From 325d44c6428af3e70d0b4c5d78a1e1d117895f52 Mon Sep 17 00:00:00 2001
From: syn <isaqtm@gmail.com>
Date: Fri, 27 Dec 2019 10:33:43 +0300
Subject: Add stuff

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+\documentclass[11pt]{article}
+%\usepackage[T2A]{fontenc}
+%\usepackage[utf8]{inputenc}
+%\usepackage[russian]{babel}
+\usepackage[x11names, svgnames, rgb]{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{arrows,shapes}
+
+\usepackage{scrextend}
+
+
+\input{intro}
+
+\lhead{\color{gray} Шарафатдинов Камиль 192}
+\rhead{\color{gray} ДЗ к 07.12 (\texttt{sol1202})}
+\title{ДЗ на 07.12}
+\author{Шарафатдинов Камиль БПМИ-192}
+\date{билд: \today}
+
+% -- Here bet  dragons --
+\begin{document}
+
+\maketitle
+
+\question{17}{
+    \[
+        \lim_{x \to \infty} f(x) = 0, \qquad
+        \lim_{x \to \infty} g(x) = 0, \qquad
+    \]
+    Доказать, что
+    \[
+        \begin{cases}
+            f(x), g(x) \text{ -- дифф. в окрестности } \infty \quad (1)\\
+            \exists \lim_{x \to +\infty} f(x) = 0\\
+            \exists \lim_{x \to +\infty} g(x) = 0\\
+            g'(x) \neq 0\\
+            \exists \frac{f'(x)}{g'(x)}
+        \end{cases}
+        \implies \lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{f'(x)}{g'(x)}
+    \]
+}
+
+    \[
+        (1) \Rightarrow f\braced{\frac{1}{t}}, g\braced{\frac{1}{t}} \text{ -- дифф. в окрестности } 0
+    \]
+
+    \[
+        \lim_{x \to \infty} \frac{f'(x)}{g'(x)} =
+        \lim_{y \to 0} \frac{(f(\frac{1}{t}))'}{(g(\frac{1}{t}))'} =
+        \lim_{y \to 0} \frac{-t^2}{-t^2}\frac{f'(\frac{1}{t})}{g'(\frac{1}{t})} \overset{\text{по правилу Лопиталя}}{=}
+        \lim_{y \to 0} \frac{f(\frac{1}{t})}{g(\frac{1}{t})} =
+        \lim_{x \to \infty} \frac{f(x)}{g(x)} \qed
+    \]
+
+
+\question{21}{
+    \begin{enumerate}
+        \item $
+            x^{2/3} + y^{2/3} = a^{2/3}
+        $
+        \item $\displaystyle
+            \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
+        $
+    \end{enumerate}
+}
+    1.
+    \begin{gather*}
+        x^{2/3} + y^{2/3} = a^{2/3}\\
+        \frac{2}{3}x^{-1/3} + y' \frac{2}{3} y^{-1/3} = 0\\
+        x^{-1/3} + y' y^{-1/3} = 0\\
+        y' = -\frac{y^{1/3}}{x^{1/3}} = -\sqrt[3]{\frac{y}{x}}
+    \end{gather*}
+
+    2.
