Вариант 18 (10б) Дайте определение линии уровня функции \( z=f(x, y) \). Запишите линию уровня функции \( z=3 x-2 y+1 \), проходящую через точку \( \mathrm{A}(1,2) \).

9/16/24, 1:33 PM 8. Use a change of variables to evaluate the definite integral. \( \int_{0}^{2} x \sqrt{4-x^{2}} d x \) \( \int_{0}^{2} x \sqrt{4-x^{2}} d x= \)

3. Use the value of the first integral \( I \) to evaluate the two given integrals. \( I=\int_{0}^{3}\left(x^{3}-4 x\right) d x=\frac{9}{4} \) a. \( \int_{0}^{3}\left(24 x-6 x^{3}\right) d x \) a. \( \int_{0}^{3}\left(24 x-6 x^{3}\right) d x=\frac{-27}{2} \) (Simplify your answer.) \( \int_{3}^{0}\left(4 x-x^{3}\right) d x \) b. \( \int_{3}^{0}\left(4 x-x^{3}\right) d x=\square \frac{9}{4} \) (Simplify your answer.)

5. Exprimer \( r \) en fonction de \( f \) et \( f^{\prime} \). Les fonctions \( r \) et \( \theta \) sont fixées ainsi pour la suite de la partie. (a) Montrer que \( \theta^{\prime}=-1+q \sin (\theta) \cos (\theta) \) (b) Montrer que \( r^{\prime}=q r \sin ^{2} \theta \). En déduire la monotonie de \( r \). (c) Etudier les variations de \( t \mapsto r(t) \exp (-\arctan (t)) \) et en déduire que \( 0 \leq r(t) \leq r(0) \exp (\operatorname{arct} \) que \( r \) a une limite strictement positive en \( +\infty \).

Use the substitution \( u=x^{2}+7 \) to find the following indefinite integral. Check your answer by differentiation. \[ \int 2 x\left(x^{2}+7\right)^{4} d x \] \( \int 2 x\left(x^{2}+7\right)^{4} d x=\frac{\left(x^{2}+7\right)^{5}}{5}+C \)

\( \lim _ { n i n } \frac { ( n + 1 ) ^ { 3 } - ( n + 2 ) ^ { 2 } } { ( n - 2 ) ^ { 3 } - ( n + 3 ) ^ { 3 } } \)