Curve Definition and 1000 Threads

Homework Statement Designing an on ramp for the 401 the engineer wants cars to be able to make the turn with a radius of 50 m while traveling 40km/hr in conditions with no friction. What angle must he bank the curve at to make this possible? Homework Equations FR = (m)(aR) FR = (m)(v2)/(r)The...

This is a problem from Tom Apostol's calculus book (the very first problem set in the introduction). It wants you to find $$\int_0^b (ax^m+c)\,dx$$ using Archimedes' method of exhaustion. I'm attaching a diagram and a pdf of my work for the problem since it's rather involved. I'm doing this...

Homework Statement Find equation of the curve on reflection of the ellipse \dfrac{(x-4)^2}{16} + \dfrac{(y-3)^2}{9} = 1 about the line x-y-2=0. Homework Equations The Attempt at a Solution Let the general point be P(4+4cosθ,3+3sinθ). Let the reflected point be (h,k). \dfrac{4+4cos...

Find the area bounded by the curve $$x = 6x - x^2$$ and the y axis. So can I use the even function rule to get: $$2 \int^2_0 6x - x^2 dx$$ I just need someone to check my work. $$2 [ 3x^2 - \frac{1}{3}x^3 ] | 2, 0$$ $$ 2 [ 12 - \frac{8}{3}]$$ $$2 [ \frac{36}{3} - \frac{8}{3} ]$$ $$2 * 28/3$$...

Find the area bounded by the curve $$x = 16 - y^4$$ and the y axis. I need someone to check my work. so I know this is a upside down parabola so I find the two x coordinates which are $$16 - y^4 = 0$$ $$y^4 = 16$$ $$y^2 = +- \sqrt{4}$$ $$y = +- 2$$ so I know $$\int^2_{-2} 16 - y^4 dy$$...

Homework Statement a) Show that the curve determined by: x=2(t^{3}+2)/3, y=2t^{2}, z=3t-2 intersects the surface: x^{2}+2y^{2}+3z^{2}=15 at a right angle at the point (2, 2, 1) b) Verify that the curve x^{2}-y^{2}+z^{2}=1, xy+xz=2 is tangent to the surface xyz-x^{2}-6y+6y=0 at the point...

Homework Statement An object slides without friction down a ramp, from (xi, yi) to (xf, yf). What is the equation for the shape of the ramp connecting those two points which would enable the object to reach (xf, yf) in the shortest possible time? Also, describe the shape of the ramp. Homework...

So as per van der waal's equation, below critical temperature, for each pressure and temperature, we have only 3 possible volumes for each pressure-temperature pair. My questions are as follows: 1. So in the usual vapor dome (considering a P-v surface), we have a straight horizontal line...

Homework Statement A car is resting on a curved path in point A as seen in the picture. The mass of the car is known. The angle θ and the radius of the curved path are known. The kinetic coefficient of friction is known. The car starts gliding backwards on the curved path. What is the...

Hi - I was reading Adams' "Calculus: A Complete Course" (6th edition) and he offers the following proof that the gradient of a function: I'm just wondering why the emphasis on specifying the values of x(0) and y(0), or even the need to specify t = 0 in the last statement? If \mathbf{r} is...

So I am new to relativity, and without all the proper math analyzing the equation isn't easy. So I am stuck at this current point until I learn more math, for the most part trying to gain an understanding of how it works, through thought experiments and visualizations. That being said I have...

Hi all, I am wondering about where the vapor pressure curve would be located on the PVT surface of a substance. I know that as the temperature of a liquid increases, its vapor pressure increases. This continues until the vapor pressure is equal to the atmospheric pressure, at which point...

The area of a polar curve is given by A=(1/2)∫ r2 d (theta). this can be interpreted as δA= ∏r2δ(theta)/2∏ (treating the area element as the area of a sector of a circle with angle δ(theta).) taking limit of δ(theta)→0, dA= ∏r2 d(theta)/2∏=1/2 (r2d(theta) ) there fore A=1/2∫r2 d(theta)...

Homework Statement Problem: A curve given parametrically by (x, y, z) = (2 + 3t, 2 – 2t^2, -3t – 2t^3). There is a unique point P on the curve with the property that the tangent line at P passes through the point (-10, -22, 76). Answer: P = (-4, -6, 22) What are the coordinates of...

Homework Statement Find the total length of the curve t --> (cos^3(t), sin^3(t)), and t is between 0 and ∏/2 where t is in radians. Find also the partial arc length s(t) along the curve between 0 and ∏/2 Homework Equations The length is given by: S = ∫\sqrt{xdot^2 + ydot^2} dt...

