Points Definition and 1000 Threads

Are inflection points critical points? and what about at the value that f(x) undefined? Is that critical point too?

Does any circle having irrational radius have no lattice points on its boundary ? Extended question: Is there any way to determine the number of lattice points lying on the boundary of a given circle ? *The centres of these circles are all (0,0) *

PROBLEM STATEMENT: I'm looking for a somewhat general method to find the expression for the distance (in \R^2 mortal, euclidean space) between a point in a certain curve and some point outside the line. ATTEMPTS TO SOLVE THE PROBLEM: In the case of the distance between the origin and some...

Classify the equlibrium points of the system-Pls help me, its urgent! Hi , I have my maths exam tomorow and I am not able to understand the concept to classify the equilibrium points of a system.. I will be grateful if anyone could help me with this problem Classify the equilibrium points...

Homework Statement Use implicit differentiation to find y' if (x-y)/(x+y)=(x/y)+1. Now show that there are, in fact, no points on that curve, so the derivative you calculated is meaningless. Homework Equations The Attempt at a Solution I managed to get it into the form: dy/dx =...

Homework Statement Hi guys, I'm having real difficulty with understanding accumulation points. I don' really know why that is since others seem to understand the concept fine but I'm very lost. For example, I'm practicing some questions and one of the is : If S is the set of rational...

Hello everyone, I'm an optometry student who is currently doing Optics 1, and I have a general question about focal points. I can do the math regarding F1, F2, and the nodal ray, and I am decent at drawing the ray traces, but I would like to know more about the actual real world applications...

Is there a clean may to get the fixed points for \begin{alignat*}{9} F - 2B' - cB - \frac{3}{4}AB^2 - \frac{3}{4}A^3 & = & 0 & \quad & \Rightarrow & \quad & B' & = & \frac{1}{2}F - \frac{c}{2}B - \frac{3}{8}AB^2 - \frac{3}{8}A^3\\ 2A' + cA - \frac{3}{4}A^2B - \frac{3}{4}B^3 & = & 0 & \quad &...

Suppose a set of k arbitrary points, x_i, 1<=i<=k, x_i from R^2 are selected from a line. How can it be shown that a weighted barycenter x_o=(o_i*x_i)/(o_1+o_2+...+o_k) also belongs to that line (assume o_i are arbitrary weights)? Does the choice of weights restrict the solutions (ie, a...

Does anyone know where I can find an algorithm for the points of intersection of two ellipses existing with arbitrary center points and rotations and having 0, 1, 2, 3 or 4 points of intersection?

Homework Statement Find the points on the graph y=x^3/2 - x^1/2 at which the tangent line is parallel to y-x=3. Homework Equations The Attempt at a Solution First I found that the derivative of y=x^3/2 - x^1/2 is 1x. I then rewrote the other line as y = 3+x and found the...

Hi all, If a,b,c,d are position vectors of four points A,B,C,D.The points will be coplanar if xa+yb+zc+td=0,x+y+z+t=0,provided x,y,z,t are not all 0,and they are scalars.Is this test needed to show 4 points are coplanar? If we consider two lines joining A,B and C,D then this will give us two...

y=x^5 x=y(y-1)^2 find points of intersect correct to 1 decimal point

So I know that the Lebesgue outer measure of a set of only countably many points is 0. An example of this is the rationals as a subset of the reals.I want to make sure my intuition behind this is correct. The process: Now, if we are going to take the Lebesgue outer measure of the rationals, we...

Hi I used plot digitizer software to pull data points off a picture of a graph and i need to fit a line to the points (and get an equation for the line). I know the data points fit to an s-curve from the picture. I'm thinking excel doesn't do this, or maybe I missed it? I have access to maple...

Homework Statement f(x,y) = xy(9x^2 + 3y^2 -16) Find the critical points of the function and their nature (local maximum, local minimum or saddle) Homework Equations The Attempt at a Solution I have partially differentiated the equation into: fx = 27yx^2* + 3y^3 -16y fy =...

Tricky problem worth 2 points on my final grade :) Homework Statement Homework Equations I honestly don't know how to tackle this. I think it's going to contain ƩF=mg, and some basic kinematic equations. The Attempt at a Solution Where do is start?

