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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Constructing a tangent line to a point on the curve $y = x^3$ using a compass and straight edge?

There are number of ways of constructing a tangent line to the curve $y = x^2$ using a compass and straight edge. Does anyone know of way of constructing a tangent line to the curve $y = x^3$ using a ...
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How does one find a parametrization of a curve , given the tangent vector and level set

Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ be a $\mathcal{C}^1$ function, i.e the directional derivatives exists and are continuous. Let $\vec{a}$ be a point in the non-empty regular level set $f=...
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Find the points on the curve where the tangent is horizontal

Question. Given $y^{2}=x^{3}+ax+b$, find the points on the curve where the tangent line is horizontal. Attempt. Let $f(x,y)=x^{3}-y^{2}+ax+b=0$ The tangent is horizontal at points where the gradient ...
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Is it possible for more than 2 different curves to have the same tangent point? [closed]

Question is above.Very direct. Just yes or no and why. Help please.
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Find the Point/s on the Curve $y-x^3=0$ where the normal line have a slope of $\frac{-1}{3}$.

I am bit clueless on how to start the problem. The only idea I have is to use derivatives, yet I can't continue on. I have tried researching different problems connected to it as well, but the results ...
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Scale intersecting circles fixed at pivots so that they have only one point in common

Given two points, A and B; Given two circles, having 2 points in common, I1 and I2: one circle at center C1, with radius r1, with the point A on to it and another circle at center C2, with radius r2, ...
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Is this a sufficient amount of knowledge to define a unique ellipse?

Recently, I've been trying to work out a closed formula for the Mandart inellipse of a triangle, and I made a little plaything on Desmos to streamline the process. So far I've successfully located the ...
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Is there widely accepted notion of tangent plane(or similar) for a vertex of a polyhedron?

For a vertex v of a polygon, I think it is reasonable to define its tangent line as ...
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Derivation of the formula for a tangent plane to a surface

I am trying to derive a formula for the tangent plane to a surface at $(x_0,y_0,z_0)$. I started with $F(x,y,z)=0$ for $(x,y,z)$ near and at $(x_0,y_0,z_0)$. It can be seen that any curve in the ...
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Infinite tangent line and rate of change.

We know that if the tangent line exists then the derivative exists except for vertical tangent lines. For example $f(x)=x^\frac{1}{3}$ has no derivative at $x=0$. How can we substantiate it both ...
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Difficulty understanding Spivak's description of a tangent line for functions of the form $c: \mathbb R \to \mathbb R^2$ in 4th Ed. of 'Calculus' book

In the Chapter 12 Appendix of Spivak's Calculus (4th Ed), the following paragraph is written about an arbitrary vector valued function $c$, which should be interpreted as $c: \mathbb R \to \mathbb R^2$...
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Question on tangent and normal for the curve ${x^3 \over a}+{y^3 \over b} =xy$

Find at what point on the curve $${x^3 \over a}+{y^3 \over b} =xy,$$ the tangent is parallel to one of the coordinate axes.
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Find the area bound by two intersecting circles and a tangent line to one of the circles

There is a circle with radius r1 and centre D This intersects a circle with radius r2 and centre C Tangent line AB is always tangential to circle with centre D It can be assumed the circle with centre ...
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Equation of a circle with radius and tangent [closed]

Find the equation of a circle with radius 10 and tangent to the line 3x + 2y = 5 at the point (1,1)
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Polynomial function with restrictions

Function $f\left( x \right)$ which is defined in $x>a$ and quartic function $g\left( x \right)$ with leading coefficient $-1$ satisfy following conditions. (a is a constant.) a)For all real numbers ...
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Creating a (set) of differential equations from given conditions

I am really struggling on a task about differential Equations. Which functions y(x) fulfil the following conditions: For every x > 0, the tangent on the point (x|y) of the function graph ...
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Find center of circle given circle radius, tangent line and a point lies on the circle [closed]

enter image description here Based on the image, is it possible to find the center point of circle (h,k), given the radius of circle is \sqrt{17}, the tangent line is y=-(-1/4x)+(9/2) and a point lies ...
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Find the equation of a line that is tangent to both parabolas simultaneously [closed]

Consider the parabolas $y=x^2$ and $y=x^2-2x+2$. How to find the equation of a line that is tangent to both of them at the same time? Please, walk me through the most intuitive solution.
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Relationship between $f$ and its derivative $f'$

I have been starting to learn derivatives and finding the equation of the tangent of a curve. However, I am a bit confused as to how to solve this question: Let $f$ be a differentiable function such ...
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Is there a function whose graph intersects every tangent line at exactly 2 points?

