HH_14_5

# HH_14_5 - 2 y 2-z 2 =-1 Answer Tangent plane 4 x − 2 4 y...

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(11/11/08) Math 10C. Lecture Examples. Section 14.5. Directional derivatives and gradient vectors in space Example 1 (a) What is the gradient of f ( x , y , z ) = xyz at (1,2,3)? (b) What is the directional derivative of f at (1,2,3) in the direction toward (2,3,4)? (c) What is the greatest directional derivative of f at (1,2,3)? Answer: (a) f (1 , 2 , 3) = a 6 , 3 , 2 A (b) The directional derivative of f at (1 , 2 , 3) in the direction toward (2 , 3 , 4) is 11 3 . (c) The greatest directional derivative of f at (1 , 2 , 3) is 7. Example 2 Give an equation of the tangent plane at the point (2, 2,3) on the hyperboloid of two sheets x
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Unformatted text preview: 2 + y 2-z 2 =-1 . Answer: Tangent plane: 4( x − 2) + 4( y − 2) − 6( z − 3) = 0 or 2 x + 2 y − 3 z = − 1 Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/: ‡ Section 14.7: Examples 4, 5, and 6 † Lecture notes to accompany Section 14.5 of Calculus by Hughes-Hallett et al. ‡ The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course. 1...
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