# 数学代写|多变量微积分代写multivariable calculus代考|MTH265

## 数学代写|多变量微积分代写multivariable calculus代考|Supplementary problems

Section 2.A

1. Consider the function $f: \mathbb{R} \longrightarrow \mathbb{R}, x \mapsto f(x)$, where
(a) $f(x)=x^{2 / 3}$
(b) $f(x)=x^{4 / 3}$
(c) $f(x)=x^{1 / 2}$.
Decide in each case whether the derivative of $f$ exists at $x=0$ as follows:
(i) Check whether $f$ exists at $x=0$ and at $x=0+h$ (for $h$ arbitrarily small).
(ii) Check right and left limits to see whether $f$ is continuous at $x=0$.
(iii) Check right and left limits to see whether $f$ has a derivative at $x=0$.
2. Consider the function $f: \mathbb{R} \longrightarrow \mathbb{R}, x \mapsto(x-1)^\alpha, \alpha \in \mathbb{R}$. Decide from first principles for what values of $\alpha$ the derivative of $f$ exists at $x=1$.
Section 2.B
3. Are the following functions continuous at $(0,0)$ ?
(a) $f(x, y)=\frac{x y}{x^2+y^2}, f(0,0)=0$.
(b) $f(x, y)=\frac{x y}{\sqrt{x^2+y^2}}, f(0,0)=0$.

## 数学代写|多变量微积分代写multivariable calculus代考|Extreme values of f

Occupying a central position in the vastness of the space of applications of differential calculus is the subject of optimization. At its most basic, the term refers to the task of finding those points in a function’s domain that gives rise to maxima or minima of that (scalar) function, and of determining the corresponding values of that function.

Of special interest in the study are the so-called extreme points of $f(\boldsymbol{x})$, a subset of which are the so-called critical points. These are points where the function can exhibit either a local maximum or minimum, and even a global maximum or minimum.

To set the stage we require some basic infrastructure. We start with a few essential definitions.

We have here invoked the open sphere $S_r(\boldsymbol{a})$ to represent the set of points $\boldsymbol{x}$ different from but near $\boldsymbol{a}$ (the radius $r>0$ is presumed small). We could equally well have referred to points $x$ in a larger “neighbourhood” of $\boldsymbol{a}$. However, that proves to be unnecessary and less convenient, it is enough to consider a small open sphere as we are defining local properties.

Points of local minimum (Figure 3.1(i)) and local maximum (Figure 3.1(ii)) are examples of critical points.

# 多变量微积分代考

## 数学代写|多变量微积分代写multivariable calculus代考|Supplementary problems

1. 考虑函数 $f: \mathbb{R} \longrightarrow \mathbb{R}, x \mapsto f(x)$ ，其中
(a) $f(x)=x^{2 / 3}$
(二) $f(x)=x^{4 / 3}$
(C) $f(x)=x^{1 / 2}$.
在每种情况下决定导数是否 $f$ 存在于 $x=0$ 如下:
(i) 检查是否 $f$ 存在于 $x=0$ 在 $x=0+h$ (为了 $h$ 任意小)。
(ii) 检查左右极限是否 $f$ 是连续的 $x=0$.
(iii) 检查左右极限是否 $f$ 有导数 $x=0$.
2. 考虑函数 $f: \mathbb{R} \longrightarrow \mathbb{R}, x \mapsto(x-1)^\alpha, \alpha \in \mathbb{R}$. 从第一原则决定什么价值 $\alpha$ 的导数 $f$ 存在于 $x=1$. 第 $2 . B$ 节
3. 以下函数是否连续 $(0,0)$ ?
(一个) $f(x, y)=\frac{x y}{x^2+y^2}, f(0,0)=0$.
(二) $f(x, y)=\frac{x y}{\sqrt{x^2+y^2}}, f(0,0)=0$.

## 数学代写|多变量微积分代写multivariable calculus代考|Extreme values of f

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