Questions tagged [tangent-line]
For questions on the tangent line, the unique straight line that is the best linear approximation to a function at a point.
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What's wrong in this line of reasoning (involving tangent lines to $x^2$)?
Context. I have been trying to solve the problem described below. I found a similar question and I see where I have gone wrong. I assumed the points on the parabola would be symmetric across the $y$...
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Proving that the feet of the perpendiculars from the foci on any tangent lie on the auxiliary circle using parametric coordinates
In the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, the parametric equation for a tangent is $\frac{x}{a}\cos(h)+\frac{y}{b}\sin(h)=1$, and the foci are $(ae,0)$ and $(-ae,0)$, taking the focus on the ...
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Approximation of plane curve with tangent vector
Let $c:[-1,1]\to\mathbb{R}^2$ a $\mathcal{C}^1$ planar curve and suppose that $c(0)=(0,0)$ and $c'(0,0)=(a,0)$, $a>0$. I'm trying to prove the following statement (without any success): there exist ...
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What points in the plane of the graph $y=x^3$ have three tangents to the curve passing through them?
I’m studying high school math and encountered this question in the extension section for derivatives. The text says an algebraic solution to the problem is harder but possible. There is also a similar ...
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The gradient at the meeting point
When two straight lines touch at one point, their gradients are most definitely not the same.
However, when I draw a tangent to a curve, why is the gradient of the tangent the same as the gradient at ...
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If the equation $7x + y = k$ is the equation of a line tangent to the graph of $y = 9x + 1/x^2$, what is the value of $k$ [closed]
How would I solve this? I tried everything, but I am still lost. Please help. I watched some videos, but I still don't know how to work it with this:
If the equation $7x + y = k$ is the equation of a ...
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The relation between formal group $\frac{X+Y}{1-XY}$ and algebraic group $x^2+y^2=1$ with group law $*$
$T(X,Y)=\frac{X+Y}{1-XY}$ is a power series which satisfies axiom of formal group.
My book reads this formal group comes from algebraic group $S: x^2+y^2=1$, with group law $*$:$(x_1,y_1)*(x_2,y_2)=(...
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Chord of a conic touches another conic
If $P, Q, R$ are three points on the conic $\frac{l}{r}=1+e\cos\theta$ and the tangent at $Q$ meets $SP$ and $SR$ in $M$ and $N$ so that $SM=SN=l$, where $S$ is the focus, then prove that the chord $...
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continuous differentiable, homogene, equation
Hello I have two questions:
Let $f : \mathbb{R}^n \setminus {0} \rightarrow \mathbb{R} $ be continuous differentiable and homogene of degree $k$.
Prove that we can extend $f$ to a continous ...
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Find the equation of the tangent line(s) to the curve $f(x) = \sqrt{x + 5}$, that passes through the point $(-8, 1$). [$-8$ can't get subbed in]. [closed]
Going through the process of finding the derivative, which is
$\dfrac{1}{2\sqrt{x+5}}$
,and substituting x gives an undefined slope, as you end up taking the square root of a negative number
I know ...
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Tangent to two parametric curves
If I have two parametric curves one defined as x(t) and y(t) and another x(s)+c and y(s).
My assumption is that if I set t = s = value. I can find the two slopes at these values. i.e. dy/dx = x'(t)y'(...
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Why is the distance equal to the height between these two circles?
I have a follow up question for a previous question linked here:What is the equation for the distance between these two circles?
In that question asked for an equation to find the distance between the ...
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Can torsion be negative?
If $B$ denotes the binormal vector, $S$ denotes the arc length, $\tau$ denotes torsion, and $N$ denotes principal normal vector, then,
$$\frac{dB}{dS} = -\Bigg|\frac{dB}{dS}\Bigg|N = -\tau N$$
So if $\...
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How can you find the equation of the line that is tangent at two distinct points to the curve? [closed]
Given a curve $y = x^3-x^4$, how can I find the equation of the line in the form $y=mx+b$ that is tangent to only two distinct points on the curve?
The problem given is part of the Madas Special Paper ...
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Regular differentiable curve has strong tangent
I have a doubt about exercise 1.3.7 from the book Differential Geometry of Curves and Surfaces by Do Carmo.
The exercise asks to prove that a curve $\alpha$ which is $C^1$ at $t_0$ and $\alpha'(t_0)\...
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How do I find the function and derivative of an unknown curve?
I have $x$ and $y$ values to plot the curve and I need to find a tangent line of slope 1 that intersects the curve (and the point at which it intersects). I was trying to do polynomial and exponential ...
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A chain of circles of radius $1/n^p$ is tangent to the $x$-axis. What is the horizontal length of the chain?
