Help with Calculus Homework: Find Tangent Plane, Intuitive Geometric Argument

[Marioqwe]

Homework Statement

1.Find the equation of the tangent plane at p=(0,0) on the surface z=f(x,y)=√(1-x²-y²)

p is a vector.

2.Give an intuitive geometric argument to support the result

Homework Equations

The Attempt at a Solution

So I have no problem finding the equation of the tangent line. I am supposed to use the gradient to find it. But I don't really understand the second question. What is a geometric argument?

Is it correct if I justify my answer by saying that the tangent plane is locally linear at p? Is that a geometric argument?

Somehow, I feel that it is just not enough.

Don't give me the answer please. Just a hind of what it means by "geometric argument."

Thank You

 

[Mark44]

Homework Statement

1.Find the equation of the tangent plane at p=(0,0) on the surface z=f(x,y)=√(1-x²-y²)

p is a vector.

2.Give an intuitive geometric argument to support the result

Homework Equations

The Attempt at a Solution

So I have no problem finding the equation of the tangent line. I am supposed to use the gradient to find it. But I don't really understand the second question. What is a geometric argument?

Is it correct if I justify my answer by saying that the tangent plane is locally linear at p? Is that a geometric argument?

No. The surface z = f(x, y) = sqrt(1 - x^2 - y^2) represents a geometric figure. Your justification should involve this figure. If you knew what this figure was, you could work out the equation of the tangent plane in your head, without the use of calculus.

Somehow, I feel that it is just not enough.

Don't give me the answer please. Just a hind of what it means by "geometric argument."

Thank You

 

[Marioqwe]

Thank you very much. I'm still struggling with geometric interpretations but this surely helped.

 

[Mark44]

You're welcome.