How to find the straight tangent line?

In summary, the conversation discusses finding the gradient of a function at a given point and the corresponding tangent plane. The conversation also mentions a possible error in the calculation of the tangent plane and the confusion surrounding finding the straight tangent line. The expert summarizes that in two dimensions, the tangent plane and tangent line are the same, but in higher dimensions, the tangent plane is a more general concept.
  • #1
Helloooo
6
0
Homework Statement
Find the straight tangent line at (2,-1) to the level curve of f(x,y)=x^2-y^2
Relevant Equations
f(x,y)=x^2-y^2
Point: (2,-1)
I have solved the gradient:

gradf(2,-1)=(4,2)

and have the tangent plane:

4x+2y+3=0

Somehow the answer is:

3=2x+y

And i really don´t understand why.
 
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  • #2
Helloooo said:
and have the tangent plane:

4x+2y+3=0
No you don’t. Inserting (2,-1) in the LHS gives 4*2+2(-1) = 8-2 = 6 so the required point is not on that plane.
 
  • #3
Orodruin said:
No you don’t. Inserting (2,-1) in the LHS gives 4*2+2(-1) = 8-2 = 6 so the required point is not on that plane.
Firstly i noticed an error in my writing.
The tangent plane i got was 4x+2y-3=0.
I´m sorry for that
So i redid the tangent plane to
4x+2y-6=0
However i still don´t know how to fint the straight tangent line.
Thank you
 
  • #4
Helloooo said:
However i still don´t know how to fint the straight tangent line.
That's it, right there:
Helloooo said:
4x+2y-6=0
 
  • #5
Orodruin said:
That's it, right there:
Does that mean that the tangent plane and straight tangent line is the same?
If so, why are they called differently or is it just in this particulary problem?
 
  • #6
Helloooo said:
Does that mean that the tangent plane and straight tangent line is the same?
If so, why are they called differently or is it just in this particulary problem?
In two dimensions, a tangent line and the tangent plane are the same thing. In n dimensions the tangent plane has n-1 dimensions. It is a more general concept because it is not restricted to two dimensions.
 
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  • #7
Orodruin said:
In two dimensions, a tangent line and the tangent plane are the same thing. In n dimensions the tangent plane has n-1 dimensions. It is a more general concept because it is not restricted to two dimensions.
I see, thank you so much for your help!
 

FAQ: How to find the straight tangent line?

How do I determine the slope of the tangent line?

The slope of the tangent line can be found by taking the derivative of the function at the point of interest. This will give you the slope of the tangent line at that specific point.

Can I use any point on the curve to find the tangent line?

No, the tangent line must be drawn at a specific point on the curve. This point is known as the point of tangency and is where the tangent line touches the curve at only one point.

What is the equation for a straight tangent line?

The equation for a straight tangent line is y = mx + b, where m is the slope of the tangent line and b is the y-intercept. The slope can be found using the derivative of the function, and the y-intercept can be found by plugging in the coordinates of the point of tangency.

How do I know if I have found the correct tangent line?

The tangent line should only touch the curve at one point, which is the point of tangency. Additionally, the slope of the tangent line should match the slope of the curve at that point. You can also check your answer by graphing the function and the tangent line to see if they intersect at the point of tangency.

Is there a shortcut for finding the tangent line?

Yes, there is a shortcut called the tangent line approximation or linearization. This method uses the tangent line at a nearby point to approximate the tangent line at the point of interest. It is often used in calculus to simplify calculations and find approximate solutions.

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