# UT M 408C - HW 14.6-solutions-1 (6 pages)

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## HW 14.6-solutions-1

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- Pages:
- 6
- School:
- University of Texas at Austin
- Course:
- M 408c - Differential and Integral Calculus

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lewis scl876 HW 14 6 radin 55315 This print out should have 10 questions Multiple choice questions may continue on the next column or page find all choices before answering 001 1 0 1 2 3 10 0 points Q 3 2 Find the gradient of P 2 3 f x y xy 3x y 9x2 y y 2 2xy 3x3 2 f 3x3 2xy y 2 9x2 y 3 f y 2 9x2 y 3x3 2xy 4 f 2xy 3x3 y 2 9x2 y I positive at Q 5 f 2xy 3x3 9x2 y y 2 II positive at P 6 f 2 y 9x y 2xy 3x 0 R 1 f 2 1 Which of the following properties does the derivative d f r t dt have III zero at R 3 correct 1 I only Explanation Since 2 II only f x y f f x y 3 all of them 4 I and II only we see that 5 II and III only f y 2 9x2 y 2xy 3x3 6 none of them 7 III only 8 I and III only correct Explanation By the multi variable Chain Rule keywords 002 10 0 points The contour map given below for a function f shows also a path r t traversed counterclockwise as indicated d f r t f r t r t dt Thus the sign of d f r t dt lewis scl876 HW 14 6 radin 55315 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 Q we are ascending the contours are increasing in the counter clockwise direction II FALSE at P we are descending the contours are decreasing in the counter clockwise direction III TRUE at R we are on the level we are following the contour keywords contour map contours slope curve on surface tangent Chain Rule multivariable Chain Rule 003 10 0 points Find the directional derivative fv of f x y 2 y 1 2 x at P 1 1 in the direction of the vector P Q when Q 4 5 1 fv 1 correct 5 2 fv 1 10 3 fv 1 5 2 The directional derivative of f x y at P in the direction of v P Q

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