# Calculus question

• Aug 3rd 2007, 05:55 AM
meebo0129
Calculus question
I was wondering if I did the following problem right:

Write the equation of the plane tangent to the given surface at the given point P.

xyz+ 2x^3 - y^2 + z^2 = 10; P(2, -1, -1)

So what I did was find the gradient of the above and got:

<yz + 4x, xz - 2y, xy - 2z>

Substituting P in I got, a normal vector of <9, 0, 0>

This led to a plane equation of:
9(x - 2) = 0
• Aug 3rd 2007, 06:40 AM
ThePerfectHacker
Quote:

Originally Posted by meebo0129
I was wondering if I did the following problem right:

Write the equation of the plane tangent to the given surface at the given point P.

xyz+ 2x^3 - y^2 + z^2 = 10; P(2, -1, -1)

So what I did was find the gradient of the above and got:

<yz + 4x, xz - 2y, xy - 2z>

Substituting P in I got, a normal vector of <9, 0, 0>

This led to a plane equation of:
9(x - 2) = 0

Try again, you got the gradient all wrong.
• Aug 3rd 2007, 07:22 AM
meebo0129
Okay...here's my second attempt:

(x-2)(yz-2x) + (y+1)(xz-2y) + (z+1)(xy-2z) = 0

(x-2)(1-4)+(y+1)(-2+2)+(z+1)(-2+2)=0

Then, I simplified this to:
-3(x-2)=0
x-2 = 0

x = 2 <- equation of plane

Would this be correct?
• Aug 3rd 2007, 08:00 AM
Plato
You have been told: "the gradient is all wrong."
$\displaystyle \begin{array}{rcl} \nabla F & = & \left\langle {\partial _x F,\partial _y F,\partial _z F} \right\rangle \\ & = & \left\langle {yz + 4x ,xz - 2y,xy + 2z} \right\rangle \\ \end{array}$
• Aug 3rd 2007, 09:24 AM
meebo0129
Oh, you know what? I accidentally wrote 2x^3 for the second term instead of 2x^2, which is what it was supposed to be. If this is the case, then is my original response correct?
• Aug 3rd 2007, 09:29 AM
Plato
O.K. I have corrected the above.
But note the your "z" component should be with a +.
• Aug 3rd 2007, 05:50 PM
curvature
Quote:

Originally Posted by meebo0129
Write the equation of the plane tangent to the given surface at the given point P.

xyz+ 2x^3 - y^2 + z^2 = 10; P(2, -1, -1)

I was wondering if P(2,-1,-1) is on the given surface.