Finding the tangent plane and normal line

In summary, you are asked to find the equations of the tangent plane and normal line to the graph of a function at a given point. You use the tangent plane equation z = f(a,b) + f1(a,b)*(x-a) + f2(a,b)*(y-b) and plug in the values of the function and its first derivatives at the given point to get a simplified equation. However, the assignment is asking for multiple equations, so you can manipulate the equation by multiplying both sides by a constant or rearranging terms to get different forms of the same equation.
  • #1
Kork
33
0
D is a set of all points (x,y) in R2 distinct from (0,0).
I have the funtion f: D --> R which is defined by:
f(x,y) = (2xy)/(x2+y2

Im asked to find the equations (in plural) of the tangelnt plane and the normal line to the graph of the function at (1,2)

My attempt:

I use the tangent plane z= f(a,b)+f1(a,b)*(x-a) + f2(a,b)*(y-b)

f(1,2) = 4/5
f1(1,2) = 12/25
f2(1,2) = -6/25

when I put that into the tangent plane equation and simplify I get: (12/25)x - (6/25)y+4/5

Is this my normal line?

Im very confused, because the assigment asks me to find multiple equations of the tangent plane which I have never done before.

Help is appreciated.
 
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  • #2
[/
Kork said:
D is a set of all points (x,y) in R2 distinct from (0,0).
I have the funtion f: D --> R which is defined by:
f(x,y) = (2xy)/(x2+y2

Im asked to find the equations (in plural) of the tangelnt plane and the normal line to the graph of the function at (1,2)

My attempt:

I use the tangent plane z= f(a,b)+f1(a,b)*(x-a) + f2(a,b)*(y-b)

f(1,2) = 4/5
f1(1,2) = 12/25
f2(1,2) = -6/25

when I put that into the tangent plane equation and simplify I get: (12/25)x - (6/25)y+4/5
Before, you had "z= " that. And what happened to the "- a" and "- b"?

Is this my normal line?
You just said it was the "tangent plane equation". A plane is not a line!

Im very confused, because the assigment asks me to find multiple equations of the tangent plane which I have never done before.

Help is appreciated.
Any equation can be written in a number of ways. For example, multiplying both sides by a constant will give a new equation for the same object. Or just move terms from one side to another.
 
  • #3
So if I do like this:

z*2 = ((12/25)x - (6/25)y+4/5) * 2

it can be concidered a valid answer?
 

Related to Finding the tangent plane and normal line

1. What is a tangent plane?

A tangent plane is a flat surface that touches a curved surface at only one point. It is used in calculus to approximate the behavior of a function at a specific point.

2. How is the tangent plane found?

The tangent plane can be found by taking the partial derivatives of a multivariable function at a specific point and using them to create an equation for a plane. This equation represents the tangent plane at that point.

3. What is the purpose of finding the tangent plane?

The purpose of finding the tangent plane is to approximate the behavior of a function at a specific point. This can be used to calculate slopes, rates of change, and other important information about the function.

4. What is a normal line?

A normal line is a line that is perpendicular to a tangent plane at a given point. It represents the direction in which the function is changing most rapidly at that point.

5. How is the normal line related to the tangent plane?

The normal line and the tangent plane are closely related, as they both represent the behavior of a function at a specific point. The tangent plane is a flat surface that touches the function at that point, while the normal line is a line that is perpendicular to the tangent plane at that point.

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