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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34 views

Tangent plane with three variables

The queation is : Show that the sum of the squares of the intersecting axes of the tangent plane at any point ($x_0$, $y_0$, $z_0$) of the surface $x^{2/3}$ +$y^{2/3}$ + $z^{2/3}$ = $a^{2/3}$ is ...
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4answers
39 views

Find an equation for the line tangent to the graph of $f^{-1}$ at the point $(3,1)$ if $f(x)=x^3+2x^2-x+1$

Find an equation for the line tangent to the graph of $f^{-1}$ at the point $(3,1)$ if $f(x)=x^3+2x^2-x+1$ ok, so I know that I need to take the derivative of f(x). $f'(x)=3x^2+4x-1$ The inverse ...
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2answers
62 views

Angle between tangents to the curve $x^2+3y^2=9$

Tangents drawn from the point $(\alpha,\alpha^2)$ to the curve $x^2+3y^2=9$ include an acute angle between them, then find $\alpha$. My attempt is by using the equation for pair of tangents from an ...
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0answers
57 views

Proving that two circles with one point in common have coincident tangent lines at that point

Friends: Suppose that $\Gamma_{1}$ and $\Gamma_{2}$ are two circumferences that are externally tangent (they have exactly one point in common and neither of them is contained in the region bounded by ...
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2answers
34 views

Finding the equation of the line through the points of tangency from point $(8,10)$ to circle $(x+12)^2 + (y+5)^2 = 225$

$A$ and $B$ are the points of tangency of tangents drawn from $P(8,10)$ to the circle $(x+12)^2 + (y+5)^2 = 225$. Find the equation of the line $AB$. Since $AB\perp{CP}$, where $C$ is center of ...
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0answers
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If two curves have the same tangent at $p$, then $\operatorname{mult}_p(C_1 \cap C_2) > \operatorname{ord}_p(C_1) \operatorname{ord}_p(C_2)$

I have to prove that, if two curves $C_1$ and $C_2$ have the same tangent line at point p then: $\DeclareMathOperator{\mult}{mult}$$\DeclareMathOperator{\ord}{ord}$ $$\mult_p(C_1 \cap C_2) > \ord_p(...
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2answers
31 views

Finding the tangent to a parametric curve $(t^3, t^5)$ at $(0,0)$

The curve $(t^3, t^5)$ at that point $(0,0)$ does not have a tangent vector as when you work it out, you will arrive at $(0,0)$. Question: How can you find a new parametrisation for the curve such ...
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1answer
63 views

Isomorphism between $\overline{U}_{0,4}$ and the degree $5$ Del Pezzo Surface

Tacitely, I am working over the field of complex numbers! Let $\overline{M}_{0,4}\cong\mathbb{P}^1$ be the compactification of the moduli space of $4$-pointed stable rational curves. The relevant ...
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1answer
51 views

Elementary Geometry problem - Proving the angles $A'B'C'$ and $\beta$ are the same

In the above picture, the line t is tangent to the circle at $C'$. How do I prove the angle $A'B'C'$ is equal to the angle $\beta$? I tried to do a lot of things, like tracing parallels and ...
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38 views

Line Tangent to Two Non-Equal Circles on a 2D Plane

If I have two circles, say Circle A is on the origin of a cartesian plane and Circle B is placed at a point with a known horizontal and vertical distance from the origin. The diameters of both circles ...
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24 views

Which equation is the proper equation to find the tangent line to a curve in space?

From my understanding, to write an equation of a line in 3D all you need is point and direction vector. So for a tangent line I assumed all you need is the point of tangency and the tangent direction ...
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1answer
59 views

Tangents to circles: What do I do now? (c) solved, but (a) and (b) are still open for answers… [closed]

everyone! I am confused about what to do here and how to do it. May I have some help? So, I know that $\overline{FI}$ and $\overline{IE}$ are both radii of the small circle and $\overline{JK}$ and $\...
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26 views

Writing equation in polar coordinates for tangent circle

How can I write the equation for this tangent circle? Fundamental circle is $r=3\sin(\theta)$ and I also find the tangent line for $\theta = \frac{\pi}{3}$ And the tangent line to the circle is: $y=-\...
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3answers
52 views

How do you find the formula for a tangent line if there are no points?

The question asks what is the function for the tangent line $b(x)$ if $$f(x) = \frac{e^{x}}{e^{x}+1}$$ and the part that gets me is the point given is $P = (a,f(a))$. I derived the gradient and used ...
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3answers
42 views

Derivative, slope or, the tangent of a graph with a shape which has corner like tips as in the letter V [closed]

Consider the graph of a function in the shape of the letter ‘V’, how would we be finding the derivative, slope or, the tangent of the function at the value of the function that corresponds to the tip ...
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0answers
61 views

Uniqueness in analysing coordinate geometry situations

In problems involving coordinate geometry, a lot of the time the solution provided only considers the simplest case and treats it as though the results obtained from it would work for every case. For ...
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1answer
42 views

How do I calculate the angle between the tangent to an outer circumference and a line passing through a specific point on the inner circumference?

