0
votes
2answers
19 views

proves of parametric curves via parametric equations

Hi could anyone help me with this problem. An astroid is given by the equation $$x^{2/3} + y^{2/3} = 1.$$ Prove via parametric equations that the length of a piece of a tangent line between the ...
3
votes
0answers
62 views

My orbiting body is orbiting about the wrong focus of it's elliptical orbit… why? [closed]

I am coding in c++ and am computing the position of an orbiting body as a function of time. Everything is almost working. I have a nice elliptical orbit. Except, my orbiting body speeds up as it ...
2
votes
1answer
28 views

Determination of a volume.

Consider, in the Cartesian plane, the square Q having vertices in the points $(-1, 0), (1, 0), (0, -1)$, and $(0, 1)$. The sections of a solid with planes orthogonal to $y=0$ are squares having two ...
0
votes
1answer
30 views

Given that the graph of $f$ passes through the point $(1, 6)$ and that the slope of its tangent line at $(x, f(x))$ is $2x + 1$, find $f(2)$.

As in the title - we assume that the graph of $f$ passes through $(1,6)$ (i.e. $f(1) = 6$) and that the slope of its tangent line at $(x, f(x))$ is $2x + 1$ and we are asked to find $f(2)$. How does ...
2
votes
2answers
36 views

Calculus III: Find the points of the curve…

I have to find the points of the curve $$r\left( t \right) =\left( t,{ t }^{ 2 },{ t }^{ 3 } \right) $$ where the osculating plane passes through the point $\left( 2,-\frac { 1 }{ 3 } ,-6 \right)$.
0
votes
1answer
17 views

Coordinates rotation and function change

In the Cartesian coordinates $(x,y)$, I have a vector function $\bar{f}(x)=\hat{x}A\cos(yk)$, where $A$ and $k$ are constants. I make now a 45 degrees rotation (in the same plane) to the new set of ...
1
vote
2answers
46 views

How to parameterize the maximum of a function?

$f(x) := -4(\frac{1}{2}-x)^2+1$ Is it possible to construct a parameterized version of $f$, say $f_a$, which fulfills: $f_a(0) = f_a(1) = 0$ $f_a(a) = 1$ $f_a'(x) = 0 \Leftrightarrow x=a$ $f_a''(a) ...
0
votes
1answer
19 views

Determine Center Point based on 2 separate elipses

First timer here. I've been digging back into my good old maths days but am extremely rusty (beyond belief). I got a really tricky question that i want to determine formula for so that my mate can ...
0
votes
1answer
32 views

Graphs interpretation question

Suppose we have a prarbola $y^2 = 2px$ ....this is in fact $y = \sqrt{2px}$, so we plot it like a square root function, so it has no applied values for less than zero. However I saw in my textbook ...
0
votes
2answers
21 views

Finding the equations of surfaces of revolution

I have the following question: $$\text{Sketch and find the equations of the surfaces formed by}$$ $$\text{i) }x^2 - y^2 + 1 = 0 \text{ about the y-axis}$$ $$\text{ii) }x^2 - 2y^2 + 2a^2 = 0 \text{ ...
1
vote
2answers
48 views

how to calculate the angle between the tangents of the curve?

$y=(-3/2)x$ and $y=(-2/5)x$ intersect the curve $$3x^2+4xy+5y^2-4=0$$ at points $P$ and $Q$ .find the angle between tangents drawn to curve at $P$ and $Q$ .I know a very long method of finding ...
0
votes
1answer
60 views

List of topics for basic calculus (1st,2nd,3rd semester)

I am an computer science student, currently studying in 2nd semester. Therefore my math courses are pretty weak. Although I "aced" them, I still feel I could use some extra basic calculus knowledge in ...
2
votes
3answers
63 views

The maximum from a point outside an ellipse to a ellipse.

