Question

Границы и производные

Answer 1

Ответ:

$4)\ \ \lim\limits_{x \to 0}\dfrac{e^{2x}-1}{arcsin3x}=\Big[\ \dfrac{0}{0}\ \Big]=\lim\limits_{x \to 0}\dfrac{2e^{2x}}{\dfrac{3}{1+9x^2}}=\lim\limits_{x \to 0}\dfrac{2e^{2x}\cdot (1+9x^2)}{3}=\\\\\\=\lim\limits_{x \to 0}\dfrac{2\cdot 1\cdot (1+0)}{3}=\dfrac{2}{3}$

$5)\ \ 2y=1+xy^3\ \ ,\ \ M(1;1)\\\\(2y)'=(1+xy^3)'\\\\2=0+1\cdot y^3+x\cdot 3y^2y'\\\\3xy^2\cdot y'=2-y^3\ \ \ \to \ \ \ y'=\dfrac{2-y^3}{3xy^2}\\\\\\M(1;1):\ y'\Big|_{M(1;1)}=\dfrac{2-1^3}{3\cdot 1\cdot 1^2}=\dfrac{1}{3}$

$6)\ \ \ y=\dfrac{x^4}{4}-x^3\\\\\\y'=x^3-3x^2=x^2(x-3)=0\ \ \to \ \ \ x_1=0\ ,\ x_2=3\\\\znaki\ y':\ \ \ +++(0)---(3)+++\\{}\qquad \qquad \qquad \nearrow \ \, \ (0)\, \ \ \searrow \ \ (3)\ \ \ \nearrow \\\\x_{max}=0\ \ ,\ \ y_{max}=0\\\\x_{m-n}=3\ \ ,\ \ y_{min}=\dfrac{81}{4}-27=-6,75$