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.

If $y=f(x)$ is differentiable at $a$, the equation of the tangent line to $f$ at $(a,f(a))$ is $$ T_a(x) = f(a) + f'(a)(x-a) $$ Common uses are in the definition of differentiation and finding tangent lines to circles in geometry.

The tangent line need not touch a function locally only once. Indeed, consider $s(x) = x^3\sin(1/x)$ if $x\neq 0$, $s(0)=0$. Then $s$ is differentiable at $x=0$ with tangent line $y=0$, but this intersects $s$ infinitely often in any neighborhood of zero.

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Problem with basic definition of a tangent line.

I have just started studying calculus for the first time, and here I see something called a tangent. They say, a tangent is a line that cuts a curve at exactly one point. But there are a lot of lines that can cut the same point just like shown in…
Aaryan Dewan
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How can a "proper" function have a vertical slope?

Plotting the function $f(x)=x^{1/3}$ defined for any real number $x$ gives us: Since $f$ is a function, for any given $x$ value it maps to a single y value (and not more than one $y$ value, because that would mean it's not a function as it fails…
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Why do we assume that this quadratic has one solution?

I'm asking about a question about two lines which are tangents to a circle. Most of the question is quite elementary algebra, it's just one stage I can't get my head round. Picture here: The circle $C$ has equation $(x-6)^2+(y-5)^2=17$. The lines…
Monovox
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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.
Aditya Prakash
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Minimum operations to find tangent to circle

I've been playing the game Euclidea 3, and I can't really wrap my mind around one of the minimal solutions: https://www.youtube.com/watch?v=zublg6ZevKo&feature=youtu.be&t=9 The object is to get a tangent line on right of the blue circle with only 3…
Gillespie
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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 problem is delicate, as shown by the example…
Saúl RM
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How to find the equation of a line, tangent to a circle, that passes through a given external point

I am given the equation of a circle: $(x + 2)^2 + (y + 7)^2 = 25$. The radius is $5$. Center of the circle: $(-2, -7)$. Two lines tangent to this circle pass through point $(4, -3)$, which is outside of said circle. How would I go about finding one…
Brandon T.
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Can a cubic function have two tangents at a single point?

I have a question regarding this question. The question posed is if from a point ($h,3−h$) exactly two distinct tangents are drawn to $f(x)=x^3−9x^2−px+q$ find $p$ and $q$ I've been waiting all week for someone smarter than me to answer this…
scott
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How can I find the equation of a circle given two points and a tangent line through one of the points?

I was wondering whether it was possible to find the equation of a circle given two points and the equation of a tangent line through one of the points so I produced the following problem: Find the equation of the circle which passes through $(1,7)$…
LJD200
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Formula for the area of a rhombus

Can you prove the following claim? The claim is inspired by Harcourt's theorem. In any rhombus $ABCD$ construct an arbitrary tangent to the incircle of rhombus . Let $n_1,n_2,n_3,n_4$ be a signed distances from vertices $A,B,C,D$ to tangent line…
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Proving radius of circle $\dfrac{\triangle}{a}\tan^2\dfrac{A}{2}$

If a circle be drawn touching the inscribed and circumscribed circles of a $\triangle ABC$ and the side $BC$ externally, prove that its radius is: $$r=\dfrac{\triangle}{a}\tan^2\dfrac{A}{2}$$ I tried using triangle formed by circumcenter, incenter…
mathlover
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Prove that the number of common tangents to two circles exterior to one another is 4

Given two circles outside each other, the maximum number of common tangents to the two circles is 4 according to Wikipedia. How can this be proven? That's a picture above showing what I'm talking about. How can we prove that no other common…
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Prove: Three tangents to a parabola form a triangle with an orthocenter on the directrix and a circumcircle passing through the focus

Prove the following: The intersection points of any three tangents to a parabola given by the formula $y(y-y_0)=2p(x-x_0)$ are vertices of a triangle whose orthocenter belongs to the directrix of the parabola and the circumcircle of the triangle…
Invisible
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Equation for tangent line for $f(x) = 1/\sqrt{x}$ at $x=a$

first time math stack-exchange-er here. I'm self-teaching single variable calculus using MIT's free online courses and I think I found a typo in the homework solution set (problem 1C-4 part d). I'm not confident enough in my own abilities to know…
user7875185
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Why are there only two tangents to a hyperbola from a point, instead of four?

Why are only two tangents possible to a hyperbola from a point? If we treat the hyperbola as two individual parabolas, then a point should be able to create two tangents through it for both of them, hence a total of 4.
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