Hello everyone! I'm trying to prove that the closure of $A = [-\infty,0)$ is $[-\infty,0]$. So far, I have proved that all points in $[-\infty, 0)$ are limit points of A, then I have proved that $\sup A = 0$, so it is in the closure, so $[-\infty, 0]$ subsets the closure. But how do I know...

Homework Statement What is the minimum number of points a sample space must contain in order that there exists n independent events A_1, ..., A_n , none of which has probability zero or one? Homework Equations None at this time The Attempt at a Solution I was thinking that if each A_i...

The equation for the number of possible connections between n points on a 2D plane is (n-1)*(n/2). What is the equation for the number of possible connections between n points on a 3D plane? Is it the intregal of (n-1)*(n/2)?

Homework Statement Let A,B,C be three sets of complex numbers as defined below A = {z:|z+1|\leq2+Re(z)}, B = {z:|z-1|\geq1} and C=\left\{z: \frac{|z-1|}{|z+1|}\geq 1 \right\} The number of point(s) having integral coordinates in the region A \cap B \cap C is Homework Equations...

I am trying to use Mathematica to fit a curve to these data points ListPlot[{{2*Pi/(1 - 0^2/16), 5 (3 - Log[2])}, {2*Pi/(1 - .05^2/16), 10 (3 - Log[2])}, {2*Pi/(1 - .1^2/16), 15 (3 - Log[2])}, {2*Pi/(1 - .15^2/16), 20 (3 - Log[2])}, {2*Pi/(1 - .2^2/16), 25 (3 - Log[2])}...

Hello everyone! I'm asked to find a set that is bounded and that has exactly two limit points, now this is how I am thinking. Consider the set $A_n = [0,\frac{1}{n}) \cup(2-\frac{1}{n},2]$, if $A_1 = [0,1)\cup(1,2]$, $A_2=[0,1/2)\cup (3/2,2]$. If I let $n$ grow indefinitely, I will have only...

Homework Statement Find the local max/min or saddle points of f(x,y) = (x-y)(1-xy) Homework Equations The Attempt at a Solution I expanded the equation to f(x,y) = x-y-(x^2)y+xy^2. Then I found the partial derivatives of the function. fx = 1-2xy +y^2 fy = -x^2-2xy I'm...

Homework Statement Let's say that we have a second order polynomial function, and we know all of the points where it intersects with the x and y axis. Ex: (-2; 0), (0; 2), (1; 0) How does on determine the ax^2+bx+c polynomial form based on that? Homework Equations - The Attempt at...

Find the points of discontinuity: f(x) = x + 1 , for x < 1 and 1/x for x ≥ 1? ^ supposed to be a piece-wise function. State whether f is left- or right-continuous at each point of discontinuity. I'm having difficulty figuring this out... please help?

Homework Statement I have to prove that the supremum and infimum of a set of limit points of a a sequence {an} are themselves limit points. Homework Equations The Attempt at a Solution I have been messing around with definitions but have not made any progress. Please help...

Homework Statement Here is the question along with part of the solution. I am online concerned about finding the force at C. The Attempt at a Solution For my solution I decided I would sum the moments about point B. However I ended up getting the wrong answer for the force at...

Homework Statement Find the points nearest and furthest from the origin on the intersection of a plane with a paraboloid. Plane: x+y+2z=30 Paraboloid: z=x^{2}+y^{2} Homework Equations The Attempt at a Solution Obviously the first step is to find the equation of the...

Homework Statement A projectile is fired with velocity v0 and passes through two points, both a distance h above the horizontal. The angle of the barrel of the gun is adjusted for the maximum range, find the horizontal separation of the two points. Homework Equations Max range for projectile...

Hey everyone! Recently got a question in maths which asks: "Use integral calculus to find the equation of the quartic that has stationary points of inflection at (1, 23) and (3, 15) and a y-intercept of 24" This means that the second derivative has the form (as inflection points are...

Homework Statement a particle of mass m is contrained to lie on along a frictionless,horizontal plane subject to a force given by the expression F(x)=-kx. It is projected from x=0 to the right along the positive x direction with initial kinetic energy T(o)=1/2kA^2, k and A are positive...

Homework Statement attached Homework Equations The Attempt at a Solution The first two were pretty simple, but I'm having trouble grasping why my guess for #3 was actually correct. Why are we ignoring the parallel battery/resistor combo up top and just adding the voltages of the...

