Tangent Definition and 1000 Threads

Homework Statement Show that the curve r = (t2,t3-t) Intersects itself at (1,0), and find the slopes of the tangents at this point. Homework Equations The Attempt at a Solution Okay I can show it intersects itself there, but what I am having trouble with is when they say slopes...

Find equations of the tangent lines to the graph of f(x)=\frac{x}{x-1} that pass through the point (-1, 5). Well, first I took the derivative, and afterwards, I made the connection that the derivative was a slope at any instant on the graph. By this, I inferred that f'(x) = m. I knew that the...

1. The question is What is the slope of the line tangent to f(x)=2x^2-x-7 at x=-1 2. Must be solved using derivatives 3. So basically i know the equation (f(x+deltax)-f(x)/deltax) So i plug in the problem and i get (2(x+deltax)^2-(x+deltax)-7)-(2x^2-x-7) Now i know what the...

When i was at school i used to think that any line that touches a curve at only one point is called as the tangent at that point to the curve! But after reading derivative i think this definition of tangent is not correct at all conditions! For example, x-axis cannot be called as tangent at...

That is, adding up the differential changes in angle between two arbitrarily chosen points on a function, to find the total change in angle between the tangent lines of those two points. How can this be done?

Considering f(x) = tan(x) * 5 / 8 ... how can I find the length of the curve, specifically, between (0, 0) and (1, 1) ? if anyone can help I would be happy. Thanks Keeaga

Homework Statement If the power of the secand is even and positive.. \int sec^{2k} x tan^{n} x dx = \int (sec^2 x)^{k-1} tan ^n x sec^2 x dx The Attempt at a Solution The way I see it, sec^{2k} x = sec^2 x dx * sec^k x dx the next step seems to be to break down sec^k, but on closer...

I've never been able to visualize how the tangent to a curve and the area under a curve are inverses of each other, can anyone give some intuitiveness to this?

Homework Statement See attached file. Homework Equations The Attempt at a Solution I've only been able to do part (a) of this question. I ended up with: tanz= i ({\frac{1-e^{(2iz)}}{1+e^{(2iz)}}}) I'm not sure how to approach the next two parts. If anyone could give me any...

Homework Statement A curve C is defined by the parametric equation x=t^2, y=t^3-3t. a) show that C has two tangents at the point (3,0) and find their equations. b) find the points on C where the tangent is horizontal Homework Equations y-y1=m(x1-x), (dy/dt)/(dx/dt)=m, when dy/dt=0...

How do you determine the equation of all possible tangents to a curve (say, a parabola) that pass through a given point that is not on said curve? This is more of a conceptual question, and it's not homework, so I thought it fit in this forum. I think there might be a question like this on the...

Homework Statement A curve is defined by the parametric equations: x = 2t^3 y = 2t^2 t =/ 0 1)Prove that the equation of the tangent at the point with parameter t is 2x - 3ty + 2t^3 = 0. Proven, and I've no problem with this part. 2.)The tangent at the point t = 2 meets the curve...

Homework Statement Ok so this is a question from last years past paper of my course: X= 1/2 intersects the circle that is centered at origin at two points, one of which is in the lower half plane y<0; what is the equation of the tangent tot the same circle at this point? Homework Equations...

The tangent vector is defined as : T=v/||v|| Where v is some vector. Then how is T the tangent vector to v? It's the unit vector in the direction of v right?

Homework Statement r=2-3cosθ Find the tangent line at any point, and at the point (2,∏) Find the tangent line(s) at the pole Homework Equations Do I have to use x=rcosθ and y=rsinθ to convert it to rectangular to find slopes? The Attempt at a Solution Is the point 2∏ even a...

D is a set of all points (x,y) in R2 distinct from (0,0). I have the funtion f: D --> R which is defined by: f(x,y) = (2xy)/(x2+y2 Im asked to find the equations (in plural) of the tangelnt plane and the normal line to the graph of the function at (1,2) My attempt: I use the tangent plane...

Homework Statement Given that near (1,1,1) the curve of intersection of the surfaces x^4 + y^2 + z^6 - 3xyz = 0 and xy + yz + zx - 3z^8 = 0 has the parametric equations x = f(t), y = g(t), z = t with f, g differentiable. (a) What are the values of the derivatives f'(1), g'(1)? (b)...

1. show that there is no line through the point (2,7) that is tangent to the parabola y =x^2 +x 2. y-y1=m(x-x1) 3. y'=f'(x)=2x+1 m1=2x +1 m1=2(2) +1 =5 m2=((x^2 +x)-7)/(x-2) m1=m2? ((x^2 +x)-7)/(x-2)=5 x^2-4x+3=0 2^2-4(2)+3=/=0 -1=0 I'm thinking that i would...