I recently discovered that, if a chain of circles of radius $1/n^2$, where $n\in\mathbb{N}$, is tangent to the $x$-axis, then the the horizontal length of the chain is exactly $2$.
This can be shown ...
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Find the equation of the circle touching the line $(x-2)\cos\theta+(y-2)\sin\theta=1$ for all values of $\theta$
Find the equation of the circle touching the line $(x-2)\cos\theta+(y-2)\sin\theta=1$ for all values of $\theta$
The answer is given on toppr website.
It says, $(x-2)\cos\theta+(y-2)\sin\theta=\cos^2\...
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Is there a general way to prove that PX, PY, PZ are in GP without specifically letting A to be on y-axis?
Let BC be the chord of contact of the tangents from a point A to the circle $x^2+y^2=1.$ P is any point on the arc BC. Let PX, PY, PZ be the lengths of perpendiculars from P on the AB, BC and CA ...
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tangent slope of cardioid graph
Hi i am student working on calculus and i have got question that i came up with wrong answer.
So we get $r=2(1+\cos\theta)$ cardioid function and the question is to looking for Θ angle where tangent ...
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Vertical Tangents and Derivatives
According to wikipedia (https://en.wikipedia.org/wiki/Vertical_tangent): A function ƒ has a vertical tangent at x = a if the difference quotient used to define the derivative has infinite limit:
${\...
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Proof of concurrency of transverse tangents and line joining centers of two disjoint circles
I read that "the transverse common tangents to two disjoint circles and the line joining the center of the circles are concurrent" and tried proving it.
Can I get a hint to prove that lines ...
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Tangent cone of an arbitrary algebraic curve
So, my problem is: given a real/complex (I will assume complex) algebraic curve, say $f(x,y)$, or $f(z,w)$ for $x,y\in\mathbb{R}$ or $\in\mathbb{C}$ (I would like to hear your thoughts in either case),...
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Calculating tangent at two points $(4a,8a)$
Find the equation of the tangent to the curve $ay^2=x^3$ at the points $(4a,8a)$
I have re-arranged the equation to get
$$y = \left(\frac{x^3}{a}\right)^{\frac{1}{2}}$$
Then taking its derivative I ...
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Calculating the tangent at the point x = $\theta$
I am finding the tangent at the point $\theta$ when $x = a\cos(\theta), y = b\sin(\theta)$
The tangent is calculated with the following
$$\frac{y-f(\theta)}{x-\theta}=f'(\theta)$$
at a point where $x=\...
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Calculating tangent at point x = a
The equation of the tangent to a curve $y = f(x)$ at a point where $x = a$ and $y = f(x)$ at which the derivative $f'(x)$ exists and is finite, is,
$$\frac{y-f(a)}{x-a}=f'(a)$$
Find the equation to ...
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Trouble understanding the tangent plane of a surface
$$z = z_0 +a(x-x_o)+b(y-y_0)$$
where $a$ and $b$ are the partial derivates with respect to $x$ and $y$. I dont understad why $a$ and $b$ are there since they are tangent to the plane not normal?
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How to find radius of a circle drawn inside a triangle, if we know the base and the distance from point to the circle
How to find radius of a circle drawn inside a triangle, if we know the base and the distance from point to the circle
Here we can see the problem described in the title, we know only the distances a=...
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Tangent to a graph through an exterior point
Say you are given $f(x) = 5x^2$ and you want to find a tangent line to $f$ that goes through $P(0|-10)$.
The two options you have is using the tangent line equation, $t(x) = f'(a) \cdot (x - a) + f(a)$...
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What is meant by dividing the abscissa of the point of contact?
The tangent at any point on the curve $x=at^3, y=at^4$ divides the abscissa of the point of contact in the ratio $m:n,$ then find $|m+n|$ ($m,n$ are coprime)
I don't know what the meaning of '...
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Find equation of the line tangent to the level curve of $f(x, y)$ at given point
I was given the following problem:
Let $f(x, y) = \frac{x-y}{x+y}$ and $P=(1, 1, f(1, 1))$. Find the gradient of $f$ at $P$, the equation of the plane tangent to $f$ at $P$, and the equation of the ...
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Condition for two common tangents
Let $C_1$ and $C_2$ be the centres of two circles, $r_1$ and $r_2$ be their radii respectively. I know that the condition for the existence of four common tangents between the two circles is $r_1 + ...
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Find coordinates of a point for a derivative of a parametric curve
Find the coordinates of the points at which the given parametric curve has a) a horizontal tangent and b) a vertical tangent.
The parametric curve is $$\mathscr{C}=\begin{cases}x=t^2+1\\y=2t-4\end{...
