When the tangent line to a parametric curve is horizontal or vertical?

[Miszka]

Hey! That's the second exercise I'm having problems with. I have to show the following:

1) Give a rule for determining when the tangent line to a parametric curve x = f(t), y = g(t) is horizontal and when it is vertical.
2) When is the tangent line to the curve x = t², y = t³ - t horizontal? When is it vertical?

Unfortunately I don't even know how to begin. Could you help me with that exercise as well? It's in the chapter covering Related Rates and Parametric Curves. I would be very grateful, I really want to understand that.

 

[Deleted member 4993]

Hey! That's the second exercise I'm having problems with. I have to show the following:

1) Give a rule for determining when the tangent line to a parametric curve x = f(t), y = g(t) is horizontal and when it is vertical.
2) When is the tangent line to the curve x = x², y = t³ - t horizontal? When is it vertical?

Unfortunately I don't even know how to begin. Could you help me with that exercise as well? It's in the chapter covering Related Rates and Parametric Curves. I would be very grateful, I really want to understand that.

For y = f(x):

what is the value of the slope of the "horizontal" tangent line?

what is the value of the slope of the "vertical" tangent line?

 

[pka]

Hey! That's the second exercise I'm having problems with. I have to show the following:

1) Give a rule for determining when the tangent line to a parametric curve x = f(t), y = g(t) is horizontal and when it is vertical.
2) When is the tangent line to the curve x = x², y = t³ - t horizontal? When is it vertical?

Is that a typo? should it be \(\displaystyle \large x=t^2~?\)

 

[Miszka]

Is that a typo? should it be \(\displaystyle \large x=t^2~?\)

It's a typo, sorry. I can't correct it but take that in mind please.

For y = f(x):

what is the value of the slope of the "horizontal" tangent line?

what is the value of the slope of the "vertical" tangent line?

That's when the derivative (so the slope) is equal to 0.

The slope has to be undefined.

So that's all for the first part? If so it was pretty easy, thank you. What about the next part. Should I calculate the slope of the parametric curve as dy/dx and check when it's equal to zero and then when the denominator is equal to zero?

 

[Dr.Peterson]

That's when the derivative (so the slope) is equal to 0.

The slope has to be undefined.

So that's all for the first part? If so it was pretty easy, thank you. What about the next part. Should I calculate the slope of the parametric curve as dy/dx and check when it's equal to zero and then when the denominator is equal to zero?

I think for part (1), they're looking for a condition in terms of the given functions f and g. What must be true of them or their derivatives in order for dy/dx to be zero or undefined?

Then you will be using that result to answer part (2).

 

[Steven G]

Hey! That's the second exercise I'm having problems with. I have to show the following:

1) Give a rule for determining when the tangent line to a parametric curve x = f(t), y = g(t) is horizontal and when it is vertical.
2) When is the tangent line to the curve x = t², y = t³ - t horizontal? When is it vertical?

Unfortunately I don't even know how to begin. Could you help me with that exercise as well? It's in the chapter covering Related Rates and Parametric Curves. I would be very grateful, I really want to understand that.

You say that you have no idea how to do this. You can always that from x=t². you can get that t= x^1/2 and then t³= x^3/2. So y= y = t³ - t = x^3/2 - x^1/2.
Now you can do part 2

However there ARE better ways to do this. If you have no ideas you should have thought of this one!

 

[Miszka]

I think for part (1), they're looking for a condition in terms of the given functions f and g. What must be true of them or their derivatives in order for dy/dx to be zero or undefined?

Then you will be using that result to answer part (2).

You say that you have no idea how to do this. You can always that from x=t². you can get that t= x^1/2 and then t³= x^3/2. So y= y = t³ - t = x^3/2 - x^1/2.
Now you can do part 2

However there ARE better ways to do this. If you have no ideas you should have thought of this one!

Thank you. I swear that I tried to start but wasn't able to but no it's easy.

 

[HallsofIvy]

If a curve in the xy-plane is given by x= f(t), y= g(t) then its derivative at any t is \(\displaystyle \frac{f'(t)}{g'(t)}\) where the "prime" indicates the derivative with respect to t. The tangent line is "horizontal" if and only that derivative is 0. That is, if f'(t)= 0. The tangent line is "vertical" if and only if the derivative does not exist. That is, if g'(t)= 0.