1 / 13

Understanding Tangent Planes and Normal Lines in Multivariable Calculus

This resource covers tangent planes and normal lines in the context of multivariable calculus. It includes a review of the standard equation of a plane and parametric equations for lines in space, along with practical examples of determining tangent planes and normal lines for various surfaces such as hyperboloids and paraboloids. The animated visualizations from the provided links enhance understanding. Additionally, a light-hearted anecdote highlights the concept of "normal" in a classroom setting, making the topic more relatable.

prisca
Download Presentation

Understanding Tangent Planes and Normal Lines in Multivariable Calculus

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 13.7 Tangent Planes and Normal Linesfor an animation of this topic visithttp://www.math.umn.edu/~rogness/multivar/tanplane_withvectors.shtml

  2. Recall from chapter 11: • Standard equation of a plane in Space • a(x-x1) + b(y-y1) + c (z – z1) = 0 • parametric form equations of a line in space: x = x1 + at y = y1 +bt z = z1 +ct • symmetric form of the equations of a line in space • x-x1 = y – y1 = z – z1 • a b c

  3. Example 1 For the function f(x,y,z) describe the level surfaces when f(x,y,z) = 0,4 and 10

  4. Example 1 solution For the function f(x,y,z) describe the level surface when f(x,y,z) = 0,4 and 10

  5. For animated normal vectors visit:http://www.math.umn.edu/~rogness/math2374/paraboloid_normals.htmlORhttp://www.math.umn.edu/~rogness/multivar/conenormal.html

  6. Example 2 Find an equation of the tangent plane to given the hyperboloid at the point (1,-1,4)

  7. Example 2 Solution:

  8. Example 3 Find the equation of the tangent to the given paraboloid at the point (1,1,1/2)

  9. Example 3 Solution: Find the equation of the tangent to the given paraboloid at the point (1,1,1/2). Rewrite the function as f(x,y,z) = - z

  10. Example 4 Find a set of symmetric equations for the normal line to the surface given by xyz = 12 At the point (2,-2,-3)

  11. Example 4 Solution Find a set of symmetric equations for the normal line to the surface given by xyz = 12 At the point (2,-2,-3)

  12. One day in my math class, one of my students spent the entire period standing leaning at about a 30 degree angle from standing up straight. I asked her “Why are you not standing up straight? “ She replied “Sorry, I am not feeling normal.” Of course that students name was Eileen. - Mr. Whitehead

More Related