In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
The tangent line to a point on a differentiable curve can also be thought of as the graph of the affine function that best approximates the original function at the given point.Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.
The word "tangent" comes from the Latin tangere, "to touch".
Homework Statement
Find an equation of the tangent line to this curve at the point (1, -2).
Homework Equations
The Attempt at a Solution
2y' = 3x^2+6x
y' = 3x^2+6x
y'=3/2x^2+3x
y+2=3(x-1)
y+2=3x-3
y=3x-5
Homework Statement
find the slope of the tangent line whose g(x)=x^2-4 at point (1,-3)
Homework Equations
lim f(x+Δχ) -F(c)/ (Δχ)
The Attempt at a Solution
g(x)= x^2-4
G(1+Δχ)= (1+Δχ)^2-4 ==> Δχ^2+2Δχ-3
lim (Δχ^2+2Δχ-3) - (-3)/(Δχ) = 0
but in the book the answer is 2 so what...
Homework Statement
I seem to remember that a parameterized a(t) curve in \mathbb{R}^3 that one can construct the tangent from the slope of a'(t) and the curve itself.
such that the tangent line L = a(t) + s * a'(t) to a. This is supposedly a straight line in \mathbb{R}^3.
To make a...
Homework Statement
If the tangent line to y = f (x) at (9,7) passes through the point (0,6), find the following.
(a) f (9)
(b) f ' (9)
Homework Equations
f '(x) = f(x+h) - f(x)/h ?
The Attempt at a Solution
I really have no idea how to even begin going about this one, so a...
I am having a time trying to figure out this problem given only an x value. The equation is "Find the equation of the tangent line to y= Square root of x^4-2337 ...at x=7. If anybody could help me solve this it would be a great help. Thanks
An Unknown Conic and Equation of Tangent Line **HELP**
Homework Statement
(a) Find the positive y-coordinate of the point on the conic whose x-coordinate is 27cos\theta, where 0<\theta<\pi/2
(b) The slope of the tangent line to the conic through the point (27cos\theta, y1) is -1/27cot\theta...
The curve defined by x3 +xy - y2 = 10 has a vertical tangent line when x = ?
To find when the tangent line is vertical, could I find when the slope is undefined for the original function? The way I previously tried this problem was by taking the first derivative and then finding when x = 0...
Homework Statement
Find the equation of the tangent line to the curve 3xy = x^3 + y^3 at point (2/3,4/3)
Homework Equations
The Attempt at a Solution
I found the deriviative to be 3x^2-3y / 3x - 3y^2 which i simplified to x^2 - y / x - y^2
I found the slope of at those...
Homework Statement
Let C be the intersection of the two surfaces:
S1: x^2 + 4y^2 + z^2 = 6;
s2: z = x^2 + 2y;
Show that the point (1, -1, -1) is on the curve C and find the tangent line to the curve C at the point (1, -1, -1).
Homework Equations
partial derivates, maybe the gradient...
Homework Statement
Find an equation for the line tangent to to the graph of f^-1 at the pt (3,1) if f(x)=x^3+2x^2-x+1
Homework Equations
The Attempt at a Solution
I used the Inverse Function Thm
1/(3x^2+4x)
Now do I plug in 3 to this to find slope?
1/21
y-1=1/21(x-3)...
Homework Statement
Find parametric equations of the tangent line at the point (-2,2,4) to the curve of intersection of the surface z=2x2-y2 and the plane z=4
Homework Equations
Not sure
The Attempt at a Solution
Not sure quite how to approach this. take the gradient of...
Homework Statement
Find an Equation of the tangent line to the curve y=10^x at the point (1,10)
Homework Equations
Derivative
y=mx+b The Attempt at a Solution
I am totally lost
I don't know what to do with 10^x
Please help me
this will save me failing the class
Homework Statement
On the circles y^2+x^2=1 and y^2+(x-3)^2=4
There is a line with positive slope that is tangent to both circles. Determine the points at which this tangent touches each circle.
Homework Equations
the derivative of the first circle i found:
y'=-x/y
the derivative of...
Homework Statement
The tangent line through given points (pi/4,pi/4) m=1
y= cos(y)cos(x)/sin(x)sin(y)
The Attempt at a Solution
dy/dx= d/dx[cos(y)cos(x)/sin(x)sin(y)]
First use quotient rule ?
vu'-uv'/v^2
v= sin(x)sin(y)
v'= do i need to use product rule? for product rule...
Homework Statement
Find the slope of the tangent line to the curve y=x3 at point (-1,-1)
(i) Using Definition 1
m=limx-->a\frac{f(x)-f(a)}{x-a}
(i) Using Equation 2
m=limh-->0\frac{f(a+h)-f(a)}{h}
Homework Equations
The Attempt at a Solution
For the first one, I got as far...
