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Questions tagged [tangent-line]

For questions on the tangent line, the unique straight line that touches a function locally only once.

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Find : $\lim_{n\to\infty}\sum_{k=0}^n\frac{\binom{n}{k}}{n^k(k+1)}$ [on hold]

I'm try to find this lim $\lim_{n\to\infty}\sum_{k=0}^n\frac{\binom{n}{k}}{n^{k}(k+1)}$ Is this limits can be done by integral !? Or inequality Someone help me hints me Thanks!
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On a particular drawing, a pulley wheel can be described by the equation $x^2+y^2=100$. For more please check the pic of the problems

On a particular drawing, a pulley wheel can be described by the equation $x^2+y^2=100$. For more please check the pic of the problems
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1answer
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How to calculate the vector intersecting a sphere tangent and plane

I have a sphere centred at a point (x, y, z) = (0, 0, 0) with radius r = 1. I have a point P on the outside of the sphere. How could I calculate a vector at P, which points along both the tangent ...
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536 views

Why limits give us the exact value of the slope of the tangent line?

Limits tell us how functions behave at $x\to a$, not how they behave at $x = a$. However, in limits we plug $x = a$ as an approximation of $x\to a$, so: why the limits give us the exact value of slope ...
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2answers
108 views

How to find $k$ such that the line $y=x-2-k$ is tangent to the circle given by $x^2+(y+2)^2=4$?

I have the circle $x^2+(y+2)^2=4$ and the line $y=x-2-k$. How would you find a $k$ value that would allow the second equation to sit tangent to the circle? There should, in theory, be only two ...
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0answers
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Finding points where a tangent line of a vector value function intersects a surface.

I have no idea what to do with this problem, I can't find any advise online and my book is useless. The problem involves finding points where the tangent line of a vector value function intersects ...
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How to check if a line is a tangent to a circle?

Is there a short and simple way to check if a line is a tangent to a circle, without complicated distance formulae? A solution to a question in my book says that for a circle $(x-at^2)(x-a/t^2) + y(y-...
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0answers
18 views

Normal Vector in the place

I've seen two definitions of a normal vector to a curve in $\mathbb{R^2}$. Suppose we have a parametrisation of our curve: $r(t)=(x(t),y(t)),$ Then differentiating once gives us a tangent vector, ...
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2answers
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Constructing tangent to a curve in $\mathbb{R}^2$

I've been studying Basic Mathematics for Physics courses. While teaching about derivatives my prof. said that there are actually two points the tangent at a point passes through (and those points ...
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3answers
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Tangent on a circle [closed]

In really stuck on this maths problem. Any help would be appreciated
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1answer
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Find all tangents to $f(x)$, so area between that tangent, $f(x)$ and $x$-axis will be $\frac{3}{4}$

Find all tangents to a function $f(x) = x^3$, so area between found tangent, $f(x)$ and x-axis is equal to $\frac{3}{4}$. I assume that area between $f(x)$, x-axis and tangent $ax + b$ will be same ...
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2answers
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Solutions to $(x-x_0)\cos(x_0)+\sin(x_0)-\sin(x)=0$

A tangent line to the sine function at the point $\{x_0, \sin(x_0)\}$, will intersect the sine at the points where $$(x-x_0)\cos(x_0)+\sin(x_0)-\sin(x)=0$$ The solutions to that equation looks like ...
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2answers
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Using a tangent line to approximate the value of an antiderivative given the graph of the derivative

I am wanting to solve part d. For c, I have the line's equation as $y=-3x+\cfrac{57}{5}$. I am not sure if this is the correct answer. How can I find the zero indicated in part d if I do not know how ...
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1answer
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Why is the y-intercept of this calculus problem given like that in the solution?

Given the following problem: Let $f$ be the real-valued function defined by $f(x) = \sqrt{1+6x}$. Determine the slope of the line tangent to the graph of $f$ at $x=4$. Determine the y-...
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Minimum distance between two submanifolds in $\mathbb{R}^3$ [closed]

Suppose that $S_1$ and $S_2$ are two-dimensional $C^1$ submanifolds in $\mathbb{R}^3$ and that $\ell$ is a line segment from a point $\mathbf{a}_1 \in S_1$ to $\mathbf{a}_2 \in S_2$ that has the ...
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What is the tangent at a sharp point on a curve?

