Tangent line Definition and 279 Threads

Homework Statement Find an equation of the tangent line to the curve y = x√x that is parallel to the line y = 1+3x. Homework Equations m = 3 The Attempt at a Solution Here is my attempt: dy/dx(x) * dy/dx(x^(1/2)) = (1) * (1/2x^(-1/2)) = (1/2x^(-1/2)) (1/2x^(-1/2)) = 3 →...

1. The question is What is the slope of the line tangent to f(x)=2x^2-x-7 at x=-1 2. Must be solved using derivatives 3. So basically i know the equation (f(x+deltax)-f(x)/deltax) So i plug in the problem and i get (2(x+deltax)^2-(x+deltax)-7)-(2x^2-x-7) Now i know what the...

Homework Statement Given that near (1,1,1) the curve of intersection of the surfaces x^4 + y^2 + z^6 - 3xyz = 0 and xy + yz + zx - 3z^8 = 0 has the parametric equations x = f(t), y = g(t), z = t with f, g differentiable. (a) What are the values of the derivatives f'(1), g'(1)? (b)...

Homework Statement Suppose line tangent to graph of y=f(x) at x =3 passes through (-3, 7) & (2,-1). Find f'(3), what is the equation of the tangent line to f at 3? Homework Equations I found the slope of which equals -8/5 Im not sure how to find the equation... do I do...

Homework Statement Find equation of tangent line, given x = -1. Not given y. I am used to having this when I am given both y and x. Homework Equations (x^3 - 4x + 2)(x^4 + 3x - 5)The Attempt at a Solution Differentiate (3x^2 - 4)(4x^3+3) Multiply 12x^5 - 9x^2 - 8x^3 - 12 Plug in -1, find...

Homework Statement (a) Draw a diagram to show that there are two tangent lines to the parabola y = x^2 that pass through the point (0, -4). (Do this on paper. Your teacher may ask you to turn in this work.) (b) Find the coordinates of the points where these tangent lines intersect the...

Homework Statement Find the direction of the line tangent to the curve x^4+y^4=32 at the point (2, -2) Homework Equations Anything goes, we're in vector calculus now. The Attempt at a Solution So, to find the tangent line, I was thinking of taking the gradient, but I'm not...

Homework Statement Find an equation of the tangent line to the curve at the given point. y = tan x at point (pi/4,1) Homework Equations The Attempt at a Solution step 1. find the derivative of tan x, which sec^2 x step 2. find the slope. this is where I mess up. I assume...

Homework Statement Find a > 0 such that the tangent line to the graph of f(x) = x^{2}e^{-x} at x = a passes through the origin. Homework Equations The Attempt at a Solution First I found the derivative to be: -e^{-x}(x-2)x , which is the slope of the function. I know the tangent line...

This is another question from last year's calculus exam that I'm a bit stuck on. " Let f be the real function given by f(x) = −x4+2x2+3x. (a) Determine the tangent line to the curve y = f(x) at x = 1. (b) Show that the tangent line to the curve y = f(x) at x = 1 is also the tangent line to...

Homework Statement You measure a cubic container and find it to be about 10 cm on each side. From this, you conclude that it holds 1000 cc. However, your measurement is accurate only to within ±0.1 cm. (So you can be sure that the side is between 9.9 and 10.1 cm.) What are x, f(x) , and a in...

Homework Statement Suppose the point (pi/3, pi/4) is on the curve sinx/x + siny/y = C, where C is a constant. Use the tangent line approximation to find the y-coordinate of the point on the curve with x-coordinate pi/3 + pi/180. Homework Equations TLA: f(a) + f'(a)(x-a) Where a is...

Homework Statement Find all x coordinates of points (x, y) on the curve y = (x − 2)5 /(x − 4)3 where the tangent line is horizontal Homework Equations Quotient Rule - Differentiation The Attempt at a Solution So I could guess, that if tangent line is horizontal, the equation of...

Homework Statement Find the tangent line equation for y = sin x at x = 6\,\pi. Homework Equations m_{tangent}\,=\,\lim_{h \to 0}\,\frac{f(c+h)\,-\,F(c)}{h} y\,-\,y_1\,=\,m\,(x\,-\,x_1) The Attempt at a Solution m_{tangent}\,=\,\lim_{h \to 0}\,\frac{f(c+h)\,-\,F(c)}{h}...

Given the paraboloid z = 6 - x - x2 -2y2 and the plane x = 1, find curve of intersection and the parametric equations of the tangent line to this curve at point (1,2,-4).So I plugged x=1 into the paraboloid equation and got z = 4-2y2. Then I take the derivative of the curve of intersection...

