M408D Quest Homework 11-solutions

M408D Quest Homework 11-solutions - hernandez (ah29758) –...

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Unformatted text preview: hernandez (ah29758) – M408D Quest Homework 11 – pascaleff – (54550) 1 This print-out should have 18 questions. Multiple-choice 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 3-3-2-1 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 multi-variable 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 counter-clockwise direction. III TRUE: at P we are descending - the con- tours are decreasing in the counter-clockwise 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.

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M408D Quest Homework 11-solutions - hernandez (ah29758) –...

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