Tangent Definition and 1000 Threads

Homework Statement Sketch the graphs of y=x2 and y= -x2+6x-5, and sketch the two lines that are tangent to both graphs. Find equations of these lines. Homework EquationsThe Attempt at a Solution So I know that a tangent line to both graphs means that the tangent line will touch the first graph...

I don't know how to create a commutative diagram here so I'd like to refer to Diagram (1) in this Wikipedia article. I need to discuss the application of this diagram to the tangent bundle of a smooth manifold because there are some basic points that are either glossed over or conflict in the...

Find the derivative of the function $f(y)$ $$f(y)=\tan^{-1}\left({8{y}^{3}+1}\right)$$

Homework Statement Find the slope of tangent line to curve that is intersection to the surface z= (x^2) - (y^2) with plane x =2 , at point (2,1,3) The ans given by the author is only∂z /∂y = -2 Homework EquationsThe Attempt at a Solution Is my diagram correct ? I'm wondering , why shouldn't...

I am trying to find the slope of the tangent line of this polar equation: r = 4 + sin theta, (4,0) I put the equation into wolfram alpha and it gives me a 3D plot. If someone could help me find the slope of the tangent line, I would really appreciate it. Thank you.

Homework Statement Calculate the force on the cilinder. You need the angle of the tangent and r. Homework Equations Down under here. The solution for the angle is 30°. But why does the formula I used did not work out? My solution is 40,9°, why isn't this correct for this excercise? The...

Homework Statement If the normal at P(ap^2 ,2ap) to the parabola y^2 = 4ax meets the curve again at Q(aq^2, 2aq), show that p^2 +pq+2=0 Homework Equations Point-slope form The Attempt at a Solution I tried putting y=2aq and x=aq^2 but I can seem to simplify the whole thing other than...

Hi everyone. I am told to find the equation of the tangent plane to the surface x^2 + 2xy^2 -3z^3 = 6. I do not know how to approach this problem, and I was wondering if anyone would be kind enough to help. I know that for example if I had an equation z = x^2 + y^2, with a point P(x0,y0) and...

The graph of y = x - 1 CUTS the x-axis at x = 1 while the graph of y = x2- 1 TOUCHES the x-axis at x = 1. The point at which the tangent touches the curve is shown mathematically by having two solutions of x, i.e. x = 1 (twice). Is there some deeper meaning to these two identical solutions for x?

Homework Statement http://mathwiki.ucdavis.edu/Core/Calculus/Vector_Calculus/Vector-Valued_Functions_and_Motion_in_Space/The_Unit_Tangent_and_the_Unit_Normal_Vectors In the link, I can't understand that why the Principal Unit Normal Vector is defined by N(t) = T'(t) / | T'(t) | ,can someone...

] I found in a textbook that the value does not change because the centripetal force is perpendicular to the tangential velocity. But I am confused, because a vector can have a component, which is perpendicular to the vector. So if the centripetal force is perpendicular to the tangential...

For what values of c is there a straight line that intersects the curve in four distinct places? x^4+c*x^3+12x^2-5x+2 I'm looking for a full answer (doesn't have to use the same method)

Hi all, this might be a silly question, but I was curious. In Carroll's book, the author says that, in a manifold M , for any vector k in the tangent space T_p at a point p\in M , we can find a path x^{\mu}(\lambda) that passes through p which corresponds to the geodesic for that...

Homework Statement Consider the parametric curve given by: x=6cos(2t), y=t5/2. Calculate the equation of the tangent to this curve at the point given by t=π/4, in the form y=mx+c. The tangent is given by y= Homework Equations The Attempt at a Solution [/B] the answer that I got was...

Homework Statement "Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates." ##7.##...

MENTOR Note: Moved this thread from a math forum hence no template Is it possible to find this? Really only need the semi major axis or even it's orientation. In the image below, elements in red are known.

Homework Statement In Euclidean three-space, let ##p## be the point with coordinates ##(x,y,z)=(1,0,-1)##. Consider the following curves that pass through ##p##: ##x^{i}(\lambda)=(\lambda , (\lambda -1)^{2}, -\lambda)## ##x^{i}(\mu)=(\text{cos}\ \mu , \text{sin}\ \mu , \mu - 1)##...