Homework Statement
Ok, how do i solve this question
I've attached the picture.. The answer is A but i don't understand how to get that...
The Attempt at a Solution
The solution is also given , but i don't understand the adding and subtracting 5. the Derivative of
y=ln (e^x + 5) y'=...
Find the slope of the functions graph at the given point.
F(x) = x / x-2 point (3,3)
f(x+h) - f(x) / h is what we have to use to find the answer.
so I've plugged it all in and have came to this..
((3+h) / (3+h-2)) - 3 / h
I need some help with my simplification...
Find an equation for the line tangent to the given path at the indicated value for the parameter.
x(t)=4costi-3sintj+5tk, t=\pi/3
So what I did here was take x'(t) and then plugged in \pi/3 after that to get an equation containing i, j, and k.
x'(t)= -4sinti-3costj+5k...
Homework Statement
Find the unit vectors that are parallel to the tangent line to the curve y=2sinx at the point (\pi/6, 1). Thereafter, find the unit vectors that are perpendicular to the tangent line.
Homework Equations
The Attempt at a Solution
I took the derivative of y=2sinx and...
Let f be the funtion given by f(x) = 4x^2 - x^3 , let L be the line y = 18 - 3x, where L is tangent to the graph of f.
Show that L is the tangent to the graph of y= f(x) at the point x = 3.
I'm equaling both graphs like this:
4x^2 - x^3 = 18 - 3x
and then I isolate everything...
A tangent to a curve is a line that touches the curve but at what specific point where there is a slope to the curve. Equations like y = X^2 have many slopes because the curve is shaped differently at different points. We can choose 2 points on the graph and find the slope but it will...
Homework Statement
Find an equation of the tangent line to the curve y = 5x3 at the point
(-3,-135). Enter your equation in y = form.
Homework Equations
The Attempt at a Solution
Taking the derivative of the equation I get
y = 15x2, following that I plug in the x value of...
Homework Statement
Find the equation of the tangent line to the graph of x^2 - xy + y^2 = 19 (where y=y(x)) at (3,-2).
Homework Equations
The Attempt at a Solution
So, this is what I did:
d/dx(x^2-xy+y^2) = (19)d/dx
2x*dx/dx-dy/dx+dy/dx+2y(dy/dx)=0
2x+2y(dy/dx)=0
And I...
Homework Statement
Find equation of the tangent line at t=0
x=2sint
y=tant
The Attempt at a Solution
Really don't know where to start, can someone give me the hint, don't solve it, i want to try it, just need a hint
Homework Statement
Find the smallest positive value for `theta` for which the tangent line to the curve `r = 6 e^(0.4 theta)` is horizontal
The Attempt at a Solution
I tried to use the (r'sin(theta)+rcos(theta))/(r'cos(theta) - rsin(theta)) formula to get the tangent line, which I got...
Homework Statement
Find all points on the graph of the function at which the tangent line is horizontal.
f(x) = 2sinx + (sinx)^2
Homework Equations
Hi guys I have an issue. I do not know how to approach this problem.
I know the chain rule but I still do not know how to solve the...
Homework Statement
The equation of tangent is given t:2x+3y-2=0 and the equation of elipse E:x^2+4y^2=K
Find "a" and "b" and the coordinates of touching point D.
Homework Equations
equation of elipse: b^2x^2+a^2y^2=a^2b^2
equation for touching: a^2k^2+b^2=n^2
equation for K (if...
For which value of x...horizontal Tangent Line
Homework Statement
For which value of x does f(x) = \frac{k}{ax^{2}+bx+c} have a horizontal tangent line?
Homework Equations
Quotient Rule?
F'(x) = [g(x)a'(x) - a(x)g'(x)]/g(x)^2?
The Attempt at a Solution
Am I supposed to...
Homework Statement
At which point is the tanget line to the following curve horizontal?
y= a sin^{3}\theta
x = acos^{3}\theta
Homework Equations
The Attempt at a Solution
\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}
When \frac{dy}{dx} = 0 , this means that that the tanget...
I have encountered a problem in my Calculus homework.
I have a position function, r(t)= (t^2)i + (4t)j and in my homework, I am asked to find the tangent to this curve at the point t=3. I did this by finding dy/dx, or 2t/4 @ t=3 is 6/4. However, I am also asked to relate this to the...
Given equation x^2/9 + y^2/4 = 1. Determine the two points on the ellipse having a tangent line passing through the point (0,3). Cant seem to figure this one out? I have found the derivative of the slope which is -4x/9y. I don't know how to use that slope in order to find a tangent line that...
1. Find the equations of all tangents to the graph f(x) = x^2-4x + 25 that pass through the origin.
2. The relevant equations to my knowledge are the slope formula and f ' (x)
3. First I found the derivative using various rules to be f ' (x) = 2x - 4
after this i used the slope formula and my...
