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Unformatted text preview: hernandez (ah29758) – M408D Quest Homework 11 – pascaleff – (54550) 1 This printout should have 18 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points The contour map given below for a function f shows also a path r ( t ) traversed counter clockwise as indicated. 1 2 3321 Q P R Which of the following properties does the derivative d dt f ( r ( t )) have? I zero at R , II negative at Q , III negative at P . 1. I only 2. all of them 3. III only 4. I and III only correct 5. none of them 6. II and III only 7. I and II only 8. II only Explanation: By the multivariable Chain Rule, d dt f ( r ( t )) = ( ∇ f )( r ( t )) · r ′ ( t ) . Thus the sign of d dt f ( r ( t )) will be the sign of the slope of the surface in the direction of the tangent to the curve r ( t ), and we have to know which way the curve is being traversed to know the direction the tangent points. In other words, if we think of the curve r ( t ) as defining a path on the graph of f , then we need to know the slope of the path as we travel around that path  are we going uphill, downhill, or on the level. That will depend on which way we are walking! From the contour map we see that I TRUE: at R we are on the level  we are following the contour. II FALSE: at Q we are ascending  the con tours are increasing in the counterclockwise direction. III TRUE: at P we are descending  the con tours are decreasing in the counterclockwise direction. keywords: contour map, contours, slope, curve on surface, tangent, Chain Rule, multi variable Chain Rule, 002 10.0 points Find the equation of the tangent plane to the surface 3 x 2 + 2 y 2 + 5 z 2 = 19 at the point (2 , − 1 , 1) . 1. 6 x − 2 y + 5 z = 15 2. 6 x + 2 y + 5 z = 19 3. 6 x + 2 y + 5 z = 15 hernandez (ah29758) – M408D Quest Homework 11 – pascaleff – (54550) 2 4. 3 x − 2 y + 5 z = 19 5. 6 x − 2 y + 5 z = 19 correct Explanation: Let F ( x ) = 3 x 2 + 2 y 2 + 5 z 2 . The equation to the tangent plane to the sur face at the point P (2 , − 1 , 1) is given by F x vextendsingle vextendsingle vextendsingle P ( x − 2) + F y vextendsingle vextendsingle vextendsingle P ( y + 1) + F z vextendsingle vextendsingle vextendsingle P ( z − 1) = 0 . Since F x = 6 x, F x vextendsingle vextendsingle vextendsingle P = 12 , F y = 4 y , F y vextendsingle vextendsingle vextendsingle P = − 4 , and F z = 10 z , F z vextendsingle vextendsingle vextendsingle P = 10 it follows that the equation of the tangent plane is 6 x − 2 y + 5 z = 19 . keywords: 003 10.0 points Find the directional derivative, f v , of the function f ( x, y ) = 5 + 3 x √ y at the point P (2 , 1) in the direction of the vector v = ( 3 , 4 ) ....
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This note was uploaded on 11/15/2011 for the course M 408 D taught by Professor Textbookanswers during the Spring '07 term at University of Texas.
 Spring '07
 TextbookAnswers

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