Is there some $f:\mathbb{R}\to\mathbb{R}$ differentiable at every point such that $\forall x$, the tangent to $f$ at $(x,f(x))$ intersects the graph of $f$ at $2$ points (counting (x,f(x)))? This ...
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Tangent plane of $f(x,y) := 6y+4x-x^3y$ in $(1,0)$

Let $f: \mathbb{R^2} \to \mathbb{R}$ with $f(x,y) := 6y+4x-x^3y$ I also have to determine the function definition $\tau_{(1,0)}(x,y)$ of the tangent plane to the graph of the function $f$ at the point ...
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Calculate height from tangent of a point on the circumference of a circle

I have a mechanical press which uses rotary eccentric gears and plungers to translate rotation into a linear movement of fixed length, or stroke. The machine moves linearly to a maximum distance of ...
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Finding equation of a hyperbola that is tangent to a known line.

I have the line $y = -4x + 120$. I want to find the equation of a hyperbola with the form $y = \dfrac{k}{x-a}$ (i.e. rotated $45^\circ$) that is tangent to the line at point $(15,60)$. As long as the ...
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Find the equation of a line normal to a given function, given its slope (no points.) [closed]

I have no idea how to solve this without points. Find the equation of the line, normal to $y=(2-x)^4$, which has slope equal to $-\dfrac{1}{32}$ If it helps, the solution is $y=(-1/32)x+(129/8)$ But I ...
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Find the Center Coordinate of a Sphere Given the Tangent Point

I'm struggling with a problem I'm coding on for days now. Really appreciate your inputs. Suppose you have a line segment in 3D space with endpoints (x1, y1, z1) and (x2, y2, z2). Along the segment is ...
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Prove that parallel lines drawn at the ends of diameter are tangents If one of the parallel lines is perpendicular to the diameter. [closed]

Let there be 2 parallel lines PQ and RS with a transversal intersecting PQ, RS at A,B respectively such that AB is perpendicular to PQ. Prove that AB is also perpendicular to RS. PROVE IT WITHOUT ...
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Find the conditions of $a, b$ and $c$ so the line is tangent to the parabola $y=x^2$

I tried a method for finding a tangent to a parabola described in a book. The example from the book was $y=2x-1$ is tangent to $y=x^2$. I solved it by plugging $y$ in the 2nd equation with $y$ from ...
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Tangential Circles connecting Oriented Points

I have two arbitrary points (A and B) in a plane, and I'm working in cartesian coordinates. Each of these points has an associated arbitrary unit direction vector - so from point A, for example, I ...
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Determine the equation(s) of the lines that are tangent to $y = x^2 + 3x + 1$ and pass through the origin. [closed]

I am trying to solve the question but I am not sure how to approach it. The derivative is 2x+3 but does that have any significance when solving this?
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2 answers
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Equations of tangents to an ellipse

Suppose you're given the ellipse $$ (r - H)^T Q (r - H) = 1 $$ where $r = [x, y]^T $, $Q $ is positive definite, and $H$ is the center of the ellipse, and a point $A$ lying outside the ellipse (i.e. $(...
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How to solve this geometry problem about circle?

On a circle w, we draw two pararrel tangents EA and FB where E and F are tangential point. Segment AGB is another tangent where G is the tangential point. If EA=84 and FB=189, find the radius of the ...
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Distance between two intercepts of two tangents to a circle

A circle of radius $5$ is centered at $H(10,5)$. Tangents from $A(0, 16) $ are drawn to the circle as shown in the diagram below. Find the distance $d$ between their $x$-axis intercepts. Here is ...
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Family of straight lines tangent to $e^x$

I just started to learn about Differential equations, I've been reading about families of curves, but I I just got stuck with this problem. I have been Trying to solve it, I have come to find the line ...
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Solve tangent of $y=\left(\log_{a}{x}\right)^2$ and $y=-ax+2$

How can I solve the tangent point and $a$ when $f(x)=\left(\log_{a}{x}\right)^2$ is tangent to $g(x)=-ax+2$? Although this can be solved by substituting $a=e^2$ and $x=e^{-2}$, then $f\left(e^{-2}\...
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Equivalent condition for a tangent line to conic section in real projective plane

Given a conic section $S$ in real projective plane $ \mathbb{RP}^2$ it can be represented in homogeneous coordinates by the equation \begin{equation}Ax_0^2+Bx_1^2+Cx_2^2+2Dx_0x_1+2Ex_1x_2 +2Fx_0x_2=0\...
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A High School Calculus inequality

The problem, encountered in a high school math textbook in the exercises on the MVT,goes as following: Let $f(x)=e^x-ex, x\geq 0$ and $f(\ln{2})<2$, prove that $y=(1-e)x+1$ is tangent to the graph ...
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Given an ellipse and a reference point, how to find the two lines that are tangent to the ellipse?