Homework Statement Find the point(s) on the graph of the function at which the tangent line has the indicated slope. (If an answer does not exist, enter DNE.) g(x) = (1/3)x^3 - (1/2)x^2 - 4x +8 mtan=-4 Homework Equations [b]3. The Attempt at a Solution firstly i derived...

I'm on mobile so I can't use latex. Let C: y=8x^5+5x+1 and suppose L is a line through the origin tangent to C at a point P=(a,f(a)) on C. -Find the coordinates of P -Compute the slope m sub L of L Where should I begin? I'm guessing I would need the derivative of the equation f(x)...

Homework Statement Find an equation of the tangent line at the indicated point on the graph of the function. y=f(x)=x^3/4 , (x,y)=(6,54) Homework Equations The Attempt at a Solution I did the derivative which I get 3x^2/4 and then I plugged in the 6 and get 162. Is that the...

Homework Statement Function f(x) = KxL K= 1.78 L= -1.39 Problem 1: Find f'(x). ____________________ v= 0.89 w= 0.5 "v" and "w" are two points located on the x-axis. Problem 2: Calculate f'(v). ____________________ Problem 3: Find the equation of the tangent line of f(x) over the point "w". The...

Homework Statement Consider the function f(x)=x2+5x+5 and the point A(4,5). Find a formula for the slope of a line passing through point A, and an arbitrary point on the function. Your answer should be a formula in terms of x. slope=? Homework Equations f′(x)=2x+5 The Attempt at...

If every tangent line of some curve B(s) which has the unit speed parametrization passes through a fixed point P, then the curve B(s) must be a line. As a hint, my book says p = B(s) + r(s)B'(s) where r(s) is some function. so i have that any tangent line L(t) = B(s) + t B'(s) and for...

Homework Statement So I have to find the tangent line when x=5. I use the difference quotient, but I'm stuck at the algebra? I can't seem to figure out what steps I should take. Here's what I have: [(x+h)+1/(x+h)] - (x+1/x) I've tried numerous steps on how to solve this, but can't do...

I need to find the equation of the tangent line to the curve x=t4+1, y=t3+t; t=-1 I have already found that the slope of the line is -1 by finding (dy/dt)/(dx/dt) I just need to figure out how to solve for y1 and x1 Thanks in advance

So the curve is defined as x-y^2 = 0, z= x . I'm supposed to find the tangent line to the curve. How do I find a slope in R3?

Homework Statement Find the equation for the tangent line at (2,2) f(x)=xy+y^3=12 Homework Equations The Attempt at a Solution I really feel like I know what I'm doing here, but the key disagrees. My equation for the tangent slope comes out to be: (-y)/(x+3y^2) when...

Homework Statement Find the slope of the tangent line to the curve of intersection of the vertical plane x-y+1= 0 and the surface z=x^2+ y^2 at the point (1, 2, 5) . Homework Equations The Attempt at a Solution Normal Vector 1 is i-j Normal Vector 2 is 2i+4j-k Cross them to get...

If there is an an ellipse x^2/9 + y^2/16 = 1, and the slope of the tangent is dx/dy = -16x/9y, how do you find what points at which the slope of the tangent is 1? I have no idea how to answer this and I've been trying for like an hour. Can anyone help me?

Homework Statement Center of circle is: (3,2) Tangent point: (8,4) Question: What is the equation of the tangent line? Homework Equations The Attempt at a Solution I am just not getting it. So, would the radius be 5? Now, would i go: (x-3)^2 + (y-2)^2 = 5 ...

Homework Statement What is the equation of the tangent line to the curve y = Sqrt( 2x - 1) where the tangent line is parallel to the line x - 3y = 16 The Attempt at a Solution My teacher briefly mentioned to find the derivative of the given, and then plug it into the slope-y intercept...

Does anyone know how to find the equation of a tangent line to y=e^(-2x) at the point (1,e^-2) I honestly have no idea how to even start this problem when I tried it I came up with y=1(x+.27)+ln2

Hi guys, I'm stuck with a problem here: Let a curve be given by the following parametric equations: x=t, y=t^2, z=t^3. At which points is the tangent line (of the curve) parallel to the plane x + 2y + z = 0? What is the underlying principle behind this? My thoughts: The tangent line...

Homework Statement Determine the coordinates of the points on the graph of f(x) = _/'2x+1 where the tangent line is perpendicular to the line 3x+y+4 = 0 _/' -means square root Homework Equations f(x) = _/'2x+1 3x+y+4 = 0 The Attempt at a Solution I made it equal to y like y= -3x-4...do I...

Homework Statement Homework Equations r = <x,y,z> r' = <x',y',z'> The Attempt at a Solution I started by finding the derivatives of each part of the vector and got: x= 2/sqrt(t) y= 3t^2+1 z= 3t^2-1 Then I plugged the point (5,2,0) into that and got (2/sqrt(5), 13, -1). This should be...

