Engineering Calculus Notes 326

# Engineering Calculus Notes 326 - p s,t = s,t,f s,t is a...

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314 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Practice Problems: For each given surface, express the tangent plane (a) as an equation in x , y and z (b) in parametrized form: 1. z = x 2 y 2 , (1 , 2 , 3), (2 , 1 , 3) 2. z 2 = x 2 + y 2 , (1 , 1 , 2), (2 , 1 , 5) 3. x 2 + y 2 z 2 = 1, (1 , 1 , 1), ( 3 , 0 , 2) 4. x 2 + y 2 + z 2 = 4, (1 , 1 , 2), ( 3 , 1 , 0) 5. x 3 + 3 xy + z 2 = 2, (1 , 1 3 , 0), (0 , 0 , 2) 6. x = s y = s 2 + t z = t 2 + 1 at ( 1 , 0 , 2) 7. x = u 2 v 2 y = u + v z = u 2 +4 v at ( 1 4 , 1 2 , 2) . 8. x = (2 cos v ) cos u y = (2 cos v ) sin u z = sin v at any point (give in terms of u and v ). Theory problems: 9. (a) Verify that −→
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Unformatted text preview: p ( s,t ) = ( s,t,f ( s,t )) is a regular parametrization of the graph z = f ( x,y ) of any C 1 function f ( x,y ) of two variables. (b) What is the appropriate generalization for n > 2 variables? 10. Show that the extreme values of the function b (cos θ ) −→ v + (sin θ ) −→ w b 2 occur when tan 2 θ = 2 −→ v · −→ w b −→ v b 2 − b −→ w b 2 ....
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## This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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