Curves Definition and 741 Threads

given two function f and g in the closed interval a to b, the area would be ∫(f(x) - g(x))dx from a to b if f(x)≥g(x) for all x in [a,b]. My question is.. if I were given two function f and g in a given interval [a,b], what is the best way to dertermine if f(x)≥g(x) or vise versa? Would it...

I hope you guys can help. I feel like I know how to do this, but I keep getting the same answer no matter what I do, which is wrong. Sketch the regions enclosed by the given curves. y=√x, y=1/2x, x=25 Find its area. The attempt at a solution ∫4,25 ((1/2x)-(√x))dx [(1/4x2)-(2/3x3/2)]...

Hi all, I am trying to argue that an ellipse is a good approximation for some discontinuity in a material. The ellipse and the interpolated curve I get from the photo look very much alike, but I need an actual number to show how much the two curves differ from each other. I thought of...

This is from Spivak's Calculus. In an appendix, he defines polar coordinates. One of the exercises in this appendix is showing that the lemniscate, whose polar equation is: r^2=2(a^2)*cos(2theta) is the set of points P that satisfy that the product of the distances from said point to two...

I need to deal with some curves. Nothing too fancy and nothing precise. I really need a line, something that looks like a parabola. A circle and something similar to Bézier curves would be very nice. I'm working in two dimensions, real numbers only. What I have right now is a vector of two...

How do you go about sketching y as a function of t for t≥0 y= e(0.5t + i(√7/2)t) - e(0.5t-i(√7/2)t) I know it goes through the origin, and the gradient is positive here. But I'm unsure on how to deal with the imaginary numbers when I have a graph of y vs t.

Hello all, first time poster here so please go easy :) I have to create a set of Power curves (Power Required and Power Available vs. Velocity) for a propeller-driven Ryan Navion A aircraft at intervals of 1000m. My question to you is this, due to the fact that the aircraft is in steady...

How do I know which function is the upper boundary curve and which is the lower boundary curve. For example find the area between the curves e^x and x bounded on the sides x=0 and x=1. We can draw it and we know that e^x is the upper curve and x is the lower curve. Thus the area is ∫e^x-∫x...

Hi I am pretty stuck on a proof so any help would be great: Let P and Q be two projective curves, and let p belong to both of them. Show that the intersection number of P and Q at p is equal to one iff the tangent lines to p of P and Q are distinct NB-we have defined intersection numbers...

Homework Statement Find dy/dx and determine the exact x coordinate of the stationary point for: (a) y=(4x^2+1)^5 (b) y=x^2/lnx Homework Equations The Attempt at a Solution (a) y=(4x^2+1)^5 dy/dx=40x(4x^2+1)^4 40x(4x^2+1)^4=0 Find x... How? (b) y=x^2/lnx...

Homework Statement Calculate the centripetal acceleration module in the following cases as a multiple of g = 9.8 m / s ^ 2. a) a car traveling at 100km / h on a curve of radius 50m. b) a jet plane flying at 1,500 km / h and making a turning radius of 5km. c) a stone that is rotated every...

Homework Statement Suppose you need to know an equation of the tangent plane to a surface S at the point P(2,1,3). You don't know the equation for S but you know that the curves r1(t)=<2+3t,1-t^2,3-4t+t^2> r2(u)=<1+u^2,2u^3-1,2u+1> both lie on S. Find an equation of the tangent plane at P...

I am currently reading the book "e: The Story of a Number" by Eli Maor. And I got stuck at something. In chapter 7 of the book, the author described the method Fermat used to calculate areas under curves of the form y = x^n, where n is a positive integer. I am quoting the relevant bit here...

This is not a question I need to work out but I'm trying to understand this theorem. My lecture notes state: 'This theorem states the existence of solutions to the Frenet - Serret Equations that, apart from the possibility of a rigid motion, are uniquely determined by any choice of smooth...

I have difficulty understanding the following Theorem If U is open in ℝ^2, F: U \rightarrow ℝ is a differentiable function with Lipschitz derivative, and X_c=\{x\in U|F(x)=c\}, then X_c is a smooth curve if [\operatorname{D}F(\textbf{a})] is onto for \textbf{a}\in X_c; i.e., if \big[...

