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Консультация № 161394
25.02.2009, 18:29
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Найти производную второго подрядка y'' от функции заданной параметрически. {x=t+sint, y=2-cost
Обсуждение
Неизвестный
25.02.2009, 18:48
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это ответ
Здравствуйте, Истомина Елена Андреевна!
Сначала найдём частные производные по t первого и второго порядков .
y'=sint ; y"=cost ; x'=1+cost ; x"=-sint .
Для нахождения производной от функции заданой параметрически существует специальная формулла :
(d^2)y/d(x^2)=(x'*y"-y'*x")/((x')^3) . Подставим в неё найденные частные производные и получим ответ .
(d^2)y/d(x^2)=(сost+((cost)^2)+((sint)^2))/((1+cost)^3)=(cost+1)/((1+cost)^3)=1/((1+cost)^2)=(d^2)y/d(x^2) .
cost=2-y - если заменить это на конечное выражение - то получим вторую производную выраженую через у , аналогично можно поступить из х , но обычно оставляют значения искомой производной в параметрическом виде .
Сначала найдём частные производные по t первого и второго порядков .
y'=sint ; y"=cost ; x'=1+cost ; x"=-sint .
Для нахождения производной от функции заданой параметрически существует специальная формулла :
(d^2)y/d(x^2)=(x'*y"-y'*x")/((x')^3) . Подставим в неё найденные частные производные и получим ответ .
(d^2)y/d(x^2)=(сost+((cost)^2)+((sint)^2))/((1+cost)^3)=(cost+1)/((1+cost)^3)=1/((1+cost)^2)=(d^2)y/d(x^2) .
cost=2-y - если заменить это на конечное выражение - то получим вторую производную выраженую через у , аналогично можно поступить из х , но обычно оставляют значения искомой производной в параметрическом виде .
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