Parametric equation for a tangent line (multi vars)

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Homework Help Overview

The problem involves finding the parametric equation for the tangent line to a curve defined by the equations x=t^3-1, y=t^4+1, z=t at a specific point (26, 82, 3). The subject area includes multivariable calculus and parametric equations.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the differentiation of the parametric equations and the correct substitution of the parameter t. There is confusion regarding the use of the coordinates of the point in place of the parameter t in the derivatives.

Discussion Status

There is an ongoing exploration of the correct approach to finding the tangent line, with participants providing guidance on the proper use of the parameter t. Some participants express uncertainty about their reasoning, while others clarify the distinction between the parameter and the coordinates.

Contextual Notes

Participants note that the value of t is given as 3 at the point of tangency, which is a crucial aspect of the problem that has led to confusion in the substitution process.

forest125
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Homework Statement



Find the parametric equation for the line tangent to the curve:

x=t^3-1, y=t^4+1, z=t

at the point (26, 82, 3).

Use the variable t for your parameter.


Homework Equations





The Attempt at a Solution



dx/dt=3t^2, dy/dt=4t^3, dz/dt=1

I got then that

x=26+(3*26^2)t
y=82+(4*82^3)t
z=3+t

and only the z component is correct. Where am I going wrong. Any help is really appreciated!
 
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Look, at your given point t=3. When you are working out the d/dt parts why are you substituting x and y for t?
 
You should not plug back your given point coordinates into "t" in your derived equations. Your value of t is constant. And technically it's given in your problem.
 
I substituted x and y into my derivatives to find the slope of the that component of the line at that point. I then multiply it by t, and add x and y for intercepts or to start from the origin.
 
t is t. x is x. y is y. z happens to be t. You got lucky on the z component because z=t. Just put t in where your equations say 't'. Don't substitute x for t just because it happens to be the 'x' component.
 
I don't know why I'm so hung up on this question.

For the x component for instance,

it the curve is t^3-1, the slope of the tangent is 3t^2. If we're talking about the slope of the tangent at 26 in the curve, why wouldn't we plug in 26 for t in dx/dt?

What is the next step I should take after finding the derivative?
 
Oh jesus. t=3 right. That can't change.

Wow I feel stupid. Thanks all for the help.
 

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