Homework Statement Find the length of the curve traced by the given vector function on the indicated interval. r(t)=<t, tcost, tsint> ; 0<t<pi Homework Equations s= ∫||r'(t)||dt The Attempt at a Solution r'(t) = <1, -tsint + cost, tcost + sint> s= ∫||r'(t)||dt ||r'(t)|| =...

Homework Statement On a highway curve with radius 55m, the maximum force of static friction that can act on a 795kg car going around the curve is 8740N. What speed limit should be posted for the curve so that cars can negotiate is safely? Homework Equations ƩF=ma, a=v^2/r The...

Homework Statement A banked curve has been designed so that it is safest for a vehicle going 70 mph. The topography of the land restricts the radius of the road to 300m. Assume mu(static friction) is 0.80 and mu(kinetic friction) is 0.60. A 5512 lb vehicle travels with a speed of 60 mph on...

Hi All, Today I conducted a lab in which we plotted the open circuit voltage (V[oc]) and closed circuit current (I[sc]) versus the excitation current in the rotor (I[x]). From the data (plots are attached to thread), it can be seen that saturation occurs in the V[oc] versus I[x] curve...

Hi, I was wondering if anyone can help me. I don’t have a homework problem, but a problem I have encountered at work. I am a mechanical engineer working in the railway industry and I am struggling with a problem of reconstructing the vertical geometry of a rail in terms of height and...

Homework Statement A curve of radius 67 m is banked for a design speed of 95 km/h. If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely handle the curve? Homework Equations I drew this freebody diagram. The Attempt at a...

Homework Statement Slope of the curve at the point x=0 of the function y=y(x) specified implicitly as \displaystyle \int_0^y e^{-t^2} dt + \int_0^x \cos t^2 dt = 0 is Homework Equations The Attempt at a Solution Differentiating both sides wrt x e^{-y^2} \frac{dy}{dx} + \cos x^2 = 0...

I enrolled in Calculus for my senior year in high school, so far loving it. Anyways, I have been reading ahead and figuring some things out, but on the topic of Integration, what can you use the area under the curve for? I've tried searching around in my textbook, and maybe just my google skills...

Here is the question: I have posted a link there to this topic so the OP can see my work.

Homework Statement Show that x=sin(t),y=cos⁡(t),z=sin^2 (t) is the curve of intersection of the surfaces z=x^2 and x^2+y^2=1. Homework Equations I don't think there aren't really any equations relevant for this maybe except the unit circle..? The Attempt at a Solution...

Homework Statement Consider the curve r = <cos(3t)e^(3t),sin(3t)e^(3t),e^(3t)> compute the arclength function s(t) with the initial point t = 0. Homework Equations s = integral |r'(t)|dt The Attempt at a Solution Okay so if you work all of this out it turns out it's not as bad as...

Hi, can someone help in re-parameterizing the curve δ(t)=(2/3(√(L^2+9))cos(t),1/3(√(L^2+9))sin(t),L) I found dδ/dt then I got the speed to be 1/3√(L^2+9)√(1+3sin^2(t)) L is just a constant z=L I know how to re-parameterize curves to make them parameterized by arc-length when I...

Homework Statement The problem asks us to prove the length of a curve decreases as one of its parameters, t, increases. Here is the full statement http://img94.imageshack.us/img94/6560/yf6j.jpg Homework Equations L_t=\int_0^1 ||\partial_s x(t,s)||ds is the length of the curve as a...

I'm an avid baseball fan and since I've gotten interested in physics, it's pretty cool to try to combine the two. I was wondering what forces act on a thrown baseball and how the spin and the seams of the ball cause it to curve.

I need a MATLAB expert to guide me on how to create a b-spline curve using MATLAB Software. I understand the B-spline basis function calculations for zeroth and first degree but I have no idea on how to calculate for the 2nd degree. I need a favor on that part. I am currently working on my...

Both statements 1 and 2 are given as an explanation of why the original statement is true, but I don't understand why you can use statement 2 (since in the original vector equation you do not have Sin2(t), -Cos2(t)) Show why r(t) = <e-t, Sin(t), -Cos(t)> approaches a circle as t →∞. 1. As...

Trying to understand a concept on vector calculus, the book states: If S is a surface represented by \textbf{r}(u,v) = u\textbf{i} + v\textbf{j} + f(u,v)\textbf{k} Any curve r(λ), where λ is a parameter, on the surface S can be represented by a pair of equations relating the parameters u...

Homework Statement Now use your answer from part (a)(This anwser is f'(25) = 7/10, which is correct) to find the equation of the tangent line to the curve at the point (25, f(25)). Homework Equations f(x) = 7*sqrt(x)+3 The Attempt at a Solution f'(25) = 7/10 f(25) = 38 y=mx+b...

