# Finding the tangent plane and normal line

• Kork
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.
Kork
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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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.

So if I do like this:

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

it can be concidered a valid answer?

## 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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