13 7 Tangent Planes and Normal Lines for

13. 7 Tangent Planes and Normal Lines for 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-x 1) + b(y-y 1) + c (z – z 1) = 0 parametric form equations of a line in space: x = x 1 + at y = y 1 +bt z = z 1 +ct symmetric form of the equations of a line in space x-x 1 = y – y 1 = z – z 1 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 solution For 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/math 2374/paraboloid_normals. html OR http: //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, -3)

Example 4 Solution Find a set of symmetric equations for the normal line to the surface given by xyz = 12 At the point (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