How to know which line represents tangent to a curve $y=f(x)$ (in RED) ?From the diagram , I cannot decide which line to take as tangent , all seem to touch at a single point.
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1answer
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(Calculus) Derivative Thinking Question

Recently, my Calculus and Vectors (Grade 12) teacher gave our class a thinking question/assignment to work on over the march break, and after working on for some time, I've become stuck on it. The ...
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2answers
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Why doesn't a parabola have two tangents at its vertex?

Perhaps my definition of 'tangent' is the problem but in school the tangent is always defined as a line that intersects with a curve at only one point. According to this definition the equation $y = x^...
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0answers
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A monotone function must have a tangent ray that does not cross with the function

Consider a monotonically increasing and differentiable function $y=f(x)$ that passes through the origin. $\gamma=\{(x,y)|y=f(x)\}$ is the graph of $f$. Claim: there exists a ray $R$ such that $R\cap ...
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1answer
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Parabola tangent to two lines and through two points on those lines

is it possible to calculate parabola that is tangent to two lines exactly on black points? (please see enclosed picture) And linked question is: If we assume red line is given by: $$ f_{1}(x)=S_{1}(...
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Strong/weak tangents and limit positions, with rigor

As I'm working from do Carmo's Differential Geometry of Curves and Surfaces, I have found some of his imprecise language regarding strong and weak tangents to be most irksome. I've seen similar posts ...
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Weak Tangent Problem - Reconciling Two Approaches

The problem below is given in Do Carmo's Differential Geometry of Curves and Surfaces. My question is regarding part (a). Let us first show $\alpha:I\rightarrow\mathbb{R}^3$ has weak tangent at $t_0=...
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2answers
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Is there a closed-form solution for tangent circle in lens of two other circles?

I am given real values $p, s, t, u$ and wish to find unknown values $r, v$. As shown in the diagram below, $p$ and $s$ are radii of two given circles, with centers at $(0,-p)$ and $(0,t)$. At ...
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1answer
47 views

What is $a$ in $y=ax+c$?

Why is $a$ in the $y=ax+c$ called address factor? Also, why is it equal with tangent? Thanks!
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3answers
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definition of a derivative - large values

What is the typical process of finding the tangent line using the definition of a derivative while dealing with very large values? $$\lim\limits_{h\to 0}\frac{(x+h)^{123}-x^{123}}{h}$$
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4answers
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Confusion about a tangent line approaching an asymptote

I'm working from do Carmo's Differential Geometry of Curves and Surfaces, 2ed. He tends to use language like"the curve $\alpha$ and its tangent line approach [some line] $L$" or "the curve $\alpha$ ...
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3answers
47 views

How do I find ordered pair, given slope of the tangent line?

The function is $f(x) = x^3 + 9x^2 + 36x + 10$ and the slope given is $9$. I found the derivative and set it equal to $9$, but I ended up with $x = (-9,-33)$ and the answer is $(-3,-44)$. I've asked ...
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3answers
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What is the slope of tangent line to a rotated ellipse at a specific point?

I've seen discussions about this but with too many details left out. I have an ellipse with the standard parameters: $h, k, a, b$ and a rotation angle. (I can convert all that to the general form ...
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1answer
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In a cyclic $\square ABCD$, $BC, CD$ and $DA$ are three tangents of such a circle that its center is on the side $AB$. Proving that $AD + BC = AB$

In a cyclic quadrilateral $ABCD$, $BC, CD$ and $DA$ are three tangents of a circle. The center of the circle is located on the side $AB$. Prove that $$AD + BC = AB$$ Attempt: First, I thought it to ...
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Tangents along a straight line

I was looking to solve this geometrical problem. I have a line which is subdivided so that the when it is intersected it creates tangents of equal lengths that are stacked upon each other. I'm ...
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2answers
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Proving there is no plane tangent to a graph

Define $$f(x, y) = \begin{cases} \sin(y^2/x)\sqrt{x^2 + y^2} & \text{ if } x \neq 0 \\ 0 & \text{ if } x = 0. \end{cases}$$ (a) Show $f : \mathbb{R}^{2} \rightarrow \mathbb{R}$ has ...
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1answer
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Find three points of order two on elliptic curve.

