024 100 points the contour map given below for a

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024 10.0 points The contour map given below for a function f shows also a path r ( t ) traversed counter- clockwise as indicated. 0 1 2 3 -3 -2 -1 0 Q P R Which of the following properties does the derivative d dt f ( r ( t )) have? III FALSE: at R we are on the level - we are following the contour. III positive at R . 1. I and II only 2. none of them 3. II and III only 4. I and III only 5. I only 6. all of them 7. II only correct 8. III 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 FALSE: at Q we are ascending - the con- tours are increasing in the counter-clockwise direction. II TRUE: at P we are descending - the con- tours are decreasing in the counter-clockwise direction.
moseley (cmm3869) – HW13 – Gilbert – (56380) 12 keywords: contour map, contours, slope, curve on surface, tangent, Chain Rule, multi- variable Chain Rule, 025 10.0 points Find the maximum slope on the graph of f ( x, y ) = 2 sin( xy ) at the point P (3 , 0). If the graph of z = f ( x, y ) at P = (1 , 2) has slope = 1 in the x -direction and slope = 2 in the y -direction, find the slope at P in the direction of the vector v = 3 i + 4 j . keywords: slope, gradient, trig function, max- imum slope 026 10.0 points
moseley (cmm3869) – HW13 – Gilbert – (56380) 13 When v = 3 i + 4 j , therefore, the graph of f has slope = 1 5 ( i 2 j ) · ( 3 i + 4 j ) = 1 at P in the direction of v .

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