Solved

parametric equation

Posted on 2009-04-06
6
487 Views
Last Modified: 2012-05-06
Consider the line perpendicular to the surface z = x2 + y2 at the point (3, 4, 25).

Which of the following vectors is normal to the surface at the given point?

Below is the answer key that I don't understand
Response Details:
Equation of the surface
.
at the point (3,4,25) the eqn of the tgt plane is
 
  x(3)+y(4)-(1/2)(z+25)=0  ==> 6x+8y-z=25

where did they get the formula  x(3)+y(4)-(1/2)(z+25)=0 from?
.
0
Comment
Question by:kuntilanak
  • 3
  • 2
6 Comments
 
LVL 84

Expert Comment

by:ozo
ID: 24083331
does x2 + y2 mean x squared + y squared?

If so, compare dz/dx and dz/dy at that point with
 6x+8y-z=25
0
 

Author Comment

by:kuntilanak
ID: 24083364
it means squared
0
 

Author Comment

by:kuntilanak
ID: 24083369
so is the derivative of z equals to the tangent of the plane?
0
Is Your Active Directory as Secure as You Think?

More than 75% of all records are compromised because of the loss or theft of a privileged credential. Experts have been exploring Active Directory infrastructure to identify key threats and establish best practices for keeping data safe. Attend this month’s webinar to learn more.

 
LVL 25

Accepted Solution

by:
InteractiveMind earned 500 total points
ID: 24085604
> Which of the following vectors is normal to the surface at the given point?

To find the surface normal:

Method 1.  Notice that the surface is just an inverted cone with its vertex at the origin? So the normal is quite intuitively going to be (x,y,-1/2).

Method 2.  If you can't quite spot the above, then you can work it out by first parametrising the surface:
  p(z,t) = (sqrt(z)*cos(t), sqrt(z)*sin(t), z),
then the surface normal is defined as:
  N = dp/dz x dp/dt    (where x is the vector cross product).
The result comes to (x,y,-1/2) again.


> 6x+8y-z=25

Are you sure they want you to derive it? It looks more like they're telling you what the tangent plane is, and then expect you to use it. If not though, then the tangent plane at (x_0, y_0) for a surface z=f(x,y) is defined as:

  z = f(x_0, y_0) + df/dx(x_0,y_0)(x-x_0) + df/dy(x_0,y_0)(y-y_0)

where f=x^2+y^2, and x_0=3, y_0=4 in your case. So plugging that in and rearranging leads to the required equation..
0
 

Author Comment

by:kuntilanak
ID: 24088179
>>Are you sure they want you to derive it? It looks more like they're telling you what the tangent plane is, >>and then expect you to use it. If not though, then the tangent plane at (x_0, y_0) for a surface >>z=f(x,y) is defined as:


I don't even know if they want me to derive it or not.... that's what I am asking though

>> what equation is this for
  z = f(x_0, y_0) + df/dx(x_0,y_0)(x-x_0) + df/dy(x_0,y_0)(y-y_0)
0
 
LVL 25

Expert Comment

by:InteractiveMind
ID: 24121388
"the tangent plane at (x_0, y_0) for a surface z=f(x,y)"
0

Featured Post

Is Your Active Directory as Secure as You Think?

More than 75% of all records are compromised because of the loss or theft of a privileged credential. Experts have been exploring Active Directory infrastructure to identify key threats and establish best practices for keeping data safe. Attend this month’s webinar to learn more.

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Suggested Solutions

Title # Comments Views Activity
Good problem solving books 4 379
Frequency distribution 6 42
Group Data Frequency Distribution 9 43
Triangle - calculating angles 9 63
How to Win a Jar of Candy Corn: A Scientific Approach! I love mathematics. If you love mathematics also, you may enjoy this tip on how to use math to win your own jar of candy corn and to impress your friends. As I said, I love math, but I gu…
Foreword (May 2015) This web page has appeared at Google.  It's definitely worth considering! https://www.google.com/about/careers/students/guide-to-technical-development.html How to Know You are Making a Difference at EE In August, 2013, one …
Internet Business Fax to Email Made Easy - With  eFax Corporate (http://www.enterprise.efax.com), you'll receive a dedicated online fax number, which is used the same way as a typical analog fax number. You'll receive secure faxes in your email, f…
This is a video describing the growing solar energy use in Utah. This is a topic that greatly interests me and so I decided to produce a video about it.

861 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question

Need Help in Real-Time?

Connect with top rated Experts

29 Experts available now in Live!

Get 1:1 Help Now