Help! Solving Derivatives using "Tables" and Equation on Tangent Line 1. I got a worksheet that was given to me on Derivatives to finish for homework but hardly understand it since my teacher assigned it for the same night and she had taught it I was wondering if an explanation on how to do...

Homework Statement See the attachment. I feel like my answer is right, but I keep getting told I am wrong. This is the first time I have taken any math in 5 years so I am figuring I must not be simplifying enough? But I can't see what I need to simplify. Perhaps I have made a dumb error and...

Find the equation of the line with slope -1 that is tangent to the curve y = 1/(x-1). The equation of a line with slope -1 is y = -x + k The curve is y = 1/(x-1) Set the y-values to each other: 1/(x-1) = -x + k Rearrange and set equal to 0: 1 = -x² + x + k x² - x + 1 – k = 0...

Homework Statement Okay, so I was helping someone study for the AP calculus exam, and I don't really know why we got this question wrong. It says to find the tangent line to the equation f(x)=x^4+2x^2 at x=1. Homework Equations Taylor series, we'll just use first order. f(x)|_{x=x_0}...

Homework Statement Find the slope of the tangent line to the curve: 2(x^2 + y^2)^2 = 25(x^2 - y^2) at the point (-3, -1) Homework Equations Implicit differentiation The Attempt at a Solution 2(x^2 + y^2)^2 = 25(x^2 - y^2) 1. 4(x^2 + y^2)(2x + 2y(dy/dx)) = 25(2x -...

Homework Statement Find the slope of the tangent line to the curve: sqrt(4x+2y) + sqrt(1xy) = 9.72 at the point (6,3) Homework Equations Derivative laws The Attempt at a Solution the slope of the tangent line to a curve is the Derivative of the function of the...

Homework Statement Find the lines that are tangent and normal to the curve x^2 + xy - y^2=1 at (2,3) Homework Equations Umm.. first derivative and second derivative? The Attempt at a Solution For the tangent line i found the first derivative and used the implicit differentiation(?)...

Homework Statement Consider the ellipsoid 4x^2+2y^2+z^2 = 19. Find all the points where the tangent plane to this ellipsoid is parallel to the plane 2y−8x+z = 0. Homework Equations The Attempt at a Solution So I found the normal vector to the tangent, <8x,4y,2z>. I also...

Homework Statement Let f(x,y) = 5y^(2)-(2x^(2)+xy) Then an implicit equation for the tangent plane to the graph of f at the point (0,-2) is Homework Equations The Attempt at a Solution I understand that I should take the derivative to find the gradient vector. For the...

Homework Statement The parametric form for the tangent line to the graph of y = 2x^(2)+2x-1 at x = -1 is Homework Equations The Attempt at a Solution I am confused about where to begin this problem. Any thoughts? Thanks!

How would you write a proof that proves that the minimum area between a function and its tangent line is the tangent line evaluated at point p, where p is the midpoint on a given interval? i.e. The minimum area between x^2, and its tangent line on the interval [0,1]...

Homework Statement The ellipsoid 4x^2+2y^2+z^2=16 intersects the plane y=2 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1,2,2) Homework Equations sin(t)^2 + cos(t)^2 = 1 The Attempt at a Solution After plugging 2 in for y, I get...

Homework Statement This is a very basic problem, though it did confuse me a little: Find the tangent equations to the curve x=3t^2+1 \ , \ y = 2t^3+2 which intercepts the point (4,3). Homework Equations --- The Attempt at a Solution I took \frac{dy}{dx} = t =...

Homework Statement Hello, this is a problem from the practice test for the GRE subject test. For what value of b is the line y=10x tangent to the curve y=e^{bx} at some point in the xy-plane? A) \frac{10}{e} B)10 C)10e D)e^{10} E)eHomework Equations The Attempt at a Solution For the line to be...

Homework Statement Verify that (1,0) is on the following curve and find the tangent line and normal line to the curve at the point. y=pisin(pix-y)The Attempt at a Solution i think i got it is y ' {-1/pi*cos(pi*x-y)} + pi

Homework Statement We are given the unit circle and the point (5,2). There are two lines that are tangent to the unit circle and they both intersect at the point (5,2). What are the points where these lines are tangent with the unit circle. Homework Equations Tangent line of a circle at...

First Problem Homework Statement Find the derivative of x^6+y^6=18xy Homework Equations Find derivative The Attempt at a Solution 6x^5+6y^6=18*(dy/dx) Second Problem Homework Statement Verify that (1,0) is on the following curve and find the tangent line and normal...