Let $C$ be the cubic curve defined by $y^2z = x^3 -xz^2$ where $O = (0:1:0)$ is an inflection point. Find three points of order two in the group $(C, O, +)$. I know that $2\cdot P = O$ if and only if ...
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On definition of a flex point but an inevitable consequence of multiplicity

I am studying through the book Algebraic Geometry by Garrity. There is some kind of contradictory statement in Ch.2 that I couldn't fix: In some point it says that if P is a root of multiplicity k≥2 ...
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1answer
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Length of a tangent segment to the vertex of the circumscribed angle

Problem statement: $AB$ is the diameter of circle $O$ with a radius of $12$. $P$ is a point on $AB$ between the center point $O$ and $B$ such that $PB = 8$. Find a) the length of the shortest chord ...
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Unfold a loop by perturbing a small amount in the direction of tangent vector

I am interested in designing an animation of unfolding a loop. I am trying to unfold it by taking a tangent at each point and then adding a small amount of $\delta$ in the direction of the unit ...
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1answer
35 views

Point of contact of tangents to conic section

I am new to conic sections. A tangent to any conic section is given by $$ Axx_1+Byy_1+h(x+x_1)(y+y_1)+g(x+x_1)+f(y+y_1)+c=0 $$ which applies to parabolas, circles, ellipses, hyperbolas. However the ...
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2answers
103 views

Finding lengths when circles and squares tangents. [closed]

Should one approach by coordinates or by euclidean geometry? By pure geometry, I am not able to solve.
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1answer
30 views

At what $x$-value does a line touch $f(x)$ if $f(x)$ is tangent to the graph and parallel to a segment on the interval $[0,5]$

Given $f''(x) = 3 + 4\cos(x)$, $f'(0) = 0$, and $f(0) = 0$. The line, tangent to the graph of $f(x)$ and parallel to the segment connecting the endpoints on the interval $[0,5]$, touches $f(x)$ at $...
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1answer
69 views

Find the equation of the circle between 2 tangent lines

Find the radius of the circle and its position from origin. Given - equations of tangents of the circle and point of intersection of the tangents.(It's like a pair of tangents from a circle ...
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2answers
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If the tangents are parallel at each point for two curves, then so do their principal normal and binormal vectors

In the book of Differential Geometry by Kreyszig, at page 103, it is asked that Problem 13.1: Given two twisted curves which are in a one-to-one correspondence so that at corresponding points ...
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Geometrical problem on semi-circles

Given that $AE$ is the tangent to the small semi-circle at $D$ and that arc $CD$ : arc $DB$ = $3 : 10$, find arc $AE$ : arc $EB$. How do I go about solving this? I do not know how to start.
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4answers
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Problem on tangents drawn to a circle

I am solving Co-ordinate geometry by S.L. Loney. I am stuck on a problem on circles involving tangents and chords. I am not sure, if my approach is correct to solving this problem. Any inputs, tips ...
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1answer
87 views

Area of a region delimited by chords and circular arcs.

Let $AB$ be the diameter of circle $O$, where $AB = 2$. Circle $P$ is internally tangent to circle $O$ at point $B$, and $PB$ = $\frac{2}{3}$. Two different chords $AX$ and $AY$ are drawn tangent to ...
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1answer
38 views

$A$ and $D$ are in circumference of a circle and $B$ and $C$ are its inner points such that $PA$= $12$, $\frac{AB}{CD}$ = $\frac{1}{2}$. Find $PC$

There is something misunderstanding with that question that I think it to have inadequte context or information (obviously for my little knowledge). So I couldn't solve the problem. SOURCE: ...
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1answer
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Maximum number of tangents to two circles in affine geometry

How would one prove that the maximum number of tangents to two circles is 4, without recurring to the equations of the circles? I have found several ways of determining them (most of them using ...
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2answers
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Quadric surface S tangent to plane

Suppose that the quadric surface S is given by $z = x^2 + x + 2y^2 + 3y$ and the plane is given by $x + y + z = k$, where k is a constant. Find the vector equation for the tangent line to the curve ...
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2answers
71 views

Tangent line help(without calculus)

I need to find to find a tangent line to the curve $x \over {x^2 + x + 2}$.
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3answers
36 views

Tangent Line From a Point on a Sphere and $y$-axis

Let's say I have a sphere, $$100 = x^2+y^2 +z^2 $$ This indicates that the center of our sphere is at $$(0, 0, 0)$$ and we have a radius of $$radius = 10$$ I'm under the assumption that $$P = (1, ...
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3answers
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When to use different formulas to find the slope of a tangent line

I'm having some difficulty understanding the formulas to find the slope of a tangent line. As per my textbook, the first formula we received is presented below: The tangent line to the curve $y = ...