I have an ellipse, possibly rotated and shifted from the origin, which is given by a parametrization similar to this one: $$ \begin{aligned} x &= x_0 + a\cos\theta\cos\alpha - b\sin\theta\sin\...
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Find an equation tangent to the graph of y=f(x) at the point where x=-3 if f(-3)=2 and f'(-3)=5 [stuck]

Problem: Find an equation tangent to the graph of y=f(x) at the point where x=-3 if f(-3)=2 and f'(-3)=5 What I've tried: I tried solving this the way "normal" tangent equations are found ...
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Number of tangent lines that can be drawn to $y=x^2$ from any arbitary point on $xy$- plane

I made up this question but don't know how to solve it! Consider $xy$- plane and the graph of $y=x^2$ on it. How many tangent lines can be drawn from any arbitrary point (say $(x_0,y_o)$) to the ...
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Is there an angle chase solution for this problem?

Given two disjointed circles $c_1$ and $c_2$, external to each other, let $A$ be the meeting of their internal tangents and let $K$ be the orthogonal projection of $A$ in one the common external ...
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How can I determine wheter or not two curves are tangents?

Let $a_0,b_0 \in \mathbb{R}, a_0^2 + b_0^2 = 1$. How can I show that the curves $$\frac{x^2}{a_0^2} + \frac{y^2}{b_0^2} = 1$$ and $$x+y = 1$$ are tangents to eachother? I tried to find a point such ...
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The area bounded by the tangential lines to an even-power exponential function

The problem: Consider a function, $f(x) = x^m$, where $m$ is an even, natural number. Then, consider two tangential lines, grazing the points $(-x,y)$ and $(x,y)$, respectively. My question is, what ...
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5 answers
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Why $f(x) = x^2$ has variable derivative but its tangent has constant slope?

I'm taking Brilliant.org's calculus course, and I'm on the section called The Derivative. My (mis)understanding: A tangent line is a linear function that grazes a point, $a$, on the graph of a ...
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Does tangent line of inflection point always passes through the curve?

I saw many functions on my book and all of the tangent line of inflection point always pass through the curve, Here are examples : Example 1 : $$f(x) = x^3 \quad (x=0)$$ Tangent line at $x = 0$, $l:y=...
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How can I calculate a smooth tangent wave that has no periodic delays?

I want to use a tangent function to create a waveform that always passes itself back to -1 at the end of each cycle to 1. When a tangent function only moves between -1 and 1, there is a gap left ...
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Find parameterization of line

Find a parameterization of a line tangent to the level surface S given by $e^{xy}+z^2+y^3=2$ at the point $P=(1,0,1)$ and orthogonal to $u=(0,-2,1)$. I found an equation for the tangent plane to the ...
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tangent through [0:0:1] and regular point of a curve

I am given the curve $G = (x_1-x_2)^3 - x_0 x_1^2$ and want to compute all of its regular points whose tangents go through $[0:0:1]$. Firts, the line $L = ax_0 + bx_1 + cx_2$ has to go through $[0:0:1]...
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Mysterious ratio

I need some help or hints with a problem I can't seem to fully understand. I tried to experiment with Geogebra and I think I discovered the solution. My problem is that I don't know the reason why it ...
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Morris Klein Tangent on Parabola X intersection Question

Im currently trying to solve a Math Exercise from the book "Morris Kline - Calculus An Intuitive and Physical Approach" and iv been struggling to get a grip on it. The Problem I think i know ...
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2 votes
4 answers
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Prove that the angles between tangents to circles centered on a trapezium are equal

Suppose a trapezium $ABCD$. There are circles $m,n$ with centres midpoint of leg $BC=M_{1}$ and leg $AD=M_{2}$, respectively; and diametres $BC$ and $AD$, respectively. The point $P$ is the ...
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