In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
The tangent line to a point on a differentiable curve can also be thought of as the graph of the affine function that best approximates the original function at the given point.Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.
The word "tangent" comes from the Latin tangere, "to touch".
The problem is to find the horizontal tangent lines of an equation. Here's my attempted differentiation.
y^2 = x^3 - x + 1
{dy/dx} = (3x^2 - 1)/(2y)
Correct, or no?
i'm going to need more help going forward even if that is right, I just want to make sure it is.
For which value of x...horizontal Tangent Line
Homework Statement
For which value of x does f(x) = \frac{k}{ax^{2}+bx+c} have a horizontal tangent line?
Homework Equations
Quotient Rule?
F'(x) = [g(x)a'(x) - a(x)g'(x)]/g(x)^2?
The Attempt at a Solution
Am I supposed to...
Homework Statement
At which point is the tanget line to the following curve horizontal?
y= a sin^{3}\theta
x = acos^{3}\theta
Homework Equations
The Attempt at a Solution
\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}
When \frac{dy}{dx} = 0 , this means that that the tanget...
Is it possible to have a tangent vector field on the unit 2-sphere x^2+y^2+z^2 =1 in
3D which vanishes at exactly one point? By the Poincare-Hopf index theorem
the index of such vector field at the point where it vanishes must be 2. Is that possible? If yes, can one write an explicit formula...
Homework Statement
Find the point on the parabola y= 4x^2 + 2x - 5 where the tangent line is perpendicular to the line 3x + 2y = 7.
Homework Equations
The Attempt at a Solution
I don't know what to do since I was away the last 3 classes since I was away. Help me please.
Slope of a tangent = limit of the function?
Hi.
I want to ask .
Can it be true that
The slope of a tangent line is a limit of a function.
or
The slope of a tangent line can be considered as a limit of a function?
?
Is it the same I mean the slope of a tangent line and a...
I have encountered a problem in my Calculus homework.
I have a position function, r(t)= (t^2)i + (4t)j and in my homework, I am asked to find the tangent to this curve at the point t=3. I did this by finding dy/dx, or 2t/4 @ t=3 is 6/4. However, I am also asked to relate this to the...
Given equation x^2/9 + y^2/4 = 1. Determine the two points on the ellipse having a tangent line passing through the point (0,3). Cant seem to figure this one out? I have found the derivative of the slope which is -4x/9y. I don't know how to use that slope in order to find a tangent line that...
Affine spaces can be regarded as smooth manifolds if we take the natural topology and affine coordinate charts as atlas. So, if M is an n-dimensional affine space, then the tangent vector of a curve C: [0,1] \rightarrow M in a point p = C(t_0) can be defined as a derivation (as in any smooth...
Dear all,
I can formally understand one of the many definitions of tangent vectors to a manifold, but what are they in reality? It should depend on the nature of the points of the manifold, for example, if M={set of events of general relativity}, then vectors are velocities. Other examples...
If cosine is equal to -12/13... Find sine and tangent in Quadrant
If cosine is equal to -12/13... Find sine and tangent in Quadrant Two... What is the answer??
Find the tangent lines to the ellipse x^2 + 7y^2 = 8 at the point (3,0)
Slope-intercept form: y=mx+b
I know you have to differentiate the equation implicitly to get the slope, but you come across a zero in the denominator and that has me stumped.
1. Find the equations of all tangents to the graph f(x) = x^2-4x + 25 that pass through the origin.
2. The relevant equations to my knowledge are the slope formula and f ' (x)
3. First I found the derivative using various rules to be f ' (x) = 2x - 4
after this i used the slope formula and my...
Hello,
I'm having trouble understanding an example from my test and I would appreciate your help clarifying how to get from one step to the next.
Problem: Find an equation of the tangent line to the hyberola y = 3/x at the point (3,1)
Eqation: m = \frac{f(a+h) - f(a)}{h}
m =...
Hi, I'm having trouble plotting the tangent line on a curve (or the derivative evaluated at a point) in spice. I know how to plot the derivative of the entire curve using the d() function, but I don't know how to evaluate it at a point. Any help is much appreciated!
I need someone to help me! i have been given a center point of (3,-1,-2) of a sphere and i need to get an equation for when it is tangent to the a) the yz plane b) y=7. i know i need to use the distance formuala to attain the radius, but i am majorly confusing myself in what coordinate to pick...