+    \begin{gather*}
+            \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\\
+            \frac{2x}{a^2} + \frac{2yy'}{b^2} = 0\\
+            \frac{yy'}{b^2} = -\frac{x}{a^2}\\
+            y' = -\frac{x}{y}\cdot\frac{b^2}{a^2}
+    \end{gather*}
+
+\question{21}{
+    \[
+        r = ae^{m\varphi}
+    \]
+}
+
+    \begin{gather*}
+        r = ae^{m\varphi}\\
+        \sqrt{x^2 + y^2} = ae^{m \arctan{\frac{y}{x}}}\\
+        \frac{2x + 2yy'}{2\sqrt{x^2 + y^2}} =
+            ae^{m \arctan{\frac{y}{x}}} m (\arctan{\frac{y}{x}})' =
+            ae^{m \arctan{\frac{y}{x}}} m \frac{1}{1 + \braced{\frac{y}{x}}^2} \frac{y'x - y}{x^2}\\
+        \frac{x + yy'}{y'x - y} =
+            mae^{m \varphi}\frac{1}{\frac{x^2 + y^2}{x^2}} \frac{1}{x^2} \sqrt{x^2 + y^2} =
+            \frac{mae^{m \varphi}}{r} = \mu\\
+        \frac{x + yy'}{y'x - y} = \mu\\
+        x + yy' = \mu (y'x - y)\\
+        x + \mu y = \mu y'x - yy'\\
+        y' = \frac{x + \mu y}{\mu x - y} =
+            \frac{x + \frac{mae^{m \varphi}}{r}y}{\frac{mae^{m \varphi}}{r}x - y} =
+            \frac{xr + ymae^{m \varphi}}{xmae^{m \varphi} - yr}
+    \end{gather*}
+
+\question{23}{
+    \[
+        y_1 = ax^2, \qquad\qquad y_2 = \ln x
+    \]
+}
+    
+    Чтобы кривые касались, достаточно, чтобы их функции и производные были равны в некоторой точке:
+    \[
+        \begin{cases}
+            y_1 = y_2\\
+            y_1' = y_2'\\
+        \end{cases}
+    \]
+    \[
+        2ax = \frac{1}{x} \implies x = \sqrt\frac{1}{2a}
+    \]
+    \[
+        ax^2 = \ln x \overset{\text{подставим $x$}}{\implies}
+        \frac{1}{2} = \ln {\sqrt\frac{1}{2a}} \implies
+        \frac{1}{2a} = e \implies a = \frac{1}{2e}
+    \]
+
+\question{24}{
+    \[
+        x = \frac{2t + t^2}{1 + t^3}, \qquad y = \frac{2t - t^2}{1 + t^3}
+    \]
+}
+
+    $
+        \exists \varphi^{-1}(x): xt^3 - t^2 - 2t + x = 0
+    $ и функции дифференцируемы в окрестностях нужных точек, поэтому $f'(x_0) = \frac{x'(t_0)}{y'(t_0)}$
+
+    \[
+        f'(x) = \frac{\braced{\frac{2t - t^2}{1 + t^3}}'}{\braced{\frac{2t + t^2}{1 + t^3}}'} =
+            \frac{(2 - 2t)(1 + t^3) - 3t^3(2t - t^2)}
+                 {(2 + 2t)(1 + t^3) - 3t^3(2t + t^2)} =
+            \frac{2 - 2t + 2t^3 - 2t^4 - 6t^4 + 3t^5}
+                 {2 + 2t + 2t^3 + 2t^4 - 6t^4 - 3t^5}
+    \]
+
+    \begin{enumerate}[(a)]
+        \item $t = 0: $ $f'(x) = 1$\\
+            \[
+                1 \cdot(y - 0) + (x - 0) = 0 \Rightarrow y + x = 0
+            \]
+        \item $t = 1: $ $f'(x) = 3$\\
+            \[
+                3 \cdot(y - 1.5) + (x - 1.5) = 0 \Rightarrow 3y + x - 3 = 0
+            \]
+        \item $t = +\infty: $ $f'(x) = -1$\\
+            \[
+                -1 \cdot(y - 0) + (x - 0) = 0 \Rightarrow -y + x = 0
+            \]
+    \end{enumerate}
+
+\question{25.a}{
+    \[
+        \lim_{x \to 0} \frac{\ln \cos ax}{\ln \cos bx} = \frac{a^2}{b^2}
+    \]
+}
+    \begin{align*}
+       &\lim_{x \to 0} \frac{\ln \cos ax}{\ln \cos bx} \overset{\text{Лопиталь}}=
+        \lim_{x \to 0} \frac{-a \sin ax}{\cos ax} \frac{\cos bx}{-b \sin bx} =
+        \lim_{x \to 0} \frac{\sin ax}{\sin bx} \cdot \frac{a\cos bx}{b\cos ax} =\\\\