I am being asked to calculate level curves for the following equation: f(x,y)=e^-(2x^2+2y^2) but I do not know where to start. Any advice on first steps would be greatly appreciated.

I don't get why G=0 is a contradiction. Does it imply F=0, which cannot be true since the question stated F is non constant? Can anyone give me another proof for this first part please? As the step he made to get G would have been something I would never have thought of. By the way problem 1.4...

Homework Statement The actual numbers aren't completely relevant. I made a graphic. http://i168.photobucket.com/albums/u193/kamikazehighland/calculus.png I'm actually not in calculus or engineering. I'm actually writing a program, but I've done enough research to know how to...

1. Show that, if the velocity field (V) is a fixed (spatially constant) vector, then the characteristic curves will be a family of parallel-straight lines. 2. ut+V1ux+V2uy=f f=S-[dell dotted with V]u characteristic curves: dX/dt=V1(X,Y) & dY/dt=V2(X,Y) 3. really looking for...

Just a question that came to my mind while studying differential equations. Of-course this is a silly one (I think), but I wonder if someone can answer it! Thanks :smile:

Three questions, all related. Firstly, I'm wondering what sort of modifications to Newtonian gravity were tried to explain the flatness of various galaxy rotation curves. (References, and especially a review, would be much appreciated. I haven't been able to find anything appropriate.)...

I've got a question about interpreting these - mainly about how the horizontal axis works. See attached file - On the leftmost panel, it has a form of the wavevector q plotted from left to right, labeled as (000) on the left and (100) on the right. What does this mean? Does it mean that...

Homework Statement Find the area bounded by the curves y=x^2 and y= 2 - x^2 for 0 ≤ x ≤ 2. Homework Equations ∫top - ∫bottom The Attempt at a Solution ∫(2-x^2)dx - ∫x^2dx What I'm confused about is that the two equations only cross on [-1,1] so within the interval of the...

Homework Statement This is a bonus problem on our homework, and I'm having trouble figuring out how to setup what I need. Homework Equations Here are my best guesses: f_x=\frac{\partial f}{\partial x} f_y=\frac{\partial f}{\partial y} f_{xx}=\frac{\partial}{\partial...

In my experience, whenever we want to calculate the tangent space to a smooth manifold, we usually proceed as follows. Let M be a smooth manifold and p in M. Let \gamma: \mathbb R \to M be a smooth curve such that \gamma(0) = p and \gamma'(0) = X . We then use some defining quality of M...

Hi! I am having points set say P = { [x1,y1] , [x2,y2],[x3,y3],...,[xn,yn]} Now, i want to fit curves going through these points. Is it possible. I googled and went through something called beziers.. is that helpful here ? Or is there any direct mathematical approach to achieve...

Hello, forum! I'm a newbie here. I've been visiting this site for a while but just recently joined. Anyways, I was wondering if anyone could help with this problem. I can find the orthogonal trajectories, however, this one is killing me because there is a constant. Allow me to type it below...

Homework Statement So what I've been doing to solve these questions is to set the two equations of the curves equal to each other and solving for the variable, which gives me the points where they intersect. But I'm having problems solving for the variable in some of these. For example...

Homework Statement Find all points for which the curves x^2+y^2+z^2=3 and x^3+y^3+z^3=3 share the same tangent line. Homework Equations Sharing the same tangent line amounts to having the same derivative. The constraint then is that 3x^2+3y^2+3z^2=2x+2y+2z. The points must obviously also...

Can someone do it without using a graphing calculator? The question specifically states not to use "Trace". I don't understand how to do it algebraically, and I'd love it if someone could teach me. Please and thanks!

I have to solve the differential equation (y')^2= 4y to verify the general solution curves and singular solution curves. Determine the points (a,b) in the plane for which the initial value problem (y')^2= 4y, y(a)= b has (a) no solution , (b) infinitely many solutions (that are defined for...

Answers Vmax = 13.1 m/s Vmin = 8.1 m/s ------------------------------------------------------------------------------------------ I've spent all morning attempting part 4 of this problem :/ Can someone help me please? FBD tanθ = v2 / rg v = sqrt(rg)tanθ = sqrt [(20)(9.8)tan30] =...