Homework Statement Find the curve whose curvature is 2, passes through the point (1,0) and whose tangent vector at (1,0) is [1/2 , (√3)/2 ]. The Attempt at a Solution I know I must use the Fundamental theorem of plane curves but I don't know how to apply it correctly here. Another...

Homework Statement How can I draw a closed plane curve with positive curvature that is not convex The Attempt at a Solution I was thinking drawing it like a banana but more curved, will that do?

Problem: Use an appropraite parametrization x=f(r,\theta), y=g(r,\theta) and the corresponding Jacobian such that dx \ dy \ =|J| dr \ d\theta to find the area bounded by the curve x^{2/5}+y^{2/5}=a^{2/5} Attempt at a Solution: I'm not really sure how to find the parametrization. Once I...

Homework Statement Find the curve whose curvature is 2, passes through the point (1,0) and whose tangent vector at (1,0) is [1/2 , (√3)/2 ]. The Attempt at a Solution I know I must use the Fundamental theorem of plane curves but I don't know how to apply it correctly here.

Homework Statement find the length of the curve… r(t) = <4t, t^(2) + 1/6(t)^(3)> from 0≤t≤1 Homework Equations L(t) = ∫a to b √(dx/dt)^(2) +(dy/dt)^(2) + (dz/dt)^(2))dt The Attempt at a Solution After taking the derivative of all components of the curve and finding the magnitude…...

Homework Statement What could be the reason why the graph is formed the way it is?Homework Equations τ = 2π(L/g)^(1/2) The Attempt at a Solution I don't know how to explain it. Anyone know why one is a linear and another is a curve?

Homework Statement Find the arc length of polar curve 9+9cosθ Homework Equations L = integral of sqrt(r^2 + (dr/dθ)^2 dθ dr/dθ = -9sinθ r = 9+9cosθ )The Attempt at a Solution 1. Simplifying the integral r^2 = (9+9cosθ^2) = 81 +162cosθ + 81cos^2(θ) (dr/dθ)^2 = 81sin^2(θ)...

Homework Statement What is the velocity vertor of a particle traveling to the right along the hyperbola y=x-1 which constant 5 cm/s when the particles location is (2, ##\frac{1}{2}##)? Homework Equations The Length of path forumula. $$ s\,=\int_a^b ||r'(t)||\,dt $$ Please don't make me...

So I've been spending a lot of time lately trying to figure out why an integral will give you the area under the curve. I asked the forum and got some great answers, but all were in terms of infinite sums, dx, and infinite rectangles. I think I've come upon a more fundamental answer that I...

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

Given t\in Ithe arc length of a regular parametrized curve \alpha : I \to \mathbb{R}^3 from the point t_0 is by definition s(t) = \int^t_{t_0}|\alpha'(t)|dt where |\alpha'(t)| = \sqrt{(x'(t))^2+(y'(t))^2+(z'(t))^2} is the length of the vector \alpha'(t). Since \alpha'(t) \ne 0 the arc length s...

Given t\in Ithe arc length of a regular parametrized curve \alpha : I \to \mathbb{R}^3 from the point t_0 is by definition s(t) = \int^t_{t_0}|\alpha'(t)|dt where |\alpha'(t)| = \sqrt{(x'(t))^2+(y'(t))^2+(z'(t))^2} is the length of the vector \alpha'(t). Since \alpha'(t) \ne 0 the arc length s...

Sorry if i post this in the wrong spot. I am trying to form the curve of the half quadrant of a circle. And i wonder that how do we know which or where is our control point? For cubic bezier, the 2nd control point should be on the tangent line of the starting point and the 3rd control point...

For the curve defined by r(t) = 3*t*i + 2*t^2*j − 3*t^4*k Find the tangent vector r′(t0) at the point P(4,8,−16), given that the position vector of P is r(t0). and Find the vector equation of the tangent line to the trajectory through P. Im unsure as to how to go about solving this. I've...

A hospital would like to determine the average stay for its patients having abdominal surgery. A sample of 15 patients revealed a sample mean of 6.4 days and a sample standard deviation of 1.4 days.How do you display this in a normal curve?? any help would be appreciated. Thanks

Homework Statement An industrial settling pond has a parabolic cross section described by the equation y = \frac{x^2}{80} . the pond is 40 m across and 5 m deep at the cetner. the curved bottom surface of the pond is to be covered with a layer of clay to limit seepage from the pond. determine...

Homework Statement Find the length of the curve given by the equation:Homework Equations y= \int_{-pi/2}^x √(cos t)\, dt for x between -∏/2 and ∏/2The Attempt at a Solution y= sqrt (cos x) dy/dx= (sin x)/[-2 * sqrt(cos x)] So now applying the arc length formula of sqrt (1 + (dy/dx)^(2)), I...