I would like to calculate the angles indicated by the arrows in my sketch. All I know is the segment AB between the two circumferences and the angles Theta and Theta'. The two circles are concentric, ...
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2answers
45 views

Prove that a tangent line passes through an exterior point [closed]

Given a circle $\beta$, if $O$ is a point in the plane of $\beta$ but in the exterior of $\beta$, then there is a tangent to $\beta$ that passes through $O$. I am stuck on how to get started on this.
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1answer
48 views

How to find the polar coordinate angle of the tangent of any point on an ellipse?

I have an ellipse centered at 0,0 with a height of 75 and a width of 150. Now let say I know both the x and y and also the angle of a point on the circumference of the ellipse. I want to know what the ...
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3answers
433 views

Show that there is a simultaneously tangent line to both the curves $y = e^x$ and $y = \ln x$

I found the derivatives of both of the curves but I'm having trouble on how to move on from here: $y = e^x\implies y' = e^x$ $y = \ln x\implies y' = \frac{1}{x}$
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27 views

Finding points of a function with given tangent

The function I am dealing with is as follows: $$ f_a(x)=\frac{x+1}{a}e^{2ax} \tag{1} $$ where the parameter $a\in \mathbb R\setminus\{0\}.$ The problem I am solving is about studying the variation of ...
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1answer
35 views

Finding which points that are valid for our point hitting (4,0) when leaving the orbit in a straight line. [closed]

So basically the problem is that we have a satellite that is orbiting around this equation: $3x^2-2xy+7y^2 - 20 = 0$ And it looks like this in graph form: the orbit of the so said planet At what ...
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2answers
27 views

Can you find the equation of a straight line given it runs tangent to a circle with its origin on a given intersecting line [closed]

I should start by saying I decided to invent some questions to entertain myself. I came up with the following question. This seems like it should be possible but I really don't know. Am I just missing ...
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1answer
41 views

Calculating distance between two obstacles from a Lidar vertical scan

I have a 2D Lidar scanner that is scanning vertically (see photo Graphical Representation of Data). The center point is the scanner. The bottom is the floor, top is the ceiling, and the left and ...
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1answer
35 views

How does the tangent line according to calculus correlates to the classic intuition of a line that only passes through one point of the curve?

Sorry if my English is wrong. In calculus, given a function $f$, derivable at $x_0$, the tangent line to the curve at $x_0$ is $$t(x) = f(x_0) + f'(x_0)(x-x_0)$$ How can I convince myself that this ...
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1answer
101 views

Gradient defined on level set

Suppose I have a continuously differentiable function $f:\mathbb{R}^N\rightarrow \mathbb{R}$ where the gradient is defined everywhere. Let $c$ be some constant in the range of $f$ and let $S=\{x \in \...
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2answers
61 views

Slope of a curved line

I just started with calculus and I came across the slope of a curve. According to definition the slope of a curve at a point is equal to the slope of tangent at that point. Since tangent is a straight ...
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0answers
15 views

Equations associated with an ellipse whose transverse axis is along $Y-$ axis

For an ellipse with a transverse axis along Y-axis. I am writing some of it's associated equations and a parametric point as follows: Equation for an ellipse $$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\tag{1}$...
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1answer
30 views

Tangent to Parametric Polar Curve

If we have some $$\gamma(t)=r(t)e^{i\theta(t)}$$ Where $\gamma(t)$ is some complex parametric curve; how would one express the tangent vector to that curve, without just converting straight to ...
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1answer
48 views

3 circles with radii 66, 77, and 88 externally tangent to each other, find the radius of the circle internally tangent to the other circles.

In the diagram, we see that there are 3 circles that are all externally tangent to each other and internally tangent to a much bigger circle. The radii of the 3 smaller circles are 66, 77, and 88. ...
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3answers
91 views

Find the slope of the tangent line to the graph of the given function at the given value of x. [closed]

Find the slope of the tangent line to the graph of the given function at the given value of x. Find the equation of the tangent line. : $y=x^4-5x^3+2; x=2$ I understand that the slope of the line ...
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1answer
30 views

How do I show that $f(x+ \Delta x) \approx f(x) - \Delta x f'(x)$ [closed]

I tried using linear approximations. the tangent line $T(x)$ at a point $a$ for a function $f(x)$ is: $$ T(x)= f'(a)(x-a) + f(a) $$ For $f(x + \Delta x)$ I would have $a = - \Delta x$, $$ T(x) = f'(-\...
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3answers
76 views

Unique questions about pentagon

Consider the (non-regular) pentagon with consecutive vertices at (-1,-1), (-1,1), (0,2), (1,1), and (1,-1). a) Prove that there is no circle that is tangent to all 5 sides of the pentagon b) Is there ...
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1answer
107 views

Find the Relation between $a,b,c$

In the figure shown find the relation between $a,b,c$. My try: When two circles of radii $r_1,r_2$ touch externally, the length of their direct common tangent is $2\sqrt{r_1r_2}$ Let the radius of the ...
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0answers
34 views

