/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Untersuchen Sie die beiden Funkt... [FREE SOLUTION] | 91影视

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Chapter 24: Problem 5

Untersuchen Sie die beiden Funktionen \(f\) und \(g\), $$ \mathbb{R}^{2} \rightarrow \mathbb{R} $$ $$ f(x)= \begin{cases}\frac{x_{1} x_{2}^{3}}{\left(x_{1}+x_{2}\right)^{2}} & \text { f眉r } x \neq 0 \\ 0 & \text { f眉r } x=0\end{cases} $$ $$ g(x)= \begin{cases}\frac{x_{1}^{3}, x_{2}^{2}}{\left(x_{1}^{2}+x_{2}^{2}\right)^{2}} & \text { f眉r } x \neq 0 \\\ 0 & \text { f眉r } x=0\end{cases} $$ auf Stetigkeit im Ursprung,

Short Answer

Expert verified
Answer: Yes, both functions f and g are continuous at the origin (0, 0) as their limit as x 鈫� 0 matches their value at x = 0.

Step by step solution

01

Recap the definition of continuity at a point

To show that a function is continuous at a point, we must verify the following condition: $$ \lim_{x \to a} f(x) = f(a) $$ In this case, we want to prove continuity at the origin, so a = 0.
02

Study the continuity of f at the origin

Function f is given by: $$ f(x)= \begin{cases}\frac{x_{1} x_{2}^{3}}{\left(x_{1}+x_{2}\right)^{2}} & \text { f眉r } x \neq 0 \\\ 0 & \text { f眉r } x=0\end{cases} $$ To show that f is continuous at the origin (0, 0), we need to check if: $$ \lim_{(x_1, x_2) \to (0, 0)} f(x_1, x_2) = f(0,0) $$ First, let's find the limit of f(x) as (x_1, x_2) 鈫� (0, 0). $$ \lim_{(x_1, x_2) \to (0, 0)} \frac{x_{1} x_{2}^{3}}{\left(x_{1}+x_{2}\right)^{2}} $$ Use polar coordinates transformation by setting \(x_1 = r cos(\theta)\) and \(x_2 = r sin(\theta)\), with \(r \geq 0\): $$ \lim_{r \to 0} \frac{(r \cos(\theta)) (r \sin(\theta))^3}{(r (\cos(\theta)+\sin(\theta)))^2} $$ Simplify: $$ \lim_{r \to 0} \frac{r^4 \cos(\theta) \sin^3(\theta)}{r^2(\cos^2(\theta) + 2 \cos(\theta) \sin(\theta) + \sin^2(\theta))} $$ Cancel out r^2 and apply limit as r goes to 0: $$ \lim_{r \to 0} \frac{r^2\cos(\theta) \sin^3(\theta)}{\cos^2(\theta) + 2 \cos(\theta) \sin(\theta) + \sin^2(\theta)} = 0 $$ Since the limit as (x_1, x_2) 鈫� (0, 0) of f(x_1, x_2) is 0 and f(0, 0) is also 0, we can conclude that f is continuous at the origin.
03

Study the continuity of g at the origin

Function g is given by: $$ g(x)= \begin{cases}\frac{x_{1}^{3}, x_{2}^{2}}{\left(x_{1}^{2}+x_{2}^{2}\right)^{2}} & \text { f眉r } x \neq 0 \\\ 0 & \text { f眉r } x=0\end{cases} $$ To show that g is continuous at the origin (0, 0), we need to check if: $$ \lim_{(x_1, x_2) \to (0, 0)} g(x_1, x_2) = g(0,0) $$ First, let's find the limit of g(x) as (x_1, x_2) 鈫� (0, 0). $$ \lim_{(x_1, x_2) \to (0, 0)} \frac{x_{1}^{3}, x_{2}^{2}}{\left(x_{1}^{2}+x_{2}^{2}\right)^{2}} $$ Use polar coordinates transformation by setting \(x_1 = r cos(\theta)\) and \(x_2 = r sin(\theta)\), with \(r \geq 0\): $$ \lim_{r \to 0} \frac{(r \cos(\theta))^3 (r \sin(\theta))^2}{(r^2 (\cos^2(\theta)+\sin^2(\theta)))^2} $$ Simplify and apply limit as r goes to 0: $$ \lim_{r \to 0} \frac{r^5 \cos^3(\theta) \sin^2(\theta)}{r^4} = 0 $$ Since the limit as (x_1, x_2) 鈫� (0, 0) of g(x_1, x_2) is 0 and g(0, 0) is also 0, we can conclude that g is continuous at the origin.
04

Conclusion

Both functions f and g are continuous at the origin (0, 0) as their limit as x 鈫� 0 matches their value at x = 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limits of Functions
Understanding the concept of limits is crucial in calculus; it describes the behavior of a function as its input approaches a particular value. Think of it as sneaking up on a point without actually touching it to see what value the function should be approaching.

