directional derivative problem

Nov 2009
94
6
At what point on the paraboloid \(\displaystyle y=x^2+z^2\) is the tangent plane parallel to the plane \(\displaystyle x+2y+7z=2\)?

Really stuck on this one.

Edit: not \(\displaystyle y=x^2+y^2\)
 
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matheagle

MHF Hall of Honor
Feb 2009
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At what point on the paraboloid \(\displaystyle y=x^2+y^2\) is the tangent plane parallel to the plane \(\displaystyle x+2y+7z=2\)?

Really stuck on this one.

I assume you meant \(\displaystyle z=x^2+y^2\)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
At what point on the paraboloid \(\displaystyle y=x^2+z^2\) is the tangent plane parallel to the plane \(\displaystyle x+2y+7z=2\)?

Really stuck on this one.

Edit: not \(\displaystyle y=x^2+y^2\)
Well, that's pretty straight forward isn't it? Rewrite this as \(\displaystyle f(x,y,z)= x^2+ z^2- y= 0\) so you can think of this paraboloid as being a "level surface" for f(x,y,z). What is the gradient of f at any point (x,y,z)? What is a normal vector to the plane? where are those two vectors parallel?
 
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Nov 2009
94
6
Well, that's pretty straight forward isn't it? Rewrite this as \(\displaystyle f(x,y,z)= x^2+ z^2- y= 0\) so you can think of this paraboloid as being a "level surface" for f(x,y,z). What is the gradient of f at any point (x,y,z)? What is a normal vector to the plane? where are those two vectors parallel?
Gradient vector of f is \(\displaystyle <2x,-1,2z>\)
Normal to the plane is \(\displaystyle <1,2,7>\)

Tangent plane is parallel to that plane if the corresponding normal vectors are parallel, so
\(\displaystyle <2x_o,-1,2z_o> = c<1,2,7>\)
or equivalently \(\displaystyle <x_o,\frac{-1}{2},z_o> = k<1,2,7>\)

\(\displaystyle x_o = k \)
\(\displaystyle \frac{-1}{2} = 2k \)
\(\displaystyle z_o = 7k \)

Solving for k yields \(\displaystyle k=\frac{-1}{4}, x_o=\frac{-1}{4}, z_o=\frac{-7}{4}, y_o=\frac{50}{16} \)

Did I do that correctly?
 

HallsofIvy

MHF Helper
Apr 2005
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Yes, good! (Clapping) (Although I would have written \(\displaystyle y_0\) as \(\displaystyle \frac{25}{8}\).)