+       &\lim_{x \to 0} \frac{a(bx) \sin ax}{(ax)b \sin bx} \cdot \frac{a\cos bx}{b\cos ax} \overset{\text{1й зам. предел}}=
+        \lim_{x \to 0} \frac{a}{b} \cdot \frac{a\cos bx}{b\cos ax} = \frac{a^2}{b^2}
+    \end{align*}
+
+\question{25.b}{
+    \[
+        \lim_{x \to a} \frac{a^x - x^a}{x - a} = a^a (\ln a - 1)
+    \]
+}
+
+    \[
+        \lim_{x \to a} \frac{a^x - x^a}{x - a} =
+        \lim_{x \to a} \frac{e^{x\ln a} - x^a}{x - a} \overset{\text{Лопиталь}}=
+        \lim_{x \to a} \frac{\ln a \cdot a^x - ax^{a - 1}}{1} \overset{\text{ф-я непрерывна}}=
+        \frac{\ln a \cdot a^a - a \cdot a^{a - 1}}{1} = a^a (\ln a - 1)
+    \]
+
+\question{25.c}{
+    \[
+        \lim_{x \to 1} \frac{1}{\ln x} - \frac{1}{x - 1} = \frac{1}{2}
+    \]
+}
+
+    \begin{align*}
+        \lim_{x \to 1} \frac{1}{\ln x} - \frac{1}{x - 1} =
+        \lim_{x \to 1} \frac{x - 1 - \ln x}{(x - 1) \ln x} \overset{\text{Лопиталь}}=
+        \lim_{x \to 1} \frac{1 - \frac{1}{x}}{\ln x + \frac{x - 1}{x}} \overset{\cdot \frac{x}{x}}=
+        \lim_{x \to 1} \frac{x - 1}{x \ln x + x - 1} \overset{\text{Лопиталь}}=\\
+        =\lim_{x \to 1} \frac{1}{1 + \ln x + 1} = \frac{1}{2}
+    \end{align*}
+
+
+\question{25.c}{
+    \[
+        \frac{(1 + x)^\frac{1}{x} - e}{x} = -\frac{e}{2}
+    \]
+}
+
+    \begin{align*}
+       &\lim_{x \to 0}\frac{(1 + x)^\frac{1}{x} - e}{x} =
+        \lim_{x \to 0}\frac{e^\frac{\ln (x + 1)}{x} - e}{x} \overset{\text{Лопиталь}}=\\
+     = &\lim_{x \to 0}\frac{e^\frac{\ln (x + 1)}{x} \cdot \frac{\frac{x}{x + 1} - \ln(x + 1)}{x^2}}{1} =
+        \lim_{x \to 0}e^\frac{\ln (x + 1)}{x} \cdot \frac{\frac{x}{x + 1} - \ln(x + 1)}{x^2} \overset{\text{св-ва пределов}}=\\
+     = &\lim_{x \to 0}e^\frac{\ln (x + 1)}{x} \cdot 
+                    \lim_{x \to 0} \frac{\frac{x}{x + 1} - \ln(x + 1)}{x^2} =
+        \lim_{x \to 0}(x + 1)^\frac{1}{x} \cdot 
+                    \lim_{x \to 0} \frac{x - (x + 1)\ln(x + 1)}{(x + 1)x^2} =\\
+     = &e \lim_{x \to 0} \frac{x - (x + 1)\ln(x + 1)}{(x + 1)x^2} \overset{\text{св-ва пределов}}=
+        e \lim_{x \to 0} \frac{1}{x + 1} \lim_{x \to 0} \frac{x - (x + 1)\ln(x + 1)}{x^2} =\\
+     = &e \lim_{x \to 0} \frac{x - x\ln(x + 1) - \ln(x + 1)}{x^2} \overset{\text{св-ва пределов}}=
+        e \left( \lim_{x \to 0}\frac{x - \ln(x + 1)}{x^2} - \lim_{x \to 0}\frac{x\ln(x + 1)}{x^2} \right) \overset{\text{два Лопиталя}}=\\
+     = &e \left( \lim_{x \to 0} \frac{1 - \frac{1}{x + 1}}{2x} - \lim_{x \to 0} \frac{\frac{1}{x + 1}}{1} \right) =
+        e \left( \lim_{x \to 0} \frac{x + 1 - 1}{2x(x + 1)} - 1 \right) =
+        e \left( \lim_{x \to 0} \frac{1}{2(x + 1)} - 1 \right) =
+        e \left( \frac{1}{2} - 1 \right) = -\frac{e}{2}
+    \end{align*}
+
+\question{25.e}{
+    \[
+        \lim_{x \to 1} \frac{1}{\ln x} - \frac{1}{x - 1} = \frac{1}{2}
+    \]
+}
+    \begin{align*}
+       &\lim_{x \to 1} \frac{1}{\ln x} - \frac{1}{x - 1}=
+        \lim_{x \to 1} \frac{x - 1 - \ln x}{(x - 1)\ln x} \overset{\text{Лопиталь}}=\\
+     = &\lim_{x \to 1} \frac{1 - \frac{1}{x}}{\ln x + \frac{x - 1}{x}} =
+        \lim_{x \to 1} \frac{x - 1}{x\ln x + x - 1} \overset{\text{Лопиталь}}=\\
+     = &\lim_{x \to 1} \frac{1}{1 + \ln x + 1} = \frac{1}{2}
+    \end{align*}
+
+\end{document}
-- 
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