13.7 Tangent Planes and Normal Linesfor an animation of this topic visit

http://www.math.umn.edu/~rogness/multivar/tanplane_withvectors.shtml

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

Example 1

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

Example 1 solutionFor the function f(x,y,z) describe the level surface

when f(x,y,z) = 0,4 and 10

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

Example 2

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

Example 2 Solution:

Example 3

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

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

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)

Example 4 SolutionFind a set of symmetric equations for the normal

line to the surface given by

xyz = 12 At the point (2,-2,-3)

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