https://touchingfish.top/tags/newtons-method/Recent content in Newtons-Method on TouchingFish.topHugozh-cnThu, 15 Apr 2021 00:00:00 +0000https://touchingfish.top/calculus101/2021-cal3/Thu, 15 Apr 2021 00:00:00 +0000https://touchingfish.top/calculus101/2021-cal3/<h1 id="最优化和线性化">§最优化和线性化</h1> <ul> <li>最优化（optimization）</li> <li>线性近似（linearization）</li> <li>估算函数的零点</li> <li>牛顿法（Newton&rsquo;s method）</li> </ul> <h2 id="131-最优化">13.1 最优化</h2> <blockquote> <ol> <li>Identify all the variables you might possibly need. One of them should be the quantity you want to maximize or minimize - make sure you know which one! Let&rsquo;s call it $Q$ for now, although of course it might be another letter like $P$, $m$, or $\alpha$.</li> <li>Get a feel for the extremes of the situation, seeing how far you can push your variables. (For example, in the problem from the previous section, we saw that $x$ had to be between $2$ and $8$.)</li> <li>Write down equations relating the variables. One of them should be an equation for $Q$.</li> <li>Try to make $Q$ a function of only one variable, using all your equations to eliminate the other variables.</li> <li>Differentiate $Q$ with respect to that variable, then find the critical points; remember, these occur where the derivative is $0$ or the derivative doesn&rsquo;t exist.</li> <li>Find the values of $Q$ at all the critical points and at the endpoints. Pick out the maximum and minimum values. As a verification, use a table of signs or the sign of the second derivative to classify the critical points.</li> <li>Write out a summary of what you&rsquo;ve found, identifying the variables in words rather than symbols (wherever possible).</li> </ol> </blockquote> <ol> <li>考虑所有可能需要的变量</li> <li>确认极端变量的可能</li> <li>写出不同变量的方程</li> <li>构建单变量函数</li> <li>求导，计算临界点</li> <li>计算临界点及端点的函数值，使用一阶或二阶导数判断最大值和最小值</li> <li>结论</li> </ol> <p>注意：构建单变量函数时，有时候可以通过隐函数求导</p>