = (\frac{d^2y}{dt^2} \cdot \frac{dx}{dt} - \frac{d^2x}{dt^2} \cdot \frac{dy}{dt}) : (\frac{dx}{dt})^3[/m]
[m]\frac{d^2x}{dt^2} = -2sin(t) - 2sin(t) - 2t \cdot cos(t) = -4sin(t) - 2t \cdot cos(t)[/m]
[m]\frac{d^2y}{dt^2} = 2cos(t) + 2cos(t) + 2t (-sin(t))= 4cos(t) - 2t \cdot sin(t)[/m]
[m]\frac{d^2y}{dt^2} \cdot \frac{dx}{dt} = (4cos(t) - 2t \cdot sin(t))(2cos(t) - 2t \cdot sin (t)) = [/m]
[m] = 8cos^2(t) - 4t \cdot sin(t)cos(t) - 8t \cdot sin(t)cos(t) + 4t^2 \cdot sin^2(t) =[/m]
[m]= 4(2cos^2(t) - 3t \cdot sin(t)cos(t) + t^2 \cdot sin^2(t))[/m]
[m]\frac{d^2x}{dt^2} \cdot \frac{dy}{dt} = (-4sin(t) - 2t \cdot cos(t))(2sin(t) + 2t \cdot cos(t)) =[/m]
[m]= -8sin^2(t) - 4t \cdot sin(t)cos(t) - 8t \cdot sin(t)cos(t) - 4t^2 \cdot cos^2(t) = [/m]
[m]=-4(2sin^2(t) + 3t \cdot sin(t)cos(t) + t^2 \cdot cos^2(t))[/m]
[m](\frac{dx}{dt})^3 = (2cos(t) - 2t \cdot sin (t))^3 = 8(cos(t) - t \cdot sin (t))^3[/m]
Осталось вычесть и разделить.
[m]\frac{d^2y}{dx^2} = \frac{4(2cos^2(t) - 3t \cdot sin(t)cos(t) + t^2 \cdot sin^2(t)) + 4(2sin^2(t) + 3t \cdot sin(t)cos(t) + t^2 \cdot cos^2(t))}{8(cos(t) - t \cdot sin (t))^3} =[/m]
[m]= \frac{2cos^2(t) - 3t \cdot sin(t)cos(t) + t^2 \cdot sin^2(t) + 2sin^2(t) + 3t \cdot sin(t)cos(t) + t^2 \cdot cos^2(t)}{2(cos(t) - t \cdot sin (t))^3} =[/m]
[m]= \frac{(2cos^2(t)+ 2sin^2(t)) + (t^2 \cdot sin^2(t) + t^2 \cdot cos^2(t))}{2(cos(t) - t \cdot sin (t))^3} =\frac{2 + t^2}{2(cos(t) - t \cdot sin (t))^3}[/m]
Всё!
Все решения
[m]y`_{t}=(2t sint)`=2\cdot (t)`\cdot sint+2t\cdot (sin)`=2sint+2tcost[/m]
[m]y`_{x}=\frac{2sint+2tcost}{2cost-2tsint}[/m]
[m]y`_{x}=\frac{sint+tcost}{cost-tsint}[/m]
[m]x``_{t^2}=x``_{t}=(x`_{t})`_{t}=(2cost-2t\cdot sint)`=-2sint-2sint-2t\cdot cost=-4sint-2t\cdot cost[/m]
[m]y``_{t^2}=(y`_{t})`_{t}=(2sint+2t\cdot cost)`=2cost+2cost-2t\cdot sint=4\cdot cost-2t\cdot sint[/m]
[m]y``_{x^2}=\frac{(4\cdot cost-2t\cdot sint)\cdot (2cost-2tsint)-(-4\cdot sint-2t\cdot cost)\cdot(2sint+2t\cdot cost)}{(2cost-2t\cdot sint)^3}[/m]
[m]y``_{x^2}=\frac{8cos^2t-4tsintcost-8tsintcost+4t^2sin^2t+8sin^2t+4tsintcost+8tsintcost+4t^2sin^2t}{2^3(cost-t\cdot sint)^3}[/m]
[m]y``_{x^2}=\frac{8cos^2t+4t^2sin^2t+8sin^2t+4t^2sin^2t}{2^3(cost-t\cdot sint)^3}[/m]
[m]y``_{x^2}=\frac{(8cos^2t+8sin^2t)+(4t^2sin^2t+4t^2sin^2t)}{8(cost-tsint)^3}[/m]
[m]y``_{x^2}=\frac{8(cos^2t+sin^2t)+4t^2(sin^2t+sin^2t)}{8(cost-t\cdot sint)^3}[/m]
[m]y``_{x^2}=\frac{8+4t^2}{8(cost-t\cdot sint)^3}[/m]
