普林斯顿微积分读本 VIII

Sat, 19 Jun 2021 00:00:00 +0000

§泰勒多项式、泰勒级数与幂级数

近似值（approximations）、泰勒多项式和泰勒近似定理（Taylor approximation theorem）
近似值的精确度和泰勒定理
幂级数
泰勒级数和麦克劳林级数（Maclaurin series）
泰勒级数的收敛性问题

24.1 近似值和泰勒多项式

Here’s a nice fact: for any real number $x$, we have
$$ > e^x \cong 1 + x + {x^2\over2} + {x^3\over6} > $$
Also, the closer x is to 0, the better the approximation.

24.1.1 线性化的回顾

平滑的函数可经过多次求导（见13.2），曲线在点 $(a,f(a))$ 处的切线的方程为：

$$ y=f(a)+f'(a)(x-a) $$

称为函数 $f$ 在 $x=a$ 处的线性化（linearization）

线性近似，即在切点处附近，切线与曲线差别很小

24.1.2 二次方程近似

二次近似（quadratic approximates）

$$ y=f(a)+f'(a)(x-a)+{f''(a)\over2}(x-a)^2 $$

关于 $x$ 的二次函数

$P_2(x)=y$，代入 $x=a$ 时，$P_2(a)=f(a)$
$P_2'(x)=f'(a)+f''(a)(x-a)$，代入 $x=a$ 时，$P_2'(a)=f'(a)$
$P_2''(x)=f''(a)$，代入 $x=a$ 时，$P_2''(a)=f'(a)$
因为 $f''(a)$ 是一个常数，$P_2'''(x)=0$

The second-order correction term helps us get even closer to the curve, at least for $x$ near $a$.

普林斯顿微积分读本 III

Thu, 15 Apr 2021 00:00:00 +0000

§最优化和线性化

最优化（optimization）
线性近似（linearization）
估算函数的零点
牛顿法（Newton’s method）

13.1 最优化

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’s call it $Q$ for now, although of course it might be another letter like $P$, $m$, or $\alpha$.

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 $8$.)

Write down equations relating the variables. One of them should be an equation for $Q$.

Try to make $Q$ a function of only one variable, using all your equations to eliminate the other variables.

Differentiate $Q$ with respect to that variable, then find the critical points; remember, these occur where the derivative is $0$ or the derivative doesn’t exist.

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.

Write out a summary of what you’ve found, identifying the variables in words rather than symbols (wherever possible).

考虑所有可能需要的变量
确认极端变量的可能
写出不同变量的方程
构建单变量函数
求导，计算临界点
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
结论

注意：构建单变量函数时，有时候可以通过隐函数求导

Approximation on TouchingFish.top