In the following book, please look at equation 3.16. Why are the components of the tangent vector given by ui = dxi/dt? I understand the velocity components would be dxi/dt and the velocity vector would be a tangent vector. Is that the same reasoning the author uses? The book is normally crystal...

Homework Statement Find the slope of the tangent line to the give polar curve at the point specified by the value of theta R = 1/θ, θ=π Homework Equations ##\frac{(dr/dθ)sinθ + rcosθ}{(dr/dθ)cosθ - rsinθ}##The Attempt at a Solution The derivative of r is -1/θ2 Then plugging things into the...

Homework Statement z = 2x^2 + 5y^2 +2 C is cut by the plane x = 2 Find parametric eqns of the line tangent to C @ P(2, 1, 15) Homework Equations z = 5y^2 + 10 dz/dx = 10y dz/dx (1) = 10 The Attempt at a Solution z = 10y + 15 y = t + 1 if the slope is 10/1 then delta z = 10 and delta y = 1...

In Zee "Einstei gravity in a nutshell" section I.6, page 83, the author says about the approxiamtion of the south pole of sphere How is the first equation approximated by the second? One page later he does this expansion again. Is this thecalculus Leibnitz rule? Or some clever trick...

I am reading John M. Lee's book: Introduction to Smooth Manifolds ... I am focused on Chapter 3: Tangent Vectors ... I need some help in fully understanding Lee's conversation on computations with tangent vectors and pushforwards ... in particular I need clarification on the nature of the...

Homework Statement The points P (2,-1) and Q (3,-4) lie on the parabola y = -x2+2x-1 a) Find the slope of the secant line PQ. b) Find the slope of the tangent line to the parabola to the parabola at P. c) Find the equation of the tangent line at P. Homework EquationsThe Attempt at a Solution I...

I am reading John M. Lee's book: Introduction to Smooth Manifolds ... I am focused on Chapter 3: Tangent Vectors ... I need some help in fully understanding Lee's conversation on directional derivatives and derivations ... ... (see Lee's conversation/discussion posted below ... ... ) Lee...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 3: Advanced Calculus ... and in particular I am studying Section 3.3 Geometric Sets and Subspaces of T_p ( \mathbb{R}^n ) ... I need help with a...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 3: Advanced Calculus ... and in particular I am studying Section 3.3 Geometric Sets and Subspaces of T_p ( \mathbb{R}^n ) ... I need help with a...

Show from the definition of arctanh as the inverse function of tanh that, for $x \in (-1, 1)$ $$\tanh^{-1}{x} = \frac{1}{2}\log\left(\frac{1+x}{1-x}\right)$$ The definition of hyperbolic tangent is $\displaystyle \tanh{h} = \frac{e^x-e^{-x}}{e^{x}+e^{-x}}$ Let $\displaystyle y =...

$\displaystyle \lim_{x \to 0} \frac{x^2-\sin^2{x}}{\tan(3x^4)}$ How do you calculate this one? L'hopital gives me $\displaystyle \lim_{x \to 0} \frac{2x\cos^2(3x^4)-\sin{2x}\cos^2(3x^4)}{12x^3}$

On http://mathworld.wolfram.com/HeartCurve.html, the nicest heart-shaped curve is given by x=16sin3t y = 13 cos t-5 cos(2t) -2 cos (3t) - cos(4t) There are evidently two values of t, one in the first, one in the second quadrant, where you can get a horizontal tangent to this shape. Asking...

Given a scalar function g defined on a manifold and a curve f:λ -> xa, the change of the function along the curve is \frac{dg}{d\lambda} = \frac{dg}{dx^{a}}\frac{dx^{a}}{d\lambda} = T^{a}\frac{dg}{dx^{a}} where \frac{dx^{a}}{d\lambda} = T^{a} is the tangent to the curve. The argument that I...

Halo, my question is what is the shape of a single point? Is it round? When we find the tangent line it is in relation to the surrounding points around the point of interest. For a circle tangent line is line that cuts through a single point only but in a parabola this cannot be true because the...

Homework Statement Problem statement uploaded as image. Homework Equations Arc-length function The Attempt at a Solution Tangent vector: r=-sinh(t), cosh(t), 3 Now, I just need to reparameterize it using arclength and verify my work is unit-speed. Will someone give me a hint? Should I use...

