Questions tagged [implicit-function]
The implicit-function tag has no usage guidance.
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1answer
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Implicit equation $\ln(\frac{x}{y})-y=1$ to rectangular equation not in terms of $W(x)$
Backstory and Other Info
I'm not sure if this is possible, I'm currently a precalculus student and have a very limited understanding of much of any of this.
However, I do like to go on WolframAlpha ...
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0answers
23 views
Understanding logical implication and alternate proofs
I just don't get exactly how logical implications work.
When using truth tables (albeit impractical for large amounts of variables), it can be quite simple to show truths equate for something such as ...
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1answer
49 views
Implicit Differentiation - Different Approaches 2
Given is the function $F(x,y,z)=x^2+y^3-z$.
Determine the Jacobian matrix $Dz$ in $P=(1,1,2)$ using implicit differentiation.
My idea is to calculate $\frac{∂z}{∂x}$ in $P(1,1,2)$ and $\frac{∂z}{∂y}...
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1answer
35 views
Implicit Differentiation - Different Approaches
Given is $F(x,y)=ye^{3x}-2x^2=0$
I was asked to calculate $y’$ using implicit differentiation.
I know that $y’=-\frac{Fx}{Fy}=-\frac{\frac{∂F}{∂x}}{\frac{∂F}{∂y}}.$
So I obtained: $y’(x)=-\frac{3ye^...
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14 views
if $x^*$ and $A^*$ maximize $px - c_x x - c_a A$, then are $p(A^*)x^* - c_xx^* - c_aA^*$ and $px^*(A) - c_xx^*(A) -c_aA^*$ equivalent
where $b$ and $c$ are strictly positive constants.
Is the statement in the question correct, perhaps for some specific $x(A)$?
The question is motivated by the following paragraph from an economics ...
3
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1answer
40 views
$y=f(x) \in C^1$ is defined implicitly by $ax + by = f(x^2+y^2), f'(x) = ?$
Problem
$y=f(x) \in C^1$ is defined implicitly by $ax + by = f(x^2+y^2)$. $a$ and $b$ are constants . $f'(x) = ?$
Analysis
The answer for this exercise given by my teacher is $$f'(x) = \frac{2xf'(x^...
1
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1answer
26 views
Calculating the second order partial derivatives of $z(x,y.)$. [closed]
If we know that $z_{x}^{'} = \frac{yz}{z^{2} - xy },$, how can I calculate the second order partial derivative with respect to x, knowing that the final answer should be (as given at the back of the ...
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1answer
19 views
Calculating the first order partial derivatives of $z(x,y.)$.
My function is $z^3 - 3xyz = 1$ and I calculated $z_{x}^{'}$ and I got $z_{x}^{'} = \frac{yz + yy^{'}z}{z^{2} - xy }$. but the answer at the back of the book is $z_{x}^{'} = \frac{yz}{z^{2} - xy },$ ...
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35 views
Prove that $\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$ given a function [duplicate]
I am trying to solve the following exercise. I have the feeling that it is a very simple one, but I am lost. I do not know where to start, because you do not have an specific function. The exercise ...
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34 views
Implicit differentiation in multivariable calculus
What I don't understand is the disconnect between the reasoning for implcit derivation in single variable and multivaraible calc.
in single variable calc they define say $y = f(x)$ for a small ...
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0answers
15 views
Implicit functions and partial derivatives
I am trying to solve the following exercise about implicit functions and their partial derivatives:
If $u=x+y+z$, $v=x^2+y^2+z^2$ and $w=x^3+y^3+z^3$ prove that:
$$\frac{\partial x}{\partial u}=\left(...
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16 views
Total Differential of an Equation
I want to find the total differential of an equation which has been defined as:
$ Y = C((1-t)Y, M/P)$ where $t$ is a parameter and $M$,$P$, and $Y$ are variables. And Y is a function of C which in ...
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2answers
46 views
Implicit form for a logarithmic spiral
I was wondering if there is an implicit form for a logarithmic spiral. For example, if
$$ x=e^{-t}\cos(t)\\y=e^{-t}\sin(t)$$
we can write $x^2+y^2=e^{-2t}$ and $y/x=\tan(t)$ which yields $$x^2+y^2=...
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2answers
48 views
equation for an hexagon with rounded corners
I know a simple equation for a squircle, $x^4+y^4=a^4$
What would be the equation for an hexagon with rounded corners?
In the equation, how can I take in account for the angle of rotation of the ...
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71 views
Convergence of an implicit series
I have an implicit solution for variable $y$ as a function of time $t$ and two other variables $y_0$ and $y_f$ which are initial and final values, respectively.
$$
tC=\frac{1}{y_f^2}\ln\left(\frac{y}{...
