微积分作业代写calclulus代考| Higher-order derivatives

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微积分作业代写calclulus代考|Higher-order derivatives

each of which describes a second-order partial derivative. First, a partial derivative w.r.t. $x$, then a partial derivative w.r.t. $y$. The reader should exercise some care in interpreting the different notations.

We are now implored to explain what higher partial derivatives are. It suffices to consider a function of two variables, $f(x, y)$. If $\left.\frac{\partial f}{\partial x}\right|{\left(x{0}, y_{0}\right)}$ is the slope of the tangent to $f$ at $\left(x_{0}, y_{0}\right)$ in the direction of $x$, then, just as in the single-variable case, $\left.\frac{\partial^{2} f}{\partial x^{2}}\right|{\left(x{0}, y_{0}\right)}$ is the rate of change of the slope in this same direction. It is therefore a measure of the curvature of $f$ in this direction. On the other hand, $\left.\frac{\partial^{2} f}{\partial y \partial x}\right|{\left(x{0}, y_{0}\right)}$ is the rate of change of the $x$-directional slope in the $y$-direction.
A convenient and useful result for so-called smooth functions which, apart from their applications in applied contexts (Chapters 3 and 5 ), relieves some of the stress of interpreting notation, is the following.
Theorem 2.5
Suppose $f: \mathbb{R}^{n} \longrightarrow \mathbb{R}$ is continuous and $\frac{\partial f}{\partial x_{i}}, i=1,2, \ldots, n$ exist and are continuous in $S_{r}(\boldsymbol{x}) \subset D_{f}$ and that both $\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}$ and $\frac{\partial^{2} f}{\partial x_{j} \partial x_{i}}$ exist and are continuous at $\boldsymbol{x} \in D_{f}$. Then $\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}=\frac{\partial^{2} f}{\partial x_{j} \partial x_{i}}$ at $\boldsymbol{x} \in D_{f}$.
(For the standard proof, see a standard text book such as [1] or [2].) Note the conditions of the above theorem highlighted in Figure 2.18.

微积分作业代写calclulus代考|Mastery Check 2.16:

Determine all $C^{2}$ functions $f(x, y)$ such that
a) $\frac{\partial f}{\partial x}-2 x \sin x^{2}, \quad \frac{\partial f}{\partial y}-\cos y$.
b) $\frac{\partial f}{\partial x}=2 x+y, \quad \frac{\partial f}{\partial y}=2 y+x$.
c) $\frac{\partial f}{\partial x}=x+3 y x^{2}, \quad \frac{\partial f}{\partial y}=x^{3}+x y$.

微积分作业代写calclulus代考|Higher-order derivatives

（有关标准证明，请参阅标准教科书，例如 [1] 或 [2]。）请注意图 2.18 中突出显示的上述定理的条件。

微积分作业代写calclulus代考|Mastery Check 2.16:

b)∂F∂X=2X+和,∂F∂和=2和+X.
C）∂F∂X=X+3和X2,∂F∂和=X3+X和.