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Proving Tangent line perpendicular to the radius using Contradiction
Here is the proof by contradiction to prove tangent to the circle and radius are Perpendicular at the point of Contact. Let me know whether the proof is Complete.
Consider the circle with center $O$ ...
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discovering Mean Value Theorem
mean value theorem for single variable function is very easy and intuitive once you "see" the formula.
Actually, My question, slightly weird but helpful, is that How does someone come up ...
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Finding tangent point of a line with a given function
Given the following function:
$f(x)=x\left(1-\frac{n}{x}\right)^d$
with $n\geq1$, $x>n$, $d>>1$.
I need to find the tangent point with the line $ax+b$, with $a$ given. Note that $0<a<1$....
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Is it correct to say the differential is the equation of the tangent line?
For example. Given function
$f(x)=\sqrt[3]{x} \Rightarrow f'(x) = \frac{dy}{dx}= \frac{1}{3}x^{-2/3}$
$dy = (\frac{1}{3}x^{-2/3})dx$
compared to...
$\Delta y = \frac{1}{3}x^{-2/3}\Delta x$
I know ...
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Determine the numbers a, b and c such that it satisfies the condition
I have a function $f(x) = x^3 + ax^2 + bx+ c$ and I need to solve for the numbers $a$, $b$, and $c$. The numbers need to satisfy the following condition:
The slope of the secant line defined by points ...
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Why am I getting $\beta=90^{\circ}$
Consider the geometry below, where the small circle is touching both semi circles of radius $5$ and the side of the square. Find the radius of the small circle.
My try: $M$ and $N$ are centers of ...
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Finding the tangent line of curve $x^2y^2+5xy=14$ at $(2,1)$
I want to check if my work is correct.
Find the equation of the tangent line to the curve at (2, 1)$$𝑥^2𝑦^2 + 5𝑥𝑦 = 14(1) $$
solution:
The tangent is a straight line so it will be of the form:
$...
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How does Fermat's method of adequality actually work?
I've been looking at Fermat's method of obtaining a tangent line through this resource https://cedar.wwu.edu/cgi/viewcontent.cgi?article=1012&context=wwu_honors and it says to basically take an $...
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Perimeter of triangle $SPR$
I have to calculate perimeter of triangle $SPR$. I know, that length between $A$ and $P$ is $25$. I know that length between $B$ and $P$ is also $25$. Also in $A$ and in $B$ is angle $90$.
Any help?
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Let $r(t)$ be a parameterized curve with $r(t) = (x(t), y(t), z(t))$ and $x = (1 + \cos(t)), y = \sin(t), z = 2 \sin(t/2)$
Let $r(t)$ be a parameterized curve with $r(t) = (x(t), y(t), z(t))$ and $x = (1 + \cos(t)), y = \sin(t), z = 2 \sin(t/2)$.
Determine the unit tangent vector and curvature at a general point.
I am ...
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Drawing tangents to curves by hand
Recently I was doing a practice test that asked to draw a tangent on the graph at x = -1.5 and calculate it's gradient. However when drawing the tangent I found the gradient to be extremely inaccurate ...
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an example of tangent equation calculation, from a book, that I do not understand
I am in trouble with an example of an equation of a tangent from a book.
Here's what my book is writing (in french) :
I translate it (summarizing a bit) : take a T(X,Y) point on the tangent,
the ...
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Condition for tangency in 3D space
So I have a cone of the form $x^2+y^2 = k^2 z^2$ for some $k\in\mathbb{R}$ and some ellipse $\vec{x}(\theta)= \vec{c_0} + (\cos\theta)\, \vec{u} + (\sin\theta)\, \vec{v}$ parameterised by $\theta$, ...
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The class of curves whose derivative can be computed solely with analytic geometry
Reading old math papers and refreshing my geometry has lead me to the following:
Question:
What is the largest class of curves $f:[a,b]\to \mathbb{R}$ whose tangent line at a point $x_0\in [a,b]$ can ...
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Finding a general expression for the intersection of tangent planes to a pair of intersecting n-D hypersurfaces.
If $f=0$ and $g=0$ be 2 surfaces in $\mathbb{R}^3$ and their curve of intersection passes through the point $p$, then the tangent planes at $p$ are :
$$
\sum_{1≤i≤3} (x_i-x_{ip}) \partial_{x_i}f(p)=0
$...
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Tring to find out what this relationship is called, so I can figure out how it works.
Picture of what I am looking for
I have a formula where if I have a circle sitting in an saddle angled at 10 degrees and I know the distance from the bottom of the saddle to the top of the circle it ...
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Differential as a tangent to $\Delta f(x)$
I had come across the following
The number $h = x − x_0$ , that is, the increment of the argument, can be regarded
as a vector attached to the point $x_0$ and defining the transition from $x_0$ to $x ...
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