In the $xOy$ axes, Assume there is an ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$, and a point $A(0,t)$ ($t$ is a constant )outside the ellipse. Assume $P$ is a point in the ellipse. Find the ...
0
votes
1answer
34 views

Analytic geometry and calculus question

The tangent to the curve $ x^{0.5} + y^{0.5} = a^{0.5}$ ,$(a>0)$ intersects the $x$ axis in point $M_1$ , and the $y$ axis in point $M_2$. Prove that for any point on the curve $|OM_1|+|OM_2| = ...
1
vote
2answers
30 views

Analytic geometry and calculus combined question

Show that the equation for the tangent with the slope $m$, $(m≠0)$ to the parabola $y^2 = 4px$ is $y = mx + \frac{p}{m}$. How this is done? What is the method for proves of this kind?
2
votes
3answers
55 views

Analytic geometry and calculus mixed question

Find the normal equation to the graph $ y = 2x^2$, that goes through the point $ (1,0.25)$. How this is done? Im not even really sure what I'm being asked here...
0
votes
3answers
70 views

Calculus and analytic geometry question

Find the tangent of the angle in which the functions $x^3 $, and $x^2 $ intersect $(x≠0)$ . I find this question to be quite funny since the intersection point has two tangents going to it, with ...
1
vote
2answers
64 views

Finding the point that a normal line goes through

I have been stumped on a homework problem for quite some time and I'm hoping to get some help with it. The line from the origin to the point $(a, f(a))$ on the graph of $f(x) = \frac1{x^2}$ is ...
0
votes
1answer
18 views

Concept of parallelism in analytic terms

Below I cited a passage from Apostol's Calculus. I don't understand how to use the identity to show that two lines with equal slopes are parallel. Concepts such as perpendicularity and parallelism ...
1
vote
1answer
50 views

Rays in fat parabolas

Let $\epsilon>0$. Let $F\subseteq \mathbb{R}^2$ be the set of all points that lie at a distance less than $\epsilon$ from the curve $y=x^2$. Can $F$ contain a ray? That is, is there (for some ...
2
votes
3answers
65 views

prove that the rose (in the polar plane) has $2n$ “petals” when $n$ is even

prove that the rose $r=\cos(n\theta)$ (in the polar plane) has $2n$ "petals" when $n$ is even. How can I start this demonstration? I would appreciate your help
5
votes
1answer
86 views

Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane. Gentle hints only, please! I know that attempting to decompose R.H.S. into $$\alpha a + \beta b + ...
0
votes
1answer
61 views

A question of straight lines

If the straight lines $x+y-2=0$, $2x-y+1=0$ and $px+qy-r=0$ are concurrent, then what is the slope of the member of family of lines $2px+3qy+4r=0$ which is farthest from origin? I wrote the ...
1
vote
1answer
113 views

How to find the points of tangency of a parabola using Calculus?

How can someone find the points of tangency of a parabola in this situation? I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis ...
1
vote
3answers
149 views

Proof of Aristarchus' Inequality

Does anyone know how to prove that if $0<\alpha<\beta<\frac{\pi}{2}$ then $\frac{\sin\alpha}{\alpha}>\frac{\sin\beta}{\beta}$. Any methods/techniques may be used.
2
votes
1answer
63 views

Are my answers correct? (finding intercepts, asymptotes, and extrema)

Are my answers correct? a) (0, 4/3) and (2,0) and (-2, 0) b) Horizontal asymptote: $y = 3$, Vertical asymptotes: $x = 3, x = -3$ c) Extremum is at $(0, 4/3)$, maximum.
2
votes
2answers
361 views

How to find the orthogonal trajectories of the family of all the circles through the points $(1,1)$ and $(-1,-1)$?

I'm trying to find the orthogonal trajectories of the family of circles through the points $(1,1)$ and $(-1, -1)$. Now such a family can be given by an equation of the form $$ x^2 + y^2 + 2g(x-y) - 2 ...
0
votes
1answer
102 views

line parallel to plane, but not on plane.

I need to find a plane that goes through the points $A=(2,0,2)$ and $B=(4,1,0)$, that is parallel to the line? $$r(t) = (0,3,-2) + t\langle1,-1,1\rangle$$ or if you want it in parametric equations: ...
0
votes
1answer
264 views

Intersection of two planes and another plane parallel to the intersection.

