Parametric equation of tangent line

In summary: The book's answer is correct. In summary, the conversation discusses finding parametric equations for a tangent line to a given curve at a given point. The process involves finding a direction vector and a point, then using the formula x = r_point + t*(r'(t)), where r'(t) is the tangent vector. The mistake made was not evaluating r'(t) at the given point, resulting in incorrect equations. The correct equations are x = 5t + 1, y = 4t + 1, z = 3t + 1.
  • #1
jwxie
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Homework Statement



Find parametric equations for the tangent line to the curve with the given parametric equations at a given point.
[tex]\[x = t^5, y = t^4, z = t^3\][/tex] at point (1,1,1)

Homework Equations


The Attempt at a Solution



So we need to have direction vector, and a point.
To find the tangent vector, we may get it through taking [tex]\[\frac{\mathrm{dt} }{\mathrm{d} x,y,z}\][/tex] respectively.

So I get [tex]\[r^{'}(t) = <5t^4, 4t^3, 3t^2>\][/tex].

In the end, using the formula
[tex]\[r(t) = r_{point} + t*(r^{'}(t))\][/tex]

Putting together, I get
x = 5t^5 + 1
y = 4t^5 + 1
y = 3t^5 + 1

But the book gives
x = 5t + 1
y = 4t + 1
y = 3t + 1

What is my mistake?
Thank you for any input!
 
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  • #2
The book evaluated r'(t) at the point (1,1,1) whereas you didn't.
 
  • #3
Is it still wrong though?
I was reading the note, and apparently my professor forgot to evaluate r'(t) at the point too.
 
  • #4
Yes, if you don't do what is asked, then it is wrong!

By the way, you did NOT "take [tex]\[\frac{\mathrm{dt} }{\mathrm{d} x,y,z}\][/tex]
respectively", you took [tex]\frac{dx}{dt}[/tex], [tex]\frac{dy}{dt}[/tex], and [tex]\frac{dz}{dt}[/itex]- although it might have been better to simply say that you differentiated the "position vector", [itex]t^5\vec{i}+ t^4\vec{j}+ t^3\vec{k}[/itex] with respect to t to get the tangent vector [itex]5t^4\vec{i}+ 4t^3\vec{j}+ 3t^2\vec{k}[/itex]. At the given point, (1, 1, 1), where t= 1, that is [itex]5\vec{i}+ 4\vec{j}+ 3\vec{k}[/itex] and those become the coefficients for the parameter: x= 5t+ 1, y= 4t+ 1, z= 3t+ 1.

Since we have no idea what "note" you are reading we cannot say whether you professor forgot to evaluate at the given point. Was the result a line? If so, then he could not have.

Obviously, the equations for a tangent line must be linear which your
x = 5t^5 + 1
y = 4t^5 + 1
y = 3t^5 + 1
is not.
 

FAQ: Parametric equation of tangent line

1. What is a parametric equation of tangent line?

A parametric equation of tangent line is an equation that represents a line tangent to a curve at a specific point. It is expressed in terms of a parameter, usually denoted as t, which allows for a unique representation of the line.

2. How is the parametric equation of tangent line different from a standard equation of a line?

The parametric equation of tangent line is different from a standard equation of a line because it takes into account the slope of the curve at a specific point, rather than using a fixed slope. This allows for a more accurate representation of the line's position on the curve.

3. How do you find the parametric equation of tangent line?

To find the parametric equation of tangent line, you first need to find the slope of the curve at the point of tangency. This can be done by taking the derivative of the curve at that point. Then, you can use the point-slope formula to create the equation, with the slope and the coordinates of the point of tangency as inputs.

4. Why is the parametric equation of tangent line important in calculus?

The parametric equation of tangent line is important in calculus because it allows for the precise calculation of slopes and rates of change on a curve. This is essential in many applications of calculus, such as optimization and curve sketching.

5. Can the parametric equation of tangent line be used for any type of curve?

Yes, the parametric equation of tangent line can be used for any type of curve, as long as the curve has a defined slope at the point of tangency. This includes both smooth and non-smooth curves, such as circles, parabolas, and piecewise functions.

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