1
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1answer
61 views
How can I solve $x = e^{a+bx} + c?$
I need to solve this implicit equation for a physical system. I know that the similar equation $x = xe^x$ (solved with the Lambert W-function) doesn't have real roots, but since this equation has ...
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0answers
18 views
What is the class of implicit surfaces for which the distance to an external point is returned by its equation?
Let's define an implicit surface as the set of all points for which a function f : R3 -> R equals 0. A sphere around origin, for example, can be defined as ...
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3answers
43 views
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 ...
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1answer
30 views
array of $x$ and $y$ in an implicit plot in Matlab
If I use implicit plot of a relation (say $x\cdot y+\sin(x+y)=0$ ) and obtain a plot. How do I get the list/array of the values of $x$ and $y$ plotted?
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Finding random points and the $x$and $y$ intercepts of two implicit curves in Matlab
I plotted two implicit functions ($f_1$ and $f_2$) in Matlab using the function
fimplicit(f1,[0 1 0 10])
fimplicit(f2,[0 1 0 10])
and I had the following figures
...
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0answers
44 views
Study of an implicit function
The problem consists of two questions :
The first asks to prove that for any real number $x$, there exists a unique real numbers $t$ such that
$$t_x^3x^2+t_x+x=0$$
I'm having problems with the ...
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0answers
32 views
Continuity of implicit function $y-\epsilon \sin{y} =x$
Suppose $0<\epsilon<1$ such that $$y-\epsilon \sin{y} =x$$
is defined as a continuous function for $x$, that is, $y=y(x)$ satisfies
$$y(x)-\epsilon \sin{y(x)} = x$$
Prove that $y(x)$ is ...
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0answers
23 views
normal vector to a hypersurface reference
I'm three years university student and I saw the coarea formule and some others (green, superficial measure, green-gauss,...) so I need to compute normal vector to hypersurface.
My semester will ...
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1answer
44 views
Given a cartesian equation get points in the plane
I hope this make sense. I'm trying to understand (I'm very newbie with curves) how could I get points from a cartesian equation.
For example, given $(x^2+y^2)^2-2a^2\cdot(x^2-y^2)-a^4+c^4=0$ that is ...
6
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1answer
161 views
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 ...
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2answers
45 views
Find $\frac{\partial y}{\partial z}$ of the surface $g(s,t)=(s^2+2t,s+t,e^{st})$ near $g(1, 1) = (3, 2, e)$.
Consider the surface given by $g(s, t) = (s^2 + 2t, s + t, e^{st})$.
Think of $y$ as a function of $x$ and $z$. Find $\dfrac{\partial y}{\partial z}(3,e)$ near $g(1, 1) = (3, 2, e)$.
really ...
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1answer
71 views
How can we find minimum radius of circle which contains $\arctan^2(x)+\arctan^2(y)=a$?
How can we find minimum radius of circle which contains $\arctan^2(x)+\arctan^2(y)=a$, where I think $a<\pi/2$?
For example I have some plots from WolframAlpha and I see it depends on $a$.
But ...
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1answer
57 views
How can we find some radius of circle so $-x\arctan(x)+0.2x-y\arctan(y)+0.9y=0$ will be fully inside this circle?
If he have this region
$$
\begin{align}
\ -x\arctan(x)+0.2x-y\arctan(y)+0.9y=0\\
\end{align}
$$
How can we find some $R$ (maybe minimum) so this region will fully inside this circle $(x-a)^2+(y-b)^2=...
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15 views
How many points are required to fix a given curve from a locus
Say I have a locus on the $x,y$ plane defined by the parametric implicit function $$F(x, y, p_1, p_2, ..., p_n)=0$$ where $p_1, p_2, .., p_n$ are parameters.
I now want to constrain $F$ to pass ...
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1answer
87 views
Implicit functions and related differential equations
I'm seeking guidance in derivation of implicit equation solutions to second degree differential equations.In the example below, differentiating twice just produced a tangle of terms which did not ...
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1answer
53 views
Lagrange inversion formula example unclear
The following example is from De Bruijn's Asymptotic methods in analysis (page 24). The considered equation is
$x^t = e^{-x}$
The author wants to transform the equation into the form: $w=z/f(z)$, ...
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1answer
86 views
How to write a system of ODEs in proper state-space form?
I have the following system of differential equations:
$\dot{x}_1=f_1\left(\dot{x}_1,x_1,x_2\right)\\
\dot{x}_2=f_2\left(\dot{x}_1,x_1,x_2\right)$
where $\dot{x}_1$ and $\dot{x}_2$ are time ...
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83 views
The solution of ODE is not the same as the implicit equation it was used for.
Lets say i have a variable named $p$. With this variable i can calculate the variables $\frac{dU}{d\varphi}$ and $\frac{dQ_{w}}{d\varphi}$ and with equation shown below also $\frac{dQ_{b}}{d\varphi}$.