Homework Statement sin(y)\frac{ \partial u}{ \partial x} + \frac{ \partial u}{ \partial y} = (xcos(y)-sin^2(y))u where ln(u(x,\frac{\pi}{2})) = x^2 + x - \frac{\pi}{2} for -1 \leq x \leq 3 determine the characteristic curves in the xy plane and draw 3 of them determine the general...

Homework Statement Hello everyone, I was wondering if someone could check my work on this problem as I'm not sure it's right. I'm in Calc II right now and we are doing finding volume bounded by curves rotated around an axis. So, here is the problem, y=cos(pix)+1, y=4x^(2)-9, x=0; about...

The definition of parallel curve is well defined, such that given two curves, they must remain equidistant to each other. For instance y = (x^2) + 4 and y = (x^2) - 8 are parallel curves in a function the maps x to y. These form parabolas whose vertical distance to one another remains...

Homework Statement Find the arc length of the following curves on the given interval by integrating with respect to x y=x^4/4+ 1/(8x^2); [1,2] Homework Equations Let f have a continuous first derivative on the interval [a,b]. The length of the curve from (a,f(a)) to (b,f(b)) is...

This is the answer key to one of my quizzes If you notice in the question, see attachment, were it says to integrate with respect to y the integral is integrating from 0 to 25 but this produces a negative area so this is technically wrong, yes? You don't just simply integrate from the lower...

defining a tangent vector v as the equivalence class of of curves: v = [\sigma] = \left. \frac{df(\sigma)}{dt} \right|_{t=0}, i want to show that this definition is independent of the member of the equivalence class that i choose. where \sigma represents a function from the reals to the...

In 3D the torsion  measures how rapidly the curve twists out of the osculating plane in which it finds itself momentarily trapped. So in 4D, would torsion measure how rapidly a curve twists out of the osculating 3-hypersurface in which it finds itself momentarily trapped? Or torsion of a...

Homework Statement [PLAIN]http://img688.imageshack.us/img688/5336/unledaty.jpg The Attempt at a Solution I underlined a = 3, which doesn't make sense seeing the foci is at (plus/minus3, 3) How can this be?? Wouldn't that make a straight line?

I need to find the point of intersection of the curves x^2 + y^2 =1, z= 0 and x=cost, y=sint, z=t. I plugged in the latter equation into the former and got (1,0,0) as an answer but I'm not exactly sure why that works, I can't visualize how plugging in the parts of a parametric equation will...

Homework Statement I need assistance in learning the proper way to calculate error/uncertainty in a few things for my undergrad thesis.Homework Equations 1) how to calculate the uncertainty in the slope and intercept of calibration curves (peak area vs mol of compound) I have made via excel...

do they exist in reality or in nature?

Delete post. Delete, please.

"smooth" curves with cusps in 3d While reviewing basic calculus, I noticed that the curve (1+t^2,t^2,t^3), which clearly has a cusp at (1,0,0), has a derivative curve (2t,2t,3t^2) which is clearly smooth. This struck me as odd since differentiation usually seems to turn cusps into...

Homework Statement Two curves are said to be orthogonal if their derivatives are opposite reciprocals at the point where the two curves intersect. Are 2x^2 + y^2 =3 and x= y^2 orthogonal?Homework Equations I'm not entirely sure what to put here, but I think one relevant thing is to say that...

Homework Statement Polar plot the following sin[t] U cos[t] sin[2t] U cos[2t] sin[3t] U cos[3t] sin[4t] U cos[4t] sin[5t] U cos[5t] Notice that cos[t] and sin[t] are the same graph rotated 90 degrees only? Interesting! Just like the cartesian graph. Now here is something more...

Hello, I have a quick question about Characteristic curves. [PLAIN]http://pokit.org/get/1958a855486487230cd4e3c0a1cc0908.jpg First: Do these curves go to infinity, i mean in theory? If I had a steeper load line, I would hit saturation later. And what about that portion of load line between...

Homework Statement Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point y = f(x) = (x)^3, at (1,1) Homework Equations Area in between two curves The Attempt at a...

Homework Statement i'm studying for an exam. and I'm pretty sure i know how do do these types of problems. this is aneven problem in the book so i wanted to know if my answer is right. Find the length of the curve for r=\sqrt{1+\cos2\theta} , \pi/2\leq\theta\leq\pi/2 Homework Equations...