# Search results for query: *

1. ### Proof involving convex function and concave function

A subset E of X is called convex if, for any ##x,y \in E## and ##t \in (0,1)## then ##(1-t)x + ty \in E##. So by the inequality I wrote since ##\alpha f(x) + (1-\alpha)f(y)## is contained in the set it is convex?
2. ### Proof involving convex function and concave function

Homework Statement [/B] Let X be a vector space over ##\mathbb{R}## and ## f: X \rightarrow \mathbb{R} ## be a convex function and ##g: X \rightarrow \mathbb{R}## be a concave function. Show: The set {##x \in X: f(x) \leq g(x)##} is convex. Homework Equations [/B] If f is convex...
3. ### One to one and onto.

Okay, I take from this because \sqrt{x_1}= \sqrt{x_2}, \sqrt{x_1}^2= \sqrt{x_2}^2, so x1=x2. So this function is one to one because I can prove that . Correct?
4. ### One to one and onto.

Homework Statement I am supposed to prove or disporve that ##f:\mathbb{R} \rightarrow \mathbb{R}## ##f(x)=\sqrt{x}## is onto. And prove or disprove that it is one to one Homework Equations The Attempt at a Solution I know for certain that this function is not onto given the codomain of real...
5. ### Arbitrary Union of Sets Question

I updated it to define An.
6. ### Arbitrary Union of Sets Question

Homework Statement For each ##n \in \mathbb{N}##, let ##A_{n}=\left\{n\right\}##. What are ##\bigcup_{n\in\mathbb{N}}A_{n}## and ##\bigcap_{n\in\mathbb{N}}A_{n}##. Homework Equations The Attempt at a Solution I know that this involves natural numbers some how, I am just confused on a...
7. ### Expressing the existence of irrational numbers

Ah I understand, this makes more sense to me now. Thank you.
8. ### Expressing the existence of irrational numbers

But one question: since i had ##\neg[(p\mid q)=x]## when the negation moves inside the expression it becomes: ##[(p\nmid q )\ne x]## Correct?

10. ### Expressing the existence of irrational numbers

Homework Statement Express the following using existential and universal quantifiers restricted to the sets of Real numbers and natural numbers Homework Equations The Attempt at a Solution I believe the existence of rational numbers can be stated as: ##(\forall n \in \Re)(\exists p,q \in...
11. ### Universal and Existential Qualifiers

Homework Statement Express the following statement using only quantifiers. (You may only use the set of Real and Natural Numbers) 1. There is no largest irrational number. Homework Equations ##\forall=## for all ##\exists##=there exists The Attempt at a Solution I express the existence of...
12. ### Intro to abstract math—basic notation

Homework Statement Simplify the following statement as much as you can: (b). ##(3<4) \wedge (3<6)## Homework Equations ##\wedge= and## The Attempt at a Solution I figured that I could just write this as ##3<4<6##, but then I considered what if I didn't know that ##4<6## If it was just...
13. ### Optimization of ellipse surrounding a circle

How about since ##\frac{x^2}{a^2}+\frac{y^2}{b^2}=1## ##(x-1)^2+y^2=1## Can I set each side equal to each other or should I solve for y^2 of the circle equation to plug into the ellipse equation.
14. ### Optimization of ellipse surrounding a circle

I know this problem is emulative of https://www.physicsforums.com/threads/optimization-minimize-area-of-an-ellipse-enclosing-a-circle.270437/ this one however I am just getting into multivariable differentiation so this is very confusing to me.
15. ### Optimization of ellipse surrounding a circle

I don't think I'm following that relationship either. How am I supposed to know the relationship between the ellipse and the circle.
16. ### Optimization of ellipse surrounding a circle

Also how do we know that the circle and ellipse are in the range [0,2] for x values?
17. ### Optimization of ellipse surrounding a circle

How exactly would I solve an auxiliary optimization to turn into a single restraint?
18. ### Optimization of ellipse surrounding a circle

Homework Statement Consider the ellipse ##\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1## that encloses the circle ##x^{2}+y^{2}=2x##. Find the values of a and b that minimize the area of the ellipse. Homework Equations ##Area=ab\pi## The Attempt at a Solution I begin by completing the square...
19. ### Finding the distance from origin to a tangent line