Hello,
I'm having trouble understanding an example from my test and I would appreciate your help clarifying how to get from one step to the next.
Problem: Find an equation of the tangent line to the hyberola y = 3/x at the point (3,1)
Eqation: m = \frac{f(a+h) - f(a)}{h}
m =...
Hi, I'm having trouble plotting the tangent line on a curve (or the derivative evaluated at a point) in spice. I know how to plot the derivative of the entire curve using the d() function, but I don't know how to evaluate it at a point. Any help is much appreciated!
Sorry if the title is a bit vague :/
The Problem: Find the point on the parabola y=1-x^2 at which the tangent line cuts from the first quadrant a triangle with the smallest area.
Relevant Equations: y = 1-x^2 ; y' = -2x ; A= 1/2bh
I'm basically stuck near square one, I found...
1. The problem statement, all variables and given/known
i have to find all tangent lines in the equation below that pass though the point (0,0)
y=x^{3}+ 6x^{2}+8i took the derivative and got this.
y=3x^{2}+12x
then i substituted the points x=0 and y=0 into the derivative equation and the...
Homework Statement
if the tangent line to the graph of y=f(x) at (2.3) has an equation x-y+1=0, then f'(2) =?
The Attempt at a Solution
We did these problems in one small session at the beginning of the semester, but my notes aren't clear and I am not sure where to begin. Do I just...
i would like to know if my answer is correct can someone tell me the results for this:
Find the equation of the tangent line to x^3-2x^2y+y^2=17 at (2,-1)
ive gotten y'=(3x^2+4xy)/(-2x^2-y)
and my line equations is
y+1=(-1/4)(x-2)
thanks in advance
Homework Statement
Find the tangent line to the curve y^2 (y^2 - 4) = x^2 (x^2 -5) at the point (0,-2).
Homework Equations
y-y1=m(x-x1)
The Attempt at a Solution
I d/dx-ed both sides and was left with dy/dx=0
so
y=-2
Does that seem right?
[SOLVED] Derivative and tangent line
Homework Statement
y = (x)/(x+x^-1)
Find an equation of the tanget line to the graph at x=2
Homework Equations
The Attempt at a Solution
I used the quotient rule and got the derivative to be 0 / (x+x^-1) = y' so then the slope is 0 of the...
[SOLVED] Tangent Line
Homework Statement
Find all tangent lines that are tangent to Y=x^2 and go through the point (2,3)
Homework Equations
The Attempt at a Solution
At first I thought I knew how to do this problem, I found the derivative of the equation got:
y' = 2x ; then...
Homework Statement
Find parametric equations for the tangent line to the curve x= t^4, y= t^4, z=t^3 at the point (1,1,1)
Homework Equations
The Attempt at a Solution
I understand everything about solving this problem with the exception of how to find what t =? to plug in. ie...
Homework Statement
suppose y=2x-3 is an en equation of the tangent line to the graph of y=f(x) at the point where x=2. find the value of f'(2) and f(2).
Homework Equations
I know how to get the tangent line, but this wording throws me off. a little help at how to solve f(2) and f'(2)...
Homework Statement
Find an equation of the tangent line to the curve: 7*x*e^(x)+8 at (0,8)
Homework Equations
Derivative I guess?
The Attempt at a Solution
I know you have to take the derivative of the equation given which I think is
7*x*e^(x) + 7*e^(x)
Then you plug it...
Homework Statement
find the equation for the tangent line.
1. (e^x)(cos x) where x=0Homework Equations
plugged 0 into the equation, (e^0)(cos 0) and got 1 for the y-coordinate. so I got the points(0,1). For the slope, I derived the equation into -(e^x)(sin x). then i plugged 0 in and got 0 for...
Homework Statement
5. Find parametric equations for the tangent line to the curve of intersection of the surfaces
z^2 = x^2 + y^2 and x^2 + 2y^2 + z^2 = 66 at the point (3, 4, 5).
The Attempt at a Solution
f(x,y,z) = x^2 + y^2 - z^2
g(x,y,z) = x^2 + 2y^2 + z^2
Partial derivz...
Homework Statement
Find the equation of the tangent line to the curve at the given point.
y = \sqrt{2x+1} , (4, 3)Homework Equations
a = 4
M =\lim_{x \to a} \frac{f(x) - f(a)}{x-a}
or
Mpq = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
The Attempt at a Solution
M =\lim_{x \to 4} \frac{f(x) -...
Hi, I was wondering if anyone could help.
Consider the curve given by: 2y^3 + (6x^2)y - 12x^2 + 6y = 1
I solved the derivative which came out to be (4x-2xy)/(x^2 + y^2 + 1
1. Write an equation of each horizontal tangent line to the curve.
2. The line through the origin with slope -1...