泰勒展开

一元泰勒展开
$$f(x) = f(x_{k}) + (x-x_{k})f’(x_{k})+\frac{1}{2!}(x-x_{k})^{2}f’’(x_{k})+o^n$$
二元泰勒展开
$$
\begin{array}{c}
f(x, y)=f\left(x_{k}, y_{k}\right)+\left(x-x_{k}\right) f_{x}^{\prime}\left(x_{k}, y_{k}\right)+\left(y-y_{k}\right) f_{y}^{\prime}\left(x_{k}, y_{k}\right) \
+\frac{1}{2!}\left(x-x_{k}\right)^{2} f_{x x}^{\prime \prime}\left(x_{k}, y_{k}\right)+\frac{1}{2!}\left(x-x_{k}\right)\left(y-y_{k}\right) f_{x y}^{\prime \prime}\left(x_{k}, y_{k}\right) \
+\frac{1}{2!}\left(x-x_{k}\right)\left(y-y_{k}\right) f_{y x}^{\prime \prime}\left(x_{k}, y_{k}\right)+\frac{1}{2!}\left(y-y_{k}\right)^{2} f_{y y}^{\prime \prime}\left(x_{k}, y_{k}\right) \
+o^{n}
\end{array}
$$
多元泰勒展开
$$
\begin{array}{c}
f\left(x^{1}, x^{2}, \ldots, x^{n}\right)=f\left(x_{k}^{1}, x_{k}^{2}, \ldots, x_{k}^{n}\right)+\sum_{i=1}^{n}\left(x^{i}-x_{k}^{i}\right) f_{x^{i}}^{\prime}\left(x_{k}^{1}, x_{k}^{2}, \ldots, x_{k}^{n}\right) \
+\frac{1}{2!} \sum_{i, j=1}^{n}\left(x^{i}-x_{k}^{i}\right)\left(x^{j}-x_{k}^{j}\right) f_{x^i x^j}^{\prime \prime}\left(x_{k}^{1}, x_{k}^{2}, \ldots, x_{k}^{n}\right) \
+o^{n}
\end{array}
$$

运动学建模

运动学公式
$$
\begin{array}{c} \\ &
f(v, \theta) = \dot{x} = v\cos \theta \\ &
g(v, \theta) = \dot{y} = v\sin \theta \\ &
\dot{\theta } = \omega
\end{array}
$$
偏导数：
$$
\begin{array}{c} \\ &
f_{v}’(v, \theta) = \cos \theta \\ &
f_{\theta }’(v, \theta) = -v\sin \theta \\ &
g_{v}’(v, \theta) = \sin \theta \\ &
g_{\theta }’(v, \theta) = v\cos \theta \\ &
\end{array}
$$

所以：

$$
\left.
\begin{aligned}
\dot{x} & \approx v_{k}\cos \theta_{k} + (v - v_{k})\cos \theta_{k} - (\theta - \theta_{k})v_{k}\sin \theta_{k} \\
\dot{y} & \approx v_{k}\sin \theta_{k} + (v - v_{k})\sin \theta_{k} + (\theta - \theta_{k})v_{k}\cos \theta_{k}
\end{aligned}
\right.
$$