Hi Community, I have this question. I start by creating the derivative, $$\d{}{x}(x^2-xy+y^2) = \d{}{x}(3) $$ and solve for $$\d{}{x}(3) $$ = 0 (Derivative of a constant is alway 0) I understand how to solve the derivative of $$\d{}{x}(x^2-xy+y^2)$$ and get $$2x-y$$ In the worked...

Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (ignoring the concept of a connection at the moment), but I'm not sure I fully understand this - is...

I am having issues figuring out how to do the "in the direction of the vector" part of my problem I have found the equation of the tangent line but i do not know how to the the next part. My question asks: Find the equation of the tangent line to the surface defined by the function f(x,y) =...

Homework Statement A cup is represented by the surface -(z-1)2 + x2 + y2 = 1 and it is on a table represented by the plane z=0 a) find the angle at which the cup intersects the table b) find the equation of the normal line to the cup at the point (0, √2 , 2) c) find the equation the tangent...

Hello! I've encountered a problem of find all points (x,y) on $f(x)=\frac{x-\sqrt{\pi}}{x+1}$ where there are tangent lines perpendicular to $y=-(1+\sqrt{\pi}x+7\pi e^{e^{{\pi}^{110}}})$ So I first found derivative and ended up with $f'(x)=\frac{1(x+1)-(x-\sqrt{\pi})(1)}{x^2+2x+1}$ and then...

Homework Statement Homework EquationsThe Attempt at a Solution what is the difference between the form z - zo = fx(x-xo) + fy(y-yo) and: Fx(xo,yo,zo) + Fy(xo,yo,zo) + Fz(xo,yo,zo)=0 ____________________________________________________________________________using the first eq : while...

Homework Statement Disclaimer: English is not my first language, so i apologize for any wrong math-terms. We look at the function f(x) = x^3. On the graph for f we have a point, P(a,a^3), where a =/= 0. The tangent to f through P cuts through f in another point, Q. Find Q and show, that the...

I was given the equation of a polynomial told to find the derivative. easy enough. Then asked to give the equation of the tangent line which I've only learned how to get in the form of the question: "find the equation of the tangent line at x=" They gave me the equation of a line parallel to...

Homework Statement A student could either push or pull, at an angle of 30 degrees from the horizontal, a 40kg crate, where the coefficient of kinetic friction is .21. The crate is moved 18m. Calculate the minimum work for pushing and pulling. Homework Equations W=F•(change in)X•cos(angle in...

My professor did this question in class and I am a little confused. I wrote it down in my notes but I kind of don't understand it. The question is: Find theta 1/4 of the way through the flight of a projectile in time He does not give us numbers. Everything has to be solved algebraically. My...

Homework Statement Find an equation for the tangent plane to a surface xz^2 +x^2y-z=-1 at the point (1,-3,2). Homework Equations (\vec{r}-\vec{r_p}) \cdot \nabla f(\vec{r_p}) = 0 The Attempt at a Solution [/B] First I found the gradient of the function \nabla f = (z^2+2xy)\hat{i} + x^2...

Find the equation of the line tangent to $$\sin\left({xy}\right)=y$$ At point $$\left(\frac{\pi}{2 },1\right)$$ Answer $y=1$ I didn't know how to deal with xy. No example given

Mod note: Thread moved from Precalc section Homework Statement F(x)=sqrt(-2x^2 +2x+4) 1.discuss variation of f and draw (c) 2.find the equation of tangent line to (c) that passes through point A(-2,0) The Attempt at a Solution I solved first part I found the domain of definition and f'(x) and...

What do the functions cosx/x and tanx/x represent?

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...

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 ##R=\frac{\left...

Homework Statement Find tangent line of y=xe^{\frac{1}{x}} at point x=\alpha and it's limit position when \alpha \rightarrow +\infty. Homework Equations Tangent of y=f(x) at point M(x_0,f(x_0)): y-y_0=f^{'}(x_0)(x-x_0) The Attempt at a Solution Applying the above equation for tangent of...

Homework Statement Show that The tangent at (c,ec) on the curve y=ex intersects the chord joining the points (c-1,ec-1) and (c+1,ec+1) at the left of x=c Homework Equations Legrange's mean value theorem The Attempt at a Solution f'(c)=ec Applying LMVT at c-1, c+1...