Tangent Vector at a specific point

I was asked to parameterize the circle edge of $d_2 =\{(x,y):x^2+y^2=9$ and $x+y\ge0\}.$ Anyways I parametrized the circle edge within the bounds but now I have to find the tangent vector at $(0, 3)$ ...
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0answers
18 views

The angle of the intersection of two graphs

I encounter some problems with understanding the explanation of the follwing problem: For which value of $\alpha$ does the graph of $f_\alpha(x)=e^{\alpha x}$ intersect the graph of $g(x) = \sqrt{x+1}...
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1answer
44 views

Lines simultaneously tangent to the curves $y = x^2$ and $y = -x^2+2x-2$

Is (are) there any straight line(s) that is (are) simultaneously tangent to both the curves given by the equations $y = x^2$ and $y = -x^2+2x-2$? My Attempt: Let $y = mx+b$ be any such (non-vertical) ...
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2answers
31 views

For which values of $m$ the line is tangent to the quadratic curve?

For which values of $m$, the line $y=2x-4$ is tangent to the curve $y=(m+3)x^2+mx ?$ We have a quadratic equation. the equation of slope of tangent line to it for specific $x$ can be find by $y'=(2m+6)...
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1answer
36 views

Justifying implicit limits

Consider the hyperbola given as: $$\frac{x^2}{4}- \frac{y^2}{12} =1$$ Divide through by $x^2$ $$ \frac14 - \frac{1}{12} \left(\frac{y}{x}\right)^2= \frac{1}{x^2}$$ Now, here is the tricky step, I take ...
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0answers
41 views

If $f:\mathbb R \to\mathbb{R}^3$ is class $C^1$ and regular at t then $f$ has a strong tangent at $t$

I'm reading Elements of Geometry for Manfredo Do Carmo and I'm stucked in this problem. The book define the strong tangent: $f$ has a strong tangent at $t$ if the line determined by $f(t+h), f(t-k)$ ...
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0answers
27 views

Contact theory affine differential geometry

I am studying affine differential geometry to plan curves, that is $[\gamma_s(s), \gamma_{ss}(s)]=1$, and I need to show the following result: "Two curves having the same affine tangent also have ...
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2answers
89 views

Can the tangent line be defined independently of the derivative?

The graph of the function $f:x \mapsto x^{1/3}$ has a 'vertical tangent' at $x=0$: Although this idea is certainly geometrically sound, from what I understand the tangent line is defined by the ...
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0answers
12 views

Clamping the end tangents of a globally interpolated B-spline?

If I have a globally interpolated B-spline (using the method found at https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/INT-APP/CURVE-INT-global.html) how can I specify the tangents at the endpoints ...
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1answer
61 views

why am I getting a different answer with $(y_1-y_2) = m(x_1-x_2)$ to when I use $y = mx + c$ ?!?

The question in the book is: 'What is the equation of the tangent of the curve with parametric equations $x = 3 - 2\sin{t}$ and $y = t\cdot \cos{t}$, at the point where $t = \pi$?' The differentiation'...
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0answers
13 views

Question: Plot tan(0) against 0 between angles of -pi and +pi. explain in detail why there are some points you cannot evaluate using your calculator.

Question: Plot tan(0) against 0 between angles of -pi and +pi. explain in detail why there are some points you cannot evaluate using your calculator. Thanks for any help :)
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1answer
41 views

Finding points on $\vec r(t) = \vec At^3 + \vec Bt^2 + \vec Ct + \vec D$ where the tangent is parallel to the line $px + qy + k = 0$

Given a line defined by the equation: $$px + qy + k = 0$$ and a parametric cubic curve defined by: $$\vec r(t) = \vec At^3 + \vec Bt^2 + \vec Ct + \vec D$$ where both curves lie in 2D space, how can I ...
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1answer
48 views

What is the difference between the two equations? [closed]

I was curious about the difference between these two equations. They seem to be almost the same function. If anyone knows I would really appreciate the help. Thank you in advance for all the help! ...
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1answer
21 views

Plotting on Matlab

I'm trying to plot the curve shown below on Matlab. $y= x \, \tan x$ and $x$ is in the range $(0, 4 \, \pi)$. The thing is I can't seem to multiply x with tan(x) without getting an error. I just need ...
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1answer
51 views

Is it possible that the tangent can look like this

Consider this: $$ K(v) = \frac{v}{v^2+9} $$ Approximate the function for v = 1 by a tangent. I first did the derivitave of the function. $$ K'(v) = \frac{-v^2+9}{(v^2+9)^2} $$ And now the tangent ...
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1answer
35 views

Tangents to two sets

Find all the affine tangents that are simultaneously tangent to the set $E$ and $H$: $$H=\{(x,y)\in \mathbb R^2:xy=-5\}, E=\{(x,y)\in \mathbb R^2:\frac{x^2}{9}+\frac{y^2}{4}=1\}$$ I know that when the ...

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