Homework Statement I have four questions based on this principal that I'm struggling with. 1. The Potential at the surface of a sphere of radius R is given by V = kcos(3θ). K is a constant. (Assume no charge inside or outside the sphere). a) Find the potential inside and outside the...

Consider the equation dy/dt = alpha - y^2 a) Find all of the critical points. How does it change as alpha < 0, alpha = 0 or alpha > 0? b) In each case of different alphas, consider the graph of f(y) vs y and determine whether each critical point is asympototically stable, semistable, or...

Can an ellipse's focal points be outside the ellipse? I have tried googling this, but without any good explanations or answers. According to my calculations, the focal points of the ellipse defined by x^{2} + \frac{y^{2}}{4} = 1 are (-\sqrt{3},0) (\sqrt{3},0)) . I maybe wrong of course...

This isn't really hw, just me being confused over some examples. I have 'learned' the basic definitions of neighborhood, limit point, closed, and closure but have some trouble accepting the following examples. 1. For Q in R, Q is not closed. The set of all limit points of Q is R, so its...

I would like to find a 3D coordinate of a point (X) on a circle, knowing two points on the circle (P1,P2) which represent the circle diameter and another point (P3) NOT on the circle but on its plane. Also known the length of the line from P2 to X, for example d. Another thing that may help, the...

Homework Statement Let X be a metric space and let E be a subset of X. Prove that E' is closed and that E and E-closure have the same set of limit points. Homework Equations The Attempt at a Solution I have proven that E' is closed. Now, from the definition of a closed subset of a metric...

Hi there, I require some advice on how to interpolate more data points on a bump profile. Basically i have a curve defined be twentyfive (x,y) data points. The bump is meant to accurately represent half of an aerofoil. I would like to accurately interpolate more data points (lets say up to...

Let F : \mathbb{R}^2 \rightarrow \mathbb{R}^2 be the map given by F(x, y) := (x^3 - xy, y^3 - xy). What are some singular points? Well, I know that for an algebraic curve, a point p_0 = (x_0, y_0) is a singular point if F_x(x_0, y_0) = 0 and F_y(x_0, y_0) = 0. However, this curve is not...

Homework Statement Prove that if E \subset X and if p is a limit point of E, then there is a sequence \{p_{n}\} in E such that p=\lim_{n\to\infty}\{p_{n}\} (I presume that there is an invisible "p_{n} \rightarrow p implies that" at the beginning of the sentence). Homework Equations -...

All complex numbers of the form $(1/n) + (i/m)$, $n,m\in\mathbb{Z}^+$.Complex numbers aren't well ordered so how is this treated?

All numbers of the form $(-1)^n + (1/m)$, $n,m\in\mathbb{Z}^+$. Is this true $(-1)^n + (1/m) = (-1,1)$? If so, the accumulation points are $x\in [-1,1]$ and the set is open.

Accumulation points of rationals and open or closed. I know the accumulation points are all real but I don't understand why. The set is neither open nor closed to but I don't truly see it. Can someone explain both?

All numbers of the form $1/n$, $(n = 1,2,3,\ldots)$.$1/n = (0,1)$ The accumulation points are $x\in [0,1]$. This set is open.

Homework Statement Homework Equations F=Keq1q2/r2 The Attempt at a Solution Using Coulomb's Law, here's what I've worked out: FAB=0.593N FAC=0.309N Fx=0.309 - 0.593cos60°= 0.0125 Fy=0.593sin60° =0.514 √(0.01252 + 0.5142) = 0.514 N/C and θ = tan-1(0.514/0.0125) = 88.6° I'm not sure why...

Homework Statement Find equation of plane containing points: (1,1,5),(3,5,3),(8,8,1),(10,2,2),(18,6,-1),(-1,-3,6) Homework Equations Find 2 vectors given 3 points, using a common point. The cross product of these 2 vectors will be the normal vector of the plane. Use normal vector...

I was trying to compare some solubilities (in water) and some boiling points, and I could not explain it for some molucules. Also, I have some doubts in the theory itself. Why is 2-pentanone more soluble than pentanoic acid? Why is a ketone more soluble than the respective aldeyde...

Homework Statement Find and characterise the stationary points for F(x,y,z) = x2 + xy + y2 - 2z2 +3x -2y +z The Attempt at a Solution I found fx, fy, fz and let them equal to 0. Solving gives me the critical point (-8/3,7/3,14). From here I'm not sure how to determine the nature...