Take the example where the function's limit as \(x_1, x_2 \to (0, 0)\) needs to be found. In simple terms, we're looking at what happens to the function's output as its inputs inch infinitely close to zero from any direction. If this limit equals the function's value at that point, we can say the function is well-behaved, or continuous, there. This leads us directly into exploring what continuity means for a function, especially at a point.
Polar Coordinates Transformation
When you hear polar coordinates, picture a point defined by how far away it is from the origin and the angle it makes with the positive x-axis鈥攖his is its r and \(\theta\) in polar terms. This approach is particularly useful when you're dealing with limits involving \(x_1\) and \(x_2\) approaching the origin simultaneously.

In such scenarios, we transform the Cartesian coordinates \(x_1\) and \(x_2\) into polar coordinates \(r\) and \(\theta\). It's like switching from thinking about 'left-right and up-down' to 'how far and at what angle'. This makes handling limits towards the origin (like our \(r \to 0\)) less about juggling two variables and more about watching one (\(r\)) shrink down.
Definition of Continuity
Imagine you're drawing a curve without lifting your pen off the paper鈥攖hat's the essence of a continuous function. To say a function is continuous at a point \(a\), we expect no breaks, jumps, or holes in its graph at that point. The function \(f(x)\) is continuous at \(a\) if the limit of \(f(x)\) as \(x\) approaches \(a\) equals the function's value at \(a\).

In mathematical speak, this is \( \lim_{x \to a} f(x) = f(a) \). Applying this principle, we showed that our functions \(f(x)\) and \(g(x)\) behaved nicely at the origin by proving their limits equalled their values at that point. This absence of surprises at the origin means you can trust these functions to be predictable and smooth there, much to the relief of anyone who's drawing their curve!
Multivariable Calculus
As you dive into multivariable calculus, you leave the world of single-line graphs and enter the land of surfaces and contours where functions have more than one input. The principles of limits and continuity remain steadfast friends, but now we're considering the behavior of a function as more than one variable approaches a certain set of values.

With multivariable calculus, our analysis included checking continuity at the origin for functions of two variables (\(x_1\) and \(x_2\)). The exercise above requires a clever mix of visualizing in higher dimensions and mathematical tricks like polar coordinates to streamline the complexity into something manageable. Essentially, multivariable calculus is about generalizing the calculus you know and love to scenarios where you're juggling more balls鈥攎ore variables鈥攁t once.

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Most popular questions from this chapter

Das ideale Gas hat die Zustandsgleichung \(p V=\) \(R T\) mit der Gaskonstanten \(R\). Pr眉fen Sie f眉r dieses System die Beziehung $$ \left(\frac{\partial p}{\partial V}\right)_{T}\left(\frac{\partial V}{\partial T}\right)_{p}\left(\frac{\partial T}{\partial p}\right)_{V}=-1 $$ explizit nach.

Gegeben ist die Funktion \(f: \mathbb{R}^{3} \rightarrow \mathbb{R}\) $$ f(x, y, z)=\mathrm{e}^{\cos ^{2}\left(x y^{3} z\right)}-\sqrt{\mathrm{e}} $$ Man begr眉nde, warum sich \(f(x, y, z)=0\) in einer Umgebung von \(P=\left(x_{0}, y_{0}, z_{0}\right)=\left(\pi, 1, \frac{1}{4}\right)\) lokal nach \(z\) aufl枚sen l盲sst, und berechne dort die partiellen Ableitungen \(z_{x}\left(x_{0}, y_{0}\right)\) und \(z_{y}\left(x_{0}, y_{0}\right)\)

Bestimmen Sie mithilfe des Newton-Verfahrens eine N盲herungsl枚sung des Gleichungssystems $$ \begin{aligned} &\sin x \cos y=0.1 \\ &x^{2}+\sin y=0.2 \end{aligned} $$ die in der N?he von \(x_{0}=y_{0}=0\) liegt (zwei Iterationsschritte),

Man bestimme und klassifiziere alle Extrema der Funktion \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) $$ f(x, y)=(1+2 x-y)^{2}+(2-x+y)^{2}+(1+x-y)^{2} $$

Man betrachte die Schar aller Strecken von \((0, t)^{\mathrm{T}}\) nach \((1-t, 0)^{\mathrm{T}}\) mit \(t \in[0,1]\) und bestimme die Einh眉llende dieser Strecken.
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