# 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.

1,051 questions
Filter by
Sorted by
Tagged with
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 ...
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 ...
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 ...
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 ...
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 ...
26 views

26 views

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 ...
15 views

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 $$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\tag{1}... 1answer 30 views ### Tangent to Parametric Polar Curve If we have some$$\gamma(t)=r(t)e^{i\theta(t)}$$Where \gamma(t) is some complex parametric curve; how would one express the tangent vector to that curve, without just converting straight to ... 1answer 48 views ### 3 circles with radii 66, 77, and 88 externally tangent to each other, find the radius of the circle internally tangent to the other circles. In the diagram, we see that there are 3 circles that are all externally tangent to each other and internally tangent to a much bigger circle. The radii of the 3 smaller circles are 66, 77, and 88. ... 3answers 91 views ### Find the slope of the tangent line to the graph of the given function at the given value of x. [closed] Find the slope of the tangent line to the graph of the given function at the given value of x. Find the equation of the tangent line. : y=x^4-5x^3+2; x=2 I understand that the slope of the line ... 1answer 30 views ### How do I show that f(x+ \Delta x) \approx f(x) - \Delta x f'(x) [closed] I tried using linear approximations. the tangent line T(x) at a point a for a function f(x) is:$$ T(x)= f'(a)(x-a) + f(a) $$For f(x + \Delta x) I would have a = - \Delta x,$$ T(x) = f'(-\...
76 views

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 ...
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 ...
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 $(0, 3)$ ...
18 views

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 ...
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 $f(t+h), f(t-k)$ ...
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 ...
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 ...
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 ...
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'...
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 :)
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 ...
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! ...
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 ...
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 ...
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 ...