So i was considering minkowski space which is a 4-d manifold, why is that we use the tangent and cotangent space, to construct tensors on the space? The definition of a manifold says that the space is locally homeomorphic to Euclidean space. So is the tangent space and cotangent space...

Homework Statement Find derivative of [SIZE="5"]tan^{-1}(\frac{3sinx}{4+5cosx}) Homework Equations deriviative of[SIZE="5"] tan^{-1}=\frac{U'}{1+U^{2}} The Attempt at a Solution I found U'= [SIZE="5"]\frac{12cosx+15}{(4+5cosx)^{2}} [SIZE="5"]1+U^{2}=1+\frac{9sin^{2}x}{(4+5cosx)^{2}} I...

Homework Statement Find the equations of the lines that pass through (0,0) and are tangent to x^2 - 4x + y^2 + 1 = 0 My confusion I've been given a problem of this sort recently, except now it involves implicit differentiation. I know "how" to get to the correct answer. I just...

Homework Statement Suppose line tangent to graph of y=f(x) at x =3 passes through (-3, 7) & (2,-1). Find f'(3), what is the equation of the tangent line to f at 3? Homework Equations I found the slope of which equals -8/5 Im not sure how to find the equation... do I do...

Homework Statement Consider the plane curve \overrightarrow{r(t)}=e^tcost(t)\hat{i}+e^tsin(t) \hat{j} Find the following when t= ∏/2 Part A: \hat{T}(t) Part B: \hat{B}(t) Part C: \hat{N}(t) Homework Equations \hat{N}(t)=\frac{\hat{T}(t)}{||\hat{T}(t)||}...

Homework Statement The equation 5x^2 - 6xy + 5y^2 = 16 represents an ellipse. Determine two points on the ellipse at which the tangent is horizontal. Homework Equations The Attempt at a Solution I find the derivative of the equation: (-10x + 6y) / (-6x + 10y) = 0 iff...

Homework Statement Question: "Find the unit tangent, normal and binormal vectors T, N, B, and the curvature of the curve x = 4t, y = -3t^2, z = -4t^3 at t = 1." Answer: T = 0.285714285714286 i - 0.428571428571429 j - 0.857142857142857 k N = -0.75644794981871 i + 0.448265451744421 -...

Homework Statement My problem is one pertaining to my Vector Calculus course. The assignment is asking us to "Find equations for the planes tangent to z = x2 + 6x + y3 that are parallel to the plane 4x − 12y + z = 7." The problem I'm having with the problem is the plural aspect. It states...

Homework Statement Find the equation of the line with slope -1 that is tangent to the curve y=1/x-1 Homework Equations y=1/x-1 The Attempt at a Solution Slope of -1 means y=-1x+k So... -1x+k = 1/x-1 I don't know how to rearrange this into a quadratic equation so that I...

Homework Statement Let c(t) be a path and T the unit tangent vector. What is \int_c \mathbf{T} \cdot d\mathbf{s} Homework Equations The unit tangent vector of c(t) is c'(t) over the magnitude of c'(t) : \mathbf{T} = \frac{c'(t)}{||c'(t)||} The length of c(t) can be represented by ...

Homework Statement attached. Homework Equations The Attempt at a Solution I thought the problem was easy, but my answer is wrong (aparently; I still disagree.) First I defined x = (y^3)(z^3) to be a surface of function F So F(x,y,z) = (y^3)(z^3) - x = 0 Then, the...

Homework Statement lim x^3-2x^2+x/tanx x->0 The Attempt at a Solution All i know is that tan is going to break up into sinx/cosx so the equation will look like this lim x^3-2x^2+x/(sinx/cosx) x->0 I haven't worked with cubic or quadratic functions yet so I don't know...

Homework Statement Find the equations of all the lines through the origin that are tangent to the curve y = (some complicated cubic function) I looked up the question in google and found a much simpler example, y = x^2 passing through (1,-1). However, I don't even get what's going on...

So my problem is this: I need to figure out the center of a circle given two points. At one of the points, I know the tangent angle. So I know (x1, y1, θ1) and (x2, y2) and need to find (xc, yc). I also need to do this on a computer so I need some sort of closed-form solution. The way I...

Homework Statement Find equations of both the tangent lines to the ellipse x2 + 4y2 = 36 that pass through the point (12, 3). Homework Equations The equation of an ellipse is x2/a2 + y2/b2 = 1. I converted the given equation to x2/36 + y2/9 = 1 by dividing each value by 36. The...

