Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [implicit-function]

This tag is for questions relating to "implicit function", a function or relation in which the dependent variable is not isolated on one side of the equation.

The notion of implicit function is of utmost importance while solving real-life problems.

A function in which the dependent variable, and independent variable(s) are not separated (isolated) on opposite sides of the equality are known as implicit function.

i.e., If it only has the form $~f(x,y)=0~$, then it is implicit.

e.g., Take $~x^2 + xy~ – y^2 = 1~$, then $~f(x,y)=x^2 + xy~ – y^2-1=0~$.

Reference:

https://en.wikipedia.org/wiki/Implicit_function

274 questions
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Dog bone-shaped curve: $|x|^x=|y|^y$

EDITED: Some of the questions are ansered, some aren't. EDITED: In order not to make this post too long, I posted another post which consists of more questions. Let $f$ be (almost) the implicit curve$$|x|^x=|y|^y$$ See the graph of the…
Tony Ma
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3 answers

Explaining the graph of $\sin(x^2) + \sin(y^2) = 1$

I had to plot the graph of the implicitly defined function $\sin^2 x + \sin^2 y = 1$ in an exam. This is not particularly difficult, but it got me wondering what the graph would look like when the exponent is taken inside, viz. $$\sin(x^2) +…
xryophile
  • 697
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2 answers

Implicit function equation $f(x) + \log(f(x)) = x$

Is there a function $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that $$ f(x) + \log(f(x)) = x $$ for all $x \in \mathbb{R}_{>0}$? I have tried rewriting it as a differential equation by introducing $g(x) := f(x) + \log(f(x)) - x$, which is…
Strichcoder
  • 2,015
9
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4 answers

Implicit differentiation - why can you substitute the expression?

One of the popular proofs of the derivative of ln(x) is by implicit differentiation. $$ \begin{align*} y &= \ln x \\ e^y &= x \tag{2} \\ e^y dy &= dx \\ \frac{dy}{dx} &= \frac{1}{e^y} \tag{4} \\ \frac{dy}{dx} &= \frac{1}{x}…
user19646426
  • 99
9
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1 answer

How to Paramaterize $2\cos(x/2)\cos(y/2)=1$?

$2cos(x/2)cos(y/2)=1$" /> This curve of $2\cos(x/2)\cos(y/2)=1$ looks like a circle squished in from the sides and top and bottom. I know how to parameterize the curve by dividing it into four 90 degree segments, but second derivatives of the…
Roger Wehage
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2 answers

Finding the asymptotic behavior of a function defined implicitly

I encountered this when trying to solve a number theory problem. I have two variables $x,y$ related by $$(\ln(x))^{y+1}=(\ln(xy))^y$$ and I want to know how big $y(x)$ is as $x\to\infty$. Ideally I want to know that $y(x)\sim\ln(x)$ or $y(x)\sim…
Derivative
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Implicit derivative, implicit integral?

An undergraduate student of mine asked me last year the following question. Let $f:\mathbb R^2\to \mathbb R$. The equation $f(x,y)=0$ defines implicitly a function $y:\mathbb R\to\mathbb R$ and we can express its derivative in terms of the partial…
Tal-Botvinnik
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7
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1 answer

Finding the area surrounded by a part of the implicit equation $\sin (y^x) = \cos (x^y)$

Finding the area surrounded by the part of the implicit equation $\sin (y^x) = \cos (x^y)$ such that $y\le 2n-x$ where $n$ is the solution to $n^n=\frac{\pi}{4}$ where $n<0.5$ bounded by the $x$ and $y$ axes. This problem has no context because I…
Dylan Levine
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7
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3 answers

What is really the TRUE definition of an implicit function?

First of all I would like to say that I have already found similar questions on stack exchange but somehow my confusion regarding the definition of an implicit function still linger. The title says it all, but here's the question: what is really the…
mathadic
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3 answers

Can a 'closed' curve be a function?

For a 'closed' curve (I don't know if there is such in mathematics called closed curve, but I mean a curve which is 'closed') e.g.1)It is well-known unit circle with equation$$x^2+y^2=1$$e.g.2)It is the function $f$ which is in my another post.I…
Tony Ma
  • 2,320
7
votes
1 answer

Length of an implicit curve

The topography of an empty lake is given by a function f(x,y)=z. Each spring the lake is filled by water. The waterstand is given by a constant z=c, where c is a constant. How long is the shore of the lake? What I am actually asking is: Is there…
Max Mathebau
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3 answers

What exactly is an implicitly defined function?.

So I've just started to get into some calculus and I recently came across the topic of implicit differentiation. I am extremely confused on what implicit functions are and there is very little information on what exactly implicitly defined…
The homeschooler
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3 answers

How or why are implicit functions actually functions?

I'm a high schooler and today my teacher taught us the implicit differentiation, in which he gave us a very brief explanation of implicit function. I didn't quite get it at that time so I decided to look it up a little and honestly, the concept of…
William
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1 answer

How can we find some radius of circle which fully contains $x\arctan(x)-ax+y\arctan(y)-by=0$?

How can we find some radius of circle with center at origin which contains $x\arctan(x)-ax+y\arctan(y)-by=0$, where $\pi/2>a>0$ and $\pi/2>b>0$. I'm not sure how can we prove that these inequalities should hold so we can have closed curve:…
Tag
  • 139
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6
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3 answers

Does conic-section has a parametric form? Is it possible to draw it using the implicit form?

There is a conic-section with this implicit form: $$ f(x,y) = ax^2 + 2bxy + cy^2 + 2dx + 2ey + f = 0 $$ I would like to write a program, which draws it, but I have no idea where to start. I could use a brute-force method, calculate it for each…
Iter Ator
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