Ok so ##\frac{\sqrt[3]{y}}{\sqrt[3]{x}}x_{0}+y_{0}+C=0## ##C=-\frac{\sqrt[3]{y}}{\sqrt[3]{x}}x_{0}-y_{0}## correct? So the beginning of this equation would be the magnitude of ##\frac{\sqrt[3]{y}}{\sqrt[3]{x}}x_{0}+y_{0}-\frac{\sqrt[3]{y}}{\sqrt[3]{x}}x_{0}-y_{0}## Over ##\sqrt{\left (...
20. ### Finding the distance from origin to a tangent line

I know it shouldn't be that hard, but I'm having such problems. So if the constant C is not zero how do I find it? From what I understand you saying I move all terms to one side to get: ##(y-y_{0})+\frac{\sqrt[3]{y}}{\sqrt[3]{x}}(x-x_{0})+c=0 ## So C is...
21. ### Finding the distance from origin to a tangent line

I still don't think I am understanding. Using the equation from wikipedia for distance: ##\frac{\left | ax_{0}+by_{0}+c \right |}{\sqrt{a^2+b^2}}## I can define "a" as the value: ##\frac{\sqrt[3]{y}}{\sqrt[3]{x}}## and "b" as the value: 1 Because in the slope form equation I have...
22. ### Finding the distance from origin to a tangent line

Ok the point slope equation would be: ##y-y_{0}=-\frac{\sqrt[3]{y}}{\sqrt[3]{x}}\left ( x-x_{0} \right )##, Am I supposed to solve for Y0 and X0? How will this form be applied to finding the distance between the points?
23. ### Finding the distance from origin to a tangent line

If I put my equation in point slope form noting that the initial point is at (0,0) and I'm going to P(x,y) I would get y-0= ##\frac{x^{\frac{1}{3}}}{y^{\frac{1}{3}}}(x-0)## Which would be the same thing as y=##\frac{x^{\frac{1}{3}}}{y^{\frac{1}{3}}}x## But then how do you find distance when...
24. ### Finding the distance from origin to a tangent line

Ok using this equation, I went ahead and tried to find the distance to the specific point. Because the beginning is the origin. I just said that y0=Y-0 and x0=X-0 making the new equation for the line: ##y=\left ( -\frac{\sqrt[3]{y}}{\sqrt[3]{x}} \right )x##. And putting it into Ax+By+C form I...
25. ### Finding the distance from origin to a tangent line

So I did what you said and defined the tangent line as ##y=\left ( -\frac{\sqrt[3]{y_{o}}}{\sqrt[3]{x_{o}}} \right )x+b##. Does that look correct? And if so then how would I solve for "b" or even apply the distance formula to an equation defined this way?
26. ### Finding the distance from origin to a tangent line

Homework Statement Find the distance between the origin and the line tangent to ##x^\frac{2}{3}+y^{\frac{2}{3}}=a^{\frac{2}{3}}## at the point P(x,y) Homework Equations [/B] Distance= ##\frac{\left |a_{0}+b_{0}+c \right |}{\sqrt{a^{2}+b^{2}}}## The Attempt at a Solution To begin I find...
27. ### Curvature and Tangent Line Distance Relationship

For differentiating the first equation I get##-\frac{\sqrt[3]{y}}{\sqrt[3]{x}}##. And when I do the second derivative, and I end up with ##\left ( \frac{1}{3} \right )\sqrt[3]{\frac{y}{x^4}}##. Using these values I get ##\frac{3\left ( \frac{x^\frac{2}{3}+y^{\frac{3}{2}}}{x^{\frac{2}{3}}}...
28. ### Curvature and Tangent Line Distance Relationship

When differentiating is it okay if I say that a^(2/3) will be a constant?
29. ### Curvature and Tangent Line Distance Relationship

Does anyone have a hint? I have been working on this problem in between my breaks at school, and can't understand where to start.
30. ### Curvature and Tangent Line Distance Relationship

Homework Statement Let T be the tangent line at the point P(x,y) to the graph of the curve ##x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}, a>0##. Show that the radius of curvature at P is three times the distance from the origin to the tangent line T. Homework Equations R=1/K...
31. ### Curvature and radius of curvature of a cartesian equation

I see what you mean. (I think) So I did what you said. ##3.56dx^{2}=(\frac{89 \sqrt{89}}{120})^{2}## ##dx=\frac{89}{24}## ##4-\frac{89}{24}=\frac{7}{24}## Using ##\frac{89}{24}##. I plug into dy to distance from y=1. ##dy=(\frac{89}{15})##. ##1+(\frac{89}{15})=\frac{104}{15}## The final...
32. ### Curvature and radius of curvature of a cartesian equation

