Finding Taylor Polynomial Approximations of Functions · 求函数的泰勒多项式近似
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| Taylor polynomial/ˈteɪlə ˌpɒlɪˈnəʊmɪəl/ | 泰勒多项式 | tài lēi duō xiàng shì |
| Maclaurin polynomial/məˈklɔːrɪn ˌpɒlɪˈnəʊmɪəl/ | 麦克劳林多项式 | mài kè láo lín duō xiàng shì |
Approximating a function with a polynomial
- Linearization (Unit 4) used a tangent line to approximate a function. Why stop at a line?
- A Taylor polynomial 泰勒多项式 matches a function's value and several derivatives at a point.
- The more derivatives it matches, the better it hugs the curve nearby.
- Polynomials are easy to compute, so they're a powerful stand-in for messy functions.
用多项式近似函数
- 线性化(第 4 单元)用切线近似函数。为何止步于一条直线?
- 泰勒多项式在一点匹配函数的值和若干导数。
- 它匹配的导数越多,就在附近越紧贴曲线。
- 多项式易算,所以它们是杂乱函数的有力替身。
Building the polynomial
- Centered at $x=a$, the $n$th-degree Taylor polynomial is:
-
$$P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^{k}=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots$$
- Each term uses the $k$th derivative at $a$, divided by $k!$, times $(x-a)^k$.
- The degree-$1$ piece is exactly the linearization; higher terms add curvature.
构造多项式
- 以 $x=a$ 为中心,$n$ 次泰勒多项式是:
-
$$P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^{k}=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots$$
- 每一项用 $a$ 处的第 $k$ 阶导数,除以 $k!$,乘以 $(x-a)^k$。
- 一次项恰是线性化;更高的项增添曲率。
The $k$th term of a Taylor polynomial centered at $a$ is... · 在 $k$处展开的泰勒多项式的第 $a$ 项是...
Derivative at $a$, over $k!$, times $(x-a)^k$. · 在 $a$ 处的导数，除以 $k!$，乘以 $(x-a)^k$。
Every derivative in a Taylor polynomial is evaluated at the ____ $a$. · 泰勒多项式中的每个导数都在 ____ $a$ 处求值。
Use $f^{(k)}(a)$, not $f^{(k)}(x)$. · 使用 $f^{(k)}(a)$，而非 $f^{(k)}(x)$。
Maclaurin: centered at zero
- A Maclaurin polynomial 麦克劳林多项式 is just a Taylor polynomial centered at $a=0$.
- $P_n(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\cdots$ — the $(x-a)$ factors become plain $x^k$.
- Common ones are worth memorizing: $e^x$, $\sin x$, $\cos x$ have clean Maclaurin polynomials.
- Same recipe, centered at the origin.
麦克劳林:以零为中心
- 麦克劳林多项式就是以 $a=0$ 为中心的泰勒多项式。
- $P_n(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\cdots$——$(x-a)$ 因子变成普通的 $x^k$。
- 常见的值得背下来:$e^x$、$\sin x$、$\cos x$ 有干净的麦克劳林多项式。
- 同样的配方,以原点为中心。
A Maclaurin polynomial is a Taylor polynomial centered at... · 麦克劳林多项式是以...为中心的泰勒多项式
Maclaurin = centered at $0$. · 麦克劳林 = 以 $0$ 为中心。
Better near the center
- The approximation is best near $x=a$ and gets worse as you move away.
- Adding more terms (higher degree) improves accuracy on a wider range.
- Match value, slope, concavity, and beyond — each derivative pins down one more feature.
- The tangent line is just the first two terms of this bigger idea.
越靠近中心越好
- 近似在 $x=a$ 附近最好,离得越远越差。
- 增加更多项(更高次)在更宽的范围上改进精度。
- 匹配值、斜率、凹凸性乃至更多——每个导数确定一个额外特征。
- 切线只是这个更大想法的前两项。
A polynomial hugging eˣ · 紧贴 eˣ 的多项式
y = a·e^{bx}
The degree-2 Maclaurin polynomial $1+x+\tfrac{x^2}{2}$ hugs $e^x$ near $0$ — more terms widen the good range. · 该2 麦克劳林多项式 $1+x+\tfrac{x^2}{2}$ 在 $e^x$ 附近贴合得更好，— 更多项会扩大其有效范围。 $0$ — 更多术语扩大了良好范围。
The degree-2 Maclaurin polynomial for $e^x$ is... · n 阶-2 麦克劳林多项式为 $e^x$ ...
All derivatives of $e^x$ at $0$ are $1$. · $e^x$ 在 $0$ 处的所有导数均为 $1$。
The Taylor term divides the $k$th derivative by $k!$ (a factorial). · 泰勒项将第 $k$ 阶导数除以 $k!$（阶乘）。
Divide by $k!$, not $k$. · 除以 $k!$，而非 $k$。
A Taylor polynomial approximation is most accurate... · 泰勒多项式近似在...时最准确
Accuracy is best near $a$ and improves with more terms. · 精度在 $a$ 附近最佳，且随项数增加而提高。
Don't drop the factorials: the $k$th term is $\frac{f^{(k)}(a)}{k!}(x-a)^k$ — divide by $k!$, not just $k$. Evaluate every derivative at the center $a$ (not at $x$). And the approximation is only good near $a$; far away, a truncated Taylor polynomial can be wildly off.
别丢阶乘:第 $k$ 项是 $\frac{f^{(k)}(a)}{k!}(x-a)^k$——除以 $k!$,不是只除以 $k$。在中心 $a$(而非 $x$)处求每个导数。而且近似只在 $a$ 附近好;远处,截断的泰勒多项式可能大错特错。
Find the degree-$2$ Maclaurin polynomial for $f(x)=e^x$.
- $f(0)=1$, $f'(0)=1$, $f''(0)=1$ (since every derivative of $e^x$ is $e^x$).
- $P_2(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2=1+x+\frac{x^2}{2}$.
- Near $0$, $e^x\approx 1+x+\tfrac{x^2}{2}$ (e.g. $e^{0.1}\approx1.105$). ✓
求 $f(x)=e^x$ 的 $2$ 次麦克劳林多项式。
- $f(0)=1$、$f'(0)=1$、$f''(0)=1$(因为 $e^x$ 的每个导数都是 $e^x$)。
- $P_2(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2=1+x+\frac{x^2}{2}$。
- 在 $0$ 附近,$e^x\approx 1+x+\tfrac{x^2}{2}$(如 $e^{0.1}\approx1.105$)。✓
A Taylor polynomial centered at $a$ is $P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k$ — matching the function's value and derivatives at $a$. A Maclaurin polynomial is the special case $a=0$. Keep the factorials, evaluate derivatives at $a$, and expect the best accuracy near the center.
以 $a$ 为中心的泰勒多项式是 $P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k$——在 $a$ 匹配函数的值和导数。麦克劳林多项式是 $a=0$ 的特例。保留阶乘,在 $a$ 处求导数,并预期在中心附近精度最好。