...
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2answers
70 views
Is there a real function $f$ on $(0..1)$ such that $x·f·\log f = 1$?
I’m looking for a real-valued function $f$ on $(0..1)$ such that $x · f · \log f = 1$. Is there such function? Is it integrable on $(0..1)$?
Why? This could yield an integrable function $f$ such that ...
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1answer
19 views
Function describing trajectory of a point on a rolling circle
Let us take a 2D unit circle $C$ with center in $M=(0,1)$ and fix a point on $C$, say $A=(0,0)$. Now we start to unroll the circle to the right and watch how $A$ moves; firstly, it turns around $M$ on ...
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0answers
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Examples of smooth implicit curves and surfaces
I am currently constructing a method to approximate implicitly given plane curves and surfaces, which are smooth and single-sheeted.
Now I have finished writing a Matlab function doing all the ...
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2answers
76 views
Integration of a function provided as an implicit function [closed]
Let a differentiable function $f$ satisfies the functional rule $$f(xy) = f(x) + f(y) + xy - x -y $$ for all values of $x,y > 0$ and $f'(1)=4$.
Based on this statement there were three questions:
...
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2answers
185 views
Solving implicit equations in R [closed]
I have an equation which looks like this:
$$\frac{1}{\sqrt{𝑥}}=−2\log\left(𝐴+\frac{𝐵}{\sqrt{𝑥}}\right)$$
where $A$ and $B$ are constants
How do I solve this in $\mathbb{R}$?
Can this be solved ...
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0answers
125 views
Solving for an implicit function numerically?
I am trying to come up with a systematic way for solving an implicit "functional" equation. That is, given
$$g(x) = f(x + a \cdot f(x))$$
how would one (numerically) recover the functional form of $...
3
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2answers
153 views
How to find the self-intersection point of $x^x=y^y(x,y>0)$? [duplicate]
As the figure below shows, the graph of the implicit function $$x^x=y^y,(x,y >0)$$ composes of a straight line and an arc, which of the two have an intersection point $P$.
How to find the ...
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2answers
505 views
Integration of implicit functions [closed]
As we know an implicit function $f(x,y)$ is an expression containing $x$ and $y$ such that none of them can be seperated out.
So my question is how can we integrate or simply find area under an ...
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2answers
63 views
Evaluating implicit functions numerically
A simple question.
Are there any methods on how to evaluate an implicit function f(x,y) = 0 numerically?
Does one fix one of the variables and then use newtons method or similar on f(x0,y) and then ...
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0answers
50 views
Tangent plane of implicit surface from normal derivative
Let $f : R^3\rightarrow R$ be the implicit equation of a surface embedded in $R^3$, where the surface is defined by the $0$-isosurface $\left\lbrace
(x,y,z)\in R^3,\ f(x,y,z)=0 \right\rbrace$.
I ...
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1answer
30 views
Bernoulli's Lemniscate: Extremes and Intersection with the axis
Given the equation:
$$(x^2+y^2)^2 - 2p^2(x^2 + y^2) = k^4 - p^4$$
Determine the extremes and intersections with the axis.
I know it requires the use of implicit functions but I'm not sure how to ...
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1answer
74 views
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 ...
1
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1answer
42 views
Solve for $x(t) : x(t)= b+at+ ∫ ( ∫\sin (wt-kx(t)) \, dt) \, dt$
$$ x(t)= b+at+\int \left (\int \sin (wt-kx(t)) \,dt \right ) dt$$
How do I extract $x(t)$ from the above equation. Any hint? Or is this equation has been already solved?
Thanks in advance for any ...
1
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1answer
103 views
Transcendental and implicit functions
Def. An implicit function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments). For example,
$$F(x,y)=0$$$$...
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0answers
23 views
Analysis, implicit functions
Let $X=(x_0,y_0)=(1,1) \in \mathcal{R}^2$. Prove that there exists $\rho>0$ and continuously differentiable functions $u,v,w: B_{\rho}(X) \to \mathcal{R} $ such that
$u(X) = 1, v(X) = 1, w(X) = -...
3
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1answer
65 views
Parametrizing $(2x+y)^2(x+y)=x$
I need to find $x=x(t)$ and $y=y(t)$ so that the implicitly defined curve on $\mathbb R^2$ $$(2x+y)^2(x+y)=x$$ is converted into an explicit function of the parameter $t$ that can be analysed using ...
2
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1answer
158 views
Is it possible to execute line integrals of non-conservative vector fields on curves defined by implicit relation such as $\sin(xy)=x+y$?
I want to execute the line integral (analytically) of a vector field over the curve defined by implicit function $\sin(xy)=x+y$ for some $x=a$ & $y=c$ to $x=b$ & $y=d$.
The difficulties I ...