I have two questions: $1)$ How can I find the line of intersection between the planes $$x+ 2y +z =4 \\ \mathrm{and} \\ 2x+y-z=5$$ $2)$ How do I find an equation for a line that goes through $A = ...
1
vote
3answers
194 views

Arc Length Formulas

Use the arc length formula to find the arc length of the upper half of the circle with center at $(0,0)$ and radius $3$. Also, find the arc length of the curve in the first question by using ...
1
vote
0answers
88 views

Describing domain of integration of triple integral

I'm struggling to visualize the following problem: This question concerns the integral $\int_{0}^{2}\int_0^{\sqrt{4-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{8-x^2-y^2}}\!z\ \mathrm{d}z\ \mathrm{d}x\ ...
11
votes
2answers
438 views

When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
0
votes
0answers
104 views

two points on a unit sphere

Consider the two vectors to the points on the unit sphere, $${\bf v}_i=(\sin\theta_i\cos\varphi_i,\sin\theta_i\sin\varphi_i,\cos\theta_i)$$ with $i=1,2$. Use the dot product to get the angle $\psi$ ...
7
votes
4answers
150 views

closest point to on $y=1/x$ to a given point

I feel like I'm missing something basic - given a point $(a,b)$ how do I find the closest point to it on the curve $y=1/x$? I tried the direct approach of pluggin in $y=1/x$ into the distance formula ...
0
votes
2answers
111 views

Find minimum distance

I came across this problem in a maths exam. I solved this by taking that a light ray passes in such a way that it takes least path. But as this was a maths exam, i was wondering if it can be solved ...
1
vote
3answers
210 views

Area Between Curves

The problem I am working on is, "In Exercises 17 and 18, find the area of the region by integrating (a) with respect to and x (b) with respect to y." The two functions: $g(y)=4-y^2$, and $f(y)=y-2$ ...
2
votes
2answers
466 views

Calculation of Area of right angled triangle - Apostol exercise 1.7 problem 2

"Prove that every right triangular region is measurable because it can be obtained as the intersection of two rectangles. Prove that every triangular region is measurable and its area is one half the ...
1
vote
2answers
246 views

Parametric Equation Problem

The problem is, "to determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain." (a) $x=t;\quad ...
1
vote
1answer
144 views

Restriction Of Parametric Functions Domain

The problem I am working on is, "Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the ...
2
votes
1answer
132 views

Sketching A Plane Curve

The problem I am working on is, "Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the ...
2
votes
3answers
111 views

Smooth curve orthogonal to all hyperbolae $xy = a$ at points of intersection.

Suppose a smooth, connected curve $C$ in $R^2$ is orthogonal to all hyperbolae $xy = a$ whenever they coincide. I'd like to find the point(s) of intersection of $C$ with the hyperbola $xy = 16$ given ...
1
vote
0answers
97 views

A function with the same slope as $b\sqrt{\frac{x^2}{a^2}-1}$ but not imaginary in [0,a]?

For some fixed $a,b \in \mathbb{R}$, $y = b\sqrt{\frac{x^2}{a^2}-1}$ is supposed to plot the boundary of an ellipse in $\left[0,a\right]$. I came up with that function but it has the defect that it ...
11
votes
5answers
3k views

Calculating the area of an irregular polygon

Given the length of the sides of an irregular polygon (no coordinates provided) how do you compute the area of the maximum area of the polygon? Thanks in advance
3
votes
1answer
928 views

Use Pappus' theorem to find the moment of a region limited by a semi-circunference.

This is part of self-study; I found this question in the book "The Calculus with Analytic Geometry" (Leithold). $R$ is the region limited by the semi-circumference $\sqrt{r^2 - x^2}$ and the ...
5
votes
1answer
1k views

Formula for curve parallel to a parabola

I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
3
votes
1answer
155 views

Simulation of bouncing circles

I want to simulate two circles bouncing off one another. For this I am not sure what I need to calculate. I couldn't find any useful information on the internet, so I have thought long and hard about ...
5
votes
3answers
240 views

What are a , b and c?

$$y = ax^2 + bx + c$$ which is tangent at the origin with the line $y=x$, It is also tangential with the line $y=2x + 3$. Determine the function! Draw a figure! My main question is this solvable? I ...
1
vote
1answer
143 views

How does Rolle's theorem apply here?

The derivation below was taken from a book on Classical Differential Geometry. It uses Rolle's theorem to find the characteristic line of a family of planes, but I don't see how it applies. Given is ...
3
votes
1answer
256 views

References for the basic theory of surfaces of revolution, cylinders and cones

I'm looking for references to books were the following types of problems about finding the equation defining a surface of revolution, a cylinder or a cone are treated. These are problems that are ...
5
votes
2answers
596 views

Find equation of quadratic when given tangents?

I know the equations of 4 lines which are tangents to a quadratic: $y=2x-10$ $y=x-4$ $y=-x-4$ $y=-2x-10$ If I know that all of these equations are tangents, how do I find the equation of the ...