Question

A tough curve question; (paraboloid - tangent)
z = x^2 + y^2/9 - 9

How do I find the equation for the tangent plane @ point (0, 3) ?

The part that confuses me with this, is that the given equation (it's an elliptic paraboloid) have three variables; not just two.

On which level is the plane? Does it matter? Can I just set z = 0 and work with x and y?

Here is a piece of my math scribbles;
http://i.imgur.com/nCctKKT.png

Am I heading the right direction? Would it be wrong to set Z to 0? Help is very much appreciated! <3

Answer 1

The tangent can be found by the following: f(a,b) + f_x(a,b)*(x-a)+f_y(a,b)*(y-b)

Answer 2

You have learned about implicit differentiation right?

Answer 3

A little - but only with two variables; not three.
How do I solve it with three? I need to find the tangent plane (z=?)

Answer 4

can you clean up the notation any? to avoid confusion

Answer 5

you have values for x,y,and z for a point in space ... find Fx, Fy, and Fz

Answer 6

I dont know if I have any specifically for Z - All I know I got a point (x=0,y=3) and there is no mention of z; because the tangent plane should be z... as far as I understand , however - I do not know how to use the original equation (z=(...)) and make it usable for x=0 and y=3

Answer 7

if z = (yada)

let f = (yada) - z, if that makes sense

Answer 8

z is defined by x and y values, so the z value at x=0, y=3 can be determined

Answer 9

Big Fx is derivation? Integration? Differential? etc.

Answer 10

z = -8 if i put 0 and 3 in it, but I still do not know how to proceed :X

Answer 11

Fx is just a partial with respect to x

Answer 12

Because I study in another language; I am not familiar with the term "partial" (mathematically)

Answer 13

the notation is amgibuous, can you properly notate: z= .....

partial means that you treat all other variables as constants.

\[z=\frac{x^2+y^2}{9}-9\]
let
\[f=\frac{x^2+y^2}{9}-9-z\]
\[f_x = \frac{2x+0}{9}-0-0\]
\[f_y=\frac{0+2y}{9}-0-0\]
\[f_z=\frac{0+0}{9}-0-1\]

Answer 14

at x=0, y=3

fx = 0
fy = 6/9 = 2/3
fz = -1

or since the direction is what we need and the magnitude is irrelevant

(0,2,-3) is in the same direction is (fx,fy,fz) and is simpler to play with i think

Answer 15

partial derivative simply means that if we are working with respect to some variable (like y), then all other variables are considered to act like a constant.

say f=2xy^2

fx = 2y^2
fy = 4xy

Answer 16

Ah - but what did you mean with Fx, Fy, Fz? (Capital instead of fx, fy, etc.)

I understand what you meant with partial now - I have done something similar in other tasks

Answer 17

I was simply trying to make x,y,and z stand out is all. It looked fine in my head :)

Answer 18

another way to approach this, is to let x=0, and determine the slope of the tangent to z, at y=3

find dz/dy when x=0

find dz/dx when y=3

Answer 19

I will do some more work, and try it out - hopefully a puzzle piece will pop into my head and allow me to learn this :)