Ok, thank you... I am not very confident with my reasoning, I feel like it could be right, but at the same time worried that I am making things up. Using the curvature equation I found that the radius of stated circle in the problem would be ##\frac{89\sqrt{89}}{120}##. Using this radius I...
33. ### Curvature and radius of curvature of a cartesian equation

Am I allowed to repost a question if I feel like I didn't get the help I needed?
34. ### Curvature and radius of curvature of a cartesian equation

I think this confusion comes from my lack of perentheses. My fault again. Or not what result are you getting? I check and still get the same thing ##\frac{\left (1+\left (\frac{5}{64}x^{\frac{3}{2}} \right )^{2} \right )^{3/2}}{\frac{15}{128}x^{\frac{1}{2}}}## Sorry if the notation is confusing.
35. ### Curvature and radius of curvature of a cartesian equation

So I kind of took what you said and reasoned with it. I noticed that center of the circle in the y direction would just be the radius so I just said (y-6.99)^2 for that part. Now for how much to shift the circle to the right or left I plugged in the value 4 for x and 1 for y so that I could see...
36. ### Curvature and radius of curvature of a cartesian equation

Sorry that was a very silly mistake when I differentiated incorrectly. ##y'=\frac{5}{64}x^{\frac{3}{2}}## ##y''=\frac{15}{128}x^{\frac{1}{2}}## So now ##\frac{((1+\frac{5}{64}x^{\frac{3}{2}})^2)^(3/2)}{\frac{15}{128}x^{\frac{1}{2}}}## Evaluating this at x=4 makes R=6.99 R^2=48.98 I...
37. ### Curvature and radius of curvature of a cartesian equation

Homework Statement A highway has an exit ramp that beings at the origin of a coordinate system and follows the curve ##y=\frac{1}{32}x^{\frac{5}{2}}## to the point (4,1). Then it take on a circular path whose curvature is that given bt the curve ##y=\frac{1}{32}x^{\frac{5}{2}}## at the point...
38. ### Distance between vector and a plane proof

Thank you for the feedback. I will make sure to use latex ask questions about future problems!
39. ### Distance between vector and a plane proof

Homework Statement Homework Equations Since this a proof. Most of the equations I needed were involved in the solution I created. The Attempt at a Solution I believe my proving is valid; however, I was wondering if I might have missed a step or did some math that didn't make sense,
40. ### Conservation of Momentum of astronaut leaving ship

Homework Statement An astronaut is pushed from her ship at a velocity of 2m/s. Her weight including her tool belt is 120kg. Remembering Newtons 3rd law, she takes 5 seconds to decide to detach her belt and throw it away from her. The 20kg tool belt leaves her suit as she throws it in front of...
41. ### Calculating Traction Force for Leg Fracture Treatment

Homework Statement When the thigh is fractured, the patient's leg must be kept under traction. One method of doing so is a variation on the Russell traction apparatus. If the physical therapist specifies that the traction force directed along the leg must be 25N, what must W be? Homework...
42. ### Conceptual Question about Static Friction

Homework Statement Today we were studying a block on an incline and determining the angle needed for the block to start sliding. The question asked is: Why is the equation for the coefficient of static friction independent of the weight of the block? Homework Equations ΣFx=Fapplied-fs=0...
43. ### Coefficient of Kinetic Friction and Variables

Homework Statement This is a theoretical question from my homework. We did a lab in class where we were using motion to see how kinetic friction is affected by different variables. We found the acceleration of a sliding wooden block (sliding on another wooded block), and then used that...
44. ### Integrating Using a Substituation

What I have is ## \sin x+\sin (\frac{\pi}{2}-x)## using the Sum to Product Identity = ##\sqrt{ 2}\cos \frac {(\frac{\pi}{2}-x)}{2}## Putting that into our integral would be: ##\int \frac {\sin (\frac {\pi}{2}-x)}{\sqrt{ 2}\cos \frac {(\frac{\pi}{2}-x)}{2}}\ dx## =##\int \frac {\cos...

so for
46. ### Integrating Using a Substituation

I incidentally made that choice to leave those bounds. I now know that now though.
47. ### Integrating Using a Substituation

Homework Statement Use the substitution ##u=\frac{\pi} {2}-x## evaluate the integral ##\int_0^\frac {\pi}{2} \frac {\sin x}{\cos x + \sin x}dx##. Homework Equations [/B] ##\cos (\frac {\pi}{2}-x)=\sin x## The Attempt at a Solution [/B]I start by plugging "u" into the equation making the...