Homework Statement Find the unit tangent vector at the indicated point of the vector function r(t) = e(19t)costi + e(19t)sintj + e(19t) kT(pi/2) = <___i+___j+___k>Homework Equations r'(t) / |r'(t)| The Attempt at a SolutionAnswers: 19e(19*∏/2)(cos(∏/2)-sin(∏/2)) /...

Homework Statement How do I solve this equation? lim x>0 (x^3 - 2x^2 + x)/tanx I don't know what to do here, please help. Thank you

Homework Statement Find equation of tangent line, given x = -1. Not given y. I am used to having this when I am given both y and x. Homework Equations (x^3 - 4x + 2)(x^4 + 3x - 5)The Attempt at a Solution Differentiate (3x^2 - 4)(4x^3+3) Multiply 12x^5 - 9x^2 - 8x^3 - 12 Plug in -1, find...

I have attached both the question and the solution. I just have questions as to why the solution is the way it is (sorry if they seem stupid but, while I get how to do it mechanically, I don't understand the fundamental reasoning as to why anything is being done): 1) Why are we taking the...

Homework Statement Find the co-ordinates of all points on the curve f(x)= x3 whose tangent lines pass through the point (a,0) Homework Equations f '(x) = nxn-1 The Attempt at a Solution I am really not sure how to attack this question. My initial thoughts are to find f '(x) then...

Homework Statement (a) Draw a diagram to show that there are two tangent lines to the parabola y = x^2 that pass through the point (0, -4). (Do this on paper. Your teacher may ask you to turn in this work.) (b) Find the coordinates of the points where these tangent lines intersect the...

Homework Statement So I'm a little confused about what a tangent space is. Is it the same as the equation of the tangent plane in lower dimensions? My notes define the tangent space as follows. Let M be a hypersurface of Rd. Let x(s) be a differentiable curve in M such that x(0)=x0 is in...

The curve $\displaystyle y-e^{(xy)} + x=0 $ has a vertical tangent at which point?? I started to differentiate it, then equating dy/dx to 0, then how should i proceed??

Homework Statement Find the direction of the line tangent to the curve x^4+y^4=32 at the point (2, -2) Homework Equations Anything goes, we're in vector calculus now. The Attempt at a Solution So, to find the tangent line, I was thinking of taking the gradient, but I'm not...

Homework Statement f (x) = e^(3x) + sin(2x) + 3x +1 (a) Find a vector V that is tangent to the graph of y = f(x) at the point ( 0, 2). (b) Find a vector N that is perpendicular to the graph of y = f(x) at the point ( 0, 2). 2. The attempt at a solution The first step I took is to...

Homework Statement Find an equation of the tangent line to the curve at the given point. y = tan x at point (pi/4,1) Homework Equations The Attempt at a Solution step 1. find the derivative of tan x, which sec^2 x step 2. find the slope. this is where I mess up. I assume...

Homework Statement Homework Equations The Attempt at a Solution I don't see how they get from step one to two. I would think both secants would cancel since one is positive and the other is negative but that doesn't happen. i think i understand the manipulations of the tangents but not...

Is it possible to find a directional derivative for a point on z = f(x,y) at a point (x,y) in a direction (u1,u2) using the plane tangent to z at (x,y)? If so, how? Thanks!

Homework Statement http://www.xtremepapers.com/Edexcel/Advanced%20Level/Mathematics/Subject%20Sorted/C3/Elmwood%20Papers/Elmwood%20B.pdf Question 8(b) Homework Equations The Attempt at a Solution Ok so I found both values of dy/dx for BOTH EQUATIONS y = x2 - 4x → 2x - 4 y = |4x - x2| →...

Homework Statement "Find the equation of the plane tangent to the surface (x^2-y^2)(x^2+y^2)=15 at the point (2,1) " If only it really were a plane and a surface, I could do that. I have a formula for that. Unfortunately, this is a curve and I'm looking for tangent line. Homework Equations...

The Following data table provides information about a crate of radishes that is sliding down the ramp of a delivery truck time position 0 ------ 0 2------ 0.6 4-------2.4 6------- 5.4 8------- 9.6 10------ 15Next it asks you to draw a position time graph, which i did, and its identical...

Homework Statement Find a > 0 such that the tangent line to the graph of f(x) = x^{2}e^{-x} at x = a passes through the origin. Homework Equations The Attempt at a Solution First I found the derivative to be: -e^{-x}(x-2)x , which is the slope of the function. I know the tangent line...