# 微积分作业代写calclulus代考| Implicit functions

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## 微积分作业代写calclulus代考|Example 2.13:

Suppose the equation $x^{3} y+2 y^{3} x=3$ defines $y$ as a function $f$ of $x$ in the neighbourhood of the point $(1,1)$. We wish to find the derivative of $f$ at $x=1$, and a linear approximation to $f$ near the point. Let $F(x, y)=x^{3} y+2 y^{3} x-3$.
Note that $F \in C^{1} \forall(x, y) \in \mathbb{R}^{2}$. Then we have
$$\frac{\partial F}{\partial x}=3 x^{2} y+2 y^{3}, \quad \frac{\partial F}{\partial y}=x^{3}+6 x y^{2}$$
We note that $\frac{\partial F}{\partial y} \neq 0$ at $(1,1)$. Thus, from our linear approximation we have $\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{(1,1)}=-\frac{3+2}{1+6}=-\frac{5}{7}$
The linear approximation is $y=-\frac{5}{7} x+c$. To determine $c$, use the fact that the line passes through $(1,1)$, giving $y=-\frac{5}{7} x+\frac{12}{7}$.

## 微积分作业代写calclulus代考|Implicit functions

Suppose we are given the following task: In each of the cases below express the variable $y$ as a function of the remaining variables:
(a) $8 y+64 x^{2}=0$;
(b) $2 y^{2}+8 y+16 z \sin x=0$;
(c) $\ln |y|+y^{3} x+20 x^{2}=w$.
I am as certain that you cannot complete task (c) as I am that you can complete tasks (a) and (b). Although task (c) is impossible, the equation suggests there is a functional relationship, in principle.

This introduces the notion of an implied or implicit function. In task (c) the equation implies that $y$ can be a function $f$ of the variables $x$ and $w$. What we shall do in this section is establish conditions under which such a function is defined, at least locally. Along the way we will get, as reward, a linear approximation to this unknown function, in terms of the independent variables near a given point, and an explicit expression, and value, for the derivative (or derivatives) of this function at that point.

As before we explain by considering examples of increasing complexity. In each case we will also discuss an analogous linear problem. Since the argument we follow is based on linearization, we hope that the parallels will facilitate reader understanding. The purist reader may frown on the questionable rigour. However, the possibility of greater appreciation for the end result is worth sacrificing some degree of mathematical sophistication.
Suppose we are given the following three problems:
1) $\mathrm{e}^{x+y}+x y=0 \Longrightarrow F(x, y)=0$

• a level curve.
2) $\mathrm{e}^{x+y+z}-(x+y+z)^{2}=1 \Longrightarrow \quad F(x, y, z)=0 \quad-$ a level surface.
$3)$
$\left{\begin{array}{r}e^{x+y+z}-(x+y+z)^{2}-1=0 \ z \sin (x y)-x \cos (z y)=0\end{array} \Longrightarrow\left{\begin{array}{l}F(x, y, z)=0 \ G(x, y, z)=0\end{array}\right.\right.$
• a curve of intersection.

## 微积分作业代写calclulus代考|Example 2.13:

∂F∂X=3X2和+2和3,∂F∂和=X3+6X和2

## 微积分作业代写calclulus代考|Implicit functions

(a)8和+64X2=0;
(二)2和2+8和+16和没有⁡X=0;
（C）ln⁡|和|+和3X+20X2=在.

1）和X+和+X和=0⟹F(X,和)=0

• 一条水平曲线。
2)和X+和+和−(X+和+和)2=1⟹F(X,和,和)=0−一个水平面。
3)
$\左{和X+和+和−(X+和+和)2−1=0 和没有⁡(X和)−X某物⁡(和和)=0\长右箭头\左{F(X,和,和)=0 G(X,和,和)=0\对。\对。$
• 一条相交曲线。