If the tangent line to y = f (x) at x = a is the same as the tangent line to y = g(x) at x = b, find two equations that must be satisfied by a and b.
I don't understand how to go about this problem. I tried putting the equations of the tangent lines together using the variables in the...
Homework Statement
Solve the following limit:
\lim_{x \rightarrow 1}(1-x)\tan{\left(\frac{\pi x}{2}\right)}
The Attempt at a Solution
I solved it using L'Hospital rule, it's equal to 2/pi, but is there any other way how to solve it? thanks a lot!
The same question would apply to
\lim_{x...
Hi, I'm programming a visual implimentation of a bezier curve for a coursework. It would be beneficial for me to find the tangent at any point t on the curve. I can calculate the position of a point t, and so can hash the problem somewhat by finding the gradient between t+- a small value, but...
Sorry title was supposed to be Conic Sections, but my I key is sticky :)
I had a question today, It went somethng like this:
An epllipse of equation ((x^2)/4) + y^2 = 1
Find the equation of the tangent which passes through point P: (4,0)
Well this was a mock exam question, where no answers...
Sorry if the title is a bit vague :/
The Problem: Find the point on the parabola y=1-x^2 at which the tangent line cuts from the first quadrant a triangle with the smallest area.
Relevant Equations: y = 1-x^2 ; y' = -2x ; A= 1/2bh
I'm basically stuck near square one, I found...
Homework Statement
The point P (1/2, 0) lies on the graph of the curve of y=sin(2x-1) Find the gradient of the tangent to the curve of P
Homework Equations
...I don't know
The Attempt at a Solution
I don't know where to start with this problem
1. The problem statement, all variables and given/known
i have to find all tangent lines in the equation below that pass though the point (0,0)
y=x^{3}+ 6x^{2}+8i took the derivative and got this.
y=3x^{2}+12x
then i substituted the points x=0 and y=0 into the derivative equation and the...
Homework Statement
if the tangent line to the graph of y=f(x) at (2.3) has an equation x-y+1=0, then f'(2) =?
The Attempt at a Solution
We did these problems in one small session at the beginning of the semester, but my notes aren't clear and I am not sure where to begin. Do I just...
Homework Statement
I'm supposed to prove, that when G is a Lie group, i:G\to G is the inverse mapping i(g)=g^{-1}, then
i_{*e} v = -v\quad\quad\forall \; v\in T_e G
where i_{*e}:T_e G \to T_e G is the tangent mapping.
Homework Equations
I'm not sure how standard the tangent mapping...
What kind of tangent distributions are not integrable? Is there concrete examples with two dimensional non-integrable distributions in three dimensions? When I draw a picture of two smooth vector fields in three dimensions, they always seem to generate some submanifold, indicating integrability.
Homework Statement
Let f be the function given by f(x) = (2x-5)/(x^2-4).
a.Find the domain of f.
b.Write an equation for each vertical and each horizontal asymptote for the graph of f.
c.Find f'(x).
d.Write an equation for the line tangent to the graph of f at the point (0,f(0)).
The...
Homework Statement
The tan(theta)=2/5
Homework Equations
The Attempt at a Solution
I forgot how, I had no clue so I looked in the solutions and it was 2.62 radians, I just don't know how they got that, I am not sure on what to do. Its for trigonometric forms of complex numbers...
Ok, I'm pretty much at whit's end trying to figure this review question out. Apparently my teacher forgot to mention that our book couldn't teach us everything we need to know for our test... Anyhow, the question is as follows, and I'm utterly at a loss as to what the answer is:
Find the...
Homework Statement
1. Let f be the function defined by f(x) = -2 + ln(x^2).
a) For what real numbers x is f defined
b) Find the zeros of f
c) Write an equation for the line tangent to the graph of f at x=1
Homework Equations
The Attempt at a Solution
a) all...
Here is what the question asks:
Find the coordinate of all points in the graph of y=3e^x - x^3 at which the tangent line passes through the point (1,0).
----------------------------
I am told that the 2 points are (-.872, -1.027) and (2.275, 13.657). Any help at all that you can give...
