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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1answer
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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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0answers
28 views
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
16 views
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 ...
5
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6answers
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 ...
3
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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
23 views
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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0answers
20 views
The curve $x^2-4y^2+c=0$ and $y^2=4x$ intersect orthogonally for what values of c [closed]
Angle of intersection of hyperbola and parabola
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3answers
62 views
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-...
0
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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
64 views
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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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
59 views
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
25 views
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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0answers
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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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7answers
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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.
4
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1answer
154 views
(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 ...
3
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2answers
94 views
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
28 views
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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vote
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:
$$
f_{1}(x)=S_{1}(...
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0answers
26 views
Strong/weak tangents and limit positions, with rigor
As I'm working from do Carmo's Differential Geometry of Curves and Surfaces, I have found some of his imprecise language regarding strong and weak tangents to be most irksome. I've seen similar posts ...
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0answers
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Weak Tangent Problem - Reconciling Two Approaches
The problem below is given in Do Carmo's Differential Geometry of Curves and Surfaces.
My question is regarding part (a).
Let us first show $\alpha:I\rightarrow\mathbb{R}^3$ has weak tangent at $t_0=...
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vote
2answers
45 views
Is there a closed-form solution for tangent circle in lens of two other circles?
I am given real values $p, s, t, u$ and wish to find unknown values $r, v$. As shown in the diagram below, $p$ and $s$ are radii of two given circles, with centers at $(0,-p)$ and $(0,t)$.
At ...
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1answer
47 views
What is $a$ in $y=ax+c$?
Why is $a$ in the $y=ax+c$ called address factor?
Also, why is it equal with tangent?
Thanks!
1
vote
3answers
69 views
definition of a derivative - large values
What is the typical process of finding the tangent line using the definition of a derivative while dealing with very large values?
$$\lim\limits_{h\to 0}\frac{(x+h)^{123}-x^{123}}{h}$$
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4answers
69 views
Confusion about a tangent line approaching an asymptote
I'm working from do Carmo's Differential Geometry of Curves and Surfaces, 2ed. He tends to use language like"the curve $\alpha$ and its tangent line approach [some line] $L$" or "the curve $\alpha$ ...
1
vote
3answers
47 views
How do I find ordered pair, given slope of the tangent line?
The function is $f(x) = x^3 + 9x^2 + 36x + 10$ and the slope given is $9$.
I found the derivative and set it equal to $9$, but I ended up with $x = (-9,-33)$ and the answer is $(-3,-44)$.
I've asked ...
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3answers
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What is the slope of tangent line to a rotated ellipse at a specific point?
I've seen discussions about this but with too many details left out. I have an ellipse with the standard parameters: $h, k, a, b$ and a rotation angle. (I can convert all that to the general form ...
3
votes
1answer
79 views
In a cyclic $\square ABCD$, $BC, CD$ and $DA$ are three tangents of such a circle that its center is on the side $AB$. Proving that $AD + BC = AB$
In a cyclic quadrilateral $ABCD$, $BC, CD$ and $DA$ are three tangents of a circle. The center of the circle is located on the side $AB$. Prove that $$AD + BC = AB$$
Attempt:
First, I thought it to ...
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0answers
23 views
Tangents along a straight line
I was looking to solve this geometrical problem. I have a line which is subdivided so that the when it is intersected it creates tangents of equal lengths that are stacked upon each other.
I'm ...
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2answers
35 views
Proving there is no plane tangent to a graph
Define $$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}$$
(a) Show $f : \mathbb{R}^{2} \rightarrow \mathbb{R}$ has ...
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1answer
31 views
Find three points of order two on elliptic curve.
Let $C$ be the cubic curve defined by $y^2z = x^3 -xz^2$ where $O = (0:1:0)$ is an inflection point. Find three points of order two in the group $(C, O, +)$.
I know that $2\cdot P = O$ if and only if ...
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0answers
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On definition of a flex point but an inevitable consequence of multiplicity
I am studying through the book Algebraic Geometry by Garrity. There is some kind of contradictory statement in Ch.2 that I couldn't fix:
In some point it says that if P is a root of multiplicity k≥2 ...
3
votes
1answer
20 views
Length of a tangent segment to the vertex of the circumscribed angle
Problem statement:
$AB$ is the diameter of circle $O$ with a radius of $12$. $P$ is a point on $AB$ between the center point $O$ and $B$ such that $PB = 8$. Find
a) the length of the shortest chord ...
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0answers
19 views
Unfold a loop by perturbing a small amount in the direction of tangent vector
I am interested in designing an animation of unfolding a loop. I am trying to unfold it by taking a tangent at each point and then adding a small amount of $\delta$ in the direction of the unit ...
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1answer
35 views
Point of contact of tangents to conic section
I am new to conic sections.
A tangent to any conic section is given by $$ 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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vote
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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3answers
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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
58 views
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 ...
2
votes
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 ...
2
votes
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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vote
2answers
29 views
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}$.
1
vote
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, ...
1
vote
3answers
38 views
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 = ...