Tangent space of single layered hyperboloid
Ok i´m given a single layered hyperboloid given by \left(\frac{x}{a}-\frac{z}{c}\right)\cdot\left(\frac{x}{a}+\frac{z}{c}\right)-\left(1-\frac{y}{b}\right)\cdot\left(1+\frac{y}{b}\right)=0
Now the Problem asks me to take this as a vanishing...
My book really doesn't go into a lot of depth but I was wondering if this is correct
If we are asked to find the tangent line of a specific value of t for a given parametric equation then we can find the equation of the tangent line in either rectangular or parametric functions.
Rectangular...
i would like to know if my answer is correct can someone tell me the results for this:
Find the equation of the tangent line to x^3-2x^2y+y^2=17 at (2,-1)
ive gotten y'=(3x^2+4xy)/(-2x^2-y)
and my line equations is
y+1=(-1/4)(x-2)
thanks in advance
[SOLVED] Integration of Inverse Tangent Function
\int \frac{e^{x}}{4+9e^{2x}}dx
Saw the problem, looked at it for a bit. Noticed that it is a inverse Tangent function. Played with some integration by parts and substitution and couldn't figure it out. Can anybody toss me a starting point on...
Why are the tangent vectors of smooth manifolds defined as mappings C^{\infty}(p)\to\mathbb{R} that have the similar properties as derivations?
If a manifold is defined as a subset of some larger euclidean space, then the tangent spaces are simply affine subspaces of the larger space, but if...
Homework Statement
\int {\frac{{dx}}{{\sqrt {x^2 - 2} \left( {x^2 - 1} \right)}}}
Homework Equations
\frac{d}{{dx}}\arctan (x)The Attempt at a Solution
\frac{d}{{dx}}\arctan (x)
seems to be part of it, I can't quite get much farther...
There is a base-quadrilateraled pyramid ABCDS , which base is quadrilateral ABCD. Inscribed sphere is tangent in point P on the ABCD wall. Proof that
<)APB+<)CPD=180'
I don't like geometry and i really don't know how to start.
Homework Statement
y = 9sin(x)cos(x)
Find all points where tangent line is horizontal.
The Attempt at a Solution
I get y' = 9cos^2x - 9sin^2X
I plug in zero for the slope and get 9 but I'm stumped after that. How can I get all the horizontal tangent line points?
Homework Statement
Find the tangent line to the curve y^2 (y^2 - 4) = x^2 (x^2 -5) at the point (0,-2).
Homework Equations
y-y1=m(x-x1)
The Attempt at a Solution
I d/dx-ed both sides and was left with dy/dx=0
so
y=-2
Does that seem right?
Find the equation of the line tangent to the curve at (5, -3)
(x-2)^2 + (y+3)^2 = 9
I solved the derivative to be dy/dx = ((-2x+4)/ (2y+6))
when i plugged in the points (5, -3) I got the slope as -6/0...How is this possible??
How can i find the equation of this curve if the slope is...
This is a general question, but what is the difference between finding the linearization and the tangent line to the same curve? And what about at a specific point?
Let \tan_1x=\tan x and \tan_{k+1}x=\tan\tan_kx.
It's fairly clear that the sequence (\lfloor\tan_n1\rfloor) = http://www.research.att.com/~njas/sequences/A000319 is chaotic, in the sense that it can diverge from (\lfloor\tan_n(1+\varepsilon)\rfloor) even for small \varepsilon.
1. Any...
[SOLVED] Derivative and tangent line
Homework Statement
y = (x)/(x+x^-1)
Find an equation of the tanget line to the graph at x=2
Homework Equations
The Attempt at a Solution
I used the quotient rule and got the derivative to be 0 / (x+x^-1) = y' so then the slope is 0 of the...
[SOLVED] Tangent Line
Homework Statement
Find all tangent lines that are tangent to Y=x^2 and go through the point (2,3)
Homework Equations
The Attempt at a Solution
At first I thought I knew how to do this problem, I found the derivative of the equation got:
y' = 2x ; then...
[SOLVED]Sine, Cosine and Tangent Trig Help
I'm in 10th grade, I was just doing my homework when this dawned on me:
How would one find Sine, Cosine and Tangent without a calculator. So if I am stuck in the desert with a stick I could find the Cosine of 73º by stick and sand method...