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2.1(9pts) Consider thc following contour plot for thc arbitrary function f(x,y): Y 0 2 Х -1...
10 6 2.4) 4 -2 -8 -10 -10 -8 -2 The contour plot above shows the level curves of some surface giv...
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10 6 2.4) 4 -2 -8 -10 -10 -8 -2 The contour plot above shows the level curves of some surface given by the function z-f(x, y). At (-8,2) the green vectors e and w are tangential to the contour there, and the blue vectors are perpendicular to the green ones. Moreover, at any point (in the frame above), we know that oflàx > 0 Which of the following vectors could represent Vf(-8, 2)...
17 marks] Consider the functionf(x, y) = (y - 1) (x- 1). (a) Find a unit normal vector to the contour line given by f=0...
17 marks] Consider the functionf(x, y) = (y - 1) (x- 1). (a) Find a unit normal vector to the contour line given by f=0.5 at the point (x,y) (1.5,2). Do not forget to check that this point is on the contour line. (b) Consider the curve L given by r(t) = 2ti+3tj with 0 ts1 and the Cartesian basis vectors i= (1,0) and j= (0,1) of R2. Determine using the chain rule and use this result to determine all...
Consider the surface given as a graph of the function g(x, y) = x∗y 2 ∗cos(y). The gradient of g ...
Consider the surface given as a graph of the function g(x, y) = x∗y 2 ∗cos(y). The gradient of g represents the direction in which g increases the fastest. Notice that this is the direction in the xy plane corresponding to the steepest slope up the surface, with magnitude equal to the slope in that direction. 1. At the point (2, π), find the gradient, and explain what it means. 2. Use it to construct a vector in the tangent...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
8 pts Question 3 Consider the function f(x,y, 2)(x 1)3(y2)3 ( 1)2(y2)2(z 3)2 (a) Compute the...
8 pts Question 3 Consider the function f(x,y, 2)(x 1)3(y2)3 ( 1)2(y2)2(z 3)2 (a) Compute the increment Af if (r,y, z) changes from (1,2,3 (b) Compute the differential df for the corresponding change in position. What does (2,3,4) to this say about the point (1, 2,3)? ( 13y2)3 ( 1)2(y 2)2(z 3)2 with C (c) Consider the contour C = a constant. Use implicit differentiation to compute dz/Ox. Your answer should be a function of z. (d) Find the unit...
PLEASE DO LETTER d.) PLEASE DO LETTER f.) The plane from e.) is 4(x-2)+6(y-1)+(z-1)=0 or 4x+6y+z=15...
PLEASE DO LETTER d.)
PLEASE DO LETTER f.)
The plane from e.) is 4(x-2)+6(y-1)+(z-1)=0 or 4x+6y+z=15
15. The temperature on an unevenly heated metal plate positioned in the first quadrant of the xy-plane is given by 25xy + 25 C(x, y) = 7 (x - 1)2 + (y - 1)2 +1° Assume that temperature is measured in degrees Celsius and that x and y are each measured in inches. (Note: At no point in the following questions should you expand...
Question 1. Consider these real-valued functions of two variables TVIn (r2y2) f (x, y)- 9(r,)2 2+...
Question 1. Consider these real-valued functions of two variables TVIn (r2y2) f (x, y)- 9(r,)2 2+4 (a) (i) What is the maximal domain, D, for the functions f and g? Write D in set notation (ii) What is the range of f and g? Is either function onto? iii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z0 0, 20 2, 204 (Note: Use set notation, and draw a single contour...
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10 6 2.4) 4 -2 -8 -10 -10 -8 -2 The contour plot above shows the level curves of some surface given by the function z-f(x, y). At (-8,2) the green vectors e and w are tangential to the contour there, and the blue vectors are perpendicular to the green ones. Moreover, at any point (in the frame above), we know that oflàx > 0 Which of the following vectors could represent Vf(-8, 2)...
17 marks] Consider the functionf(x, y) = (y - 1) (x- 1). (a) Find a unit normal vector to the contour line given by f=0.5 at the point (x,y) (1.5,2). Do not forget to check that this point is on the contour line. (b) Consider the curve L given by r(t) = 2ti+3tj with 0 ts1 and the Cartesian basis vectors i= (1,0) and j= (0,1) of R2. Determine using the chain rule and use this result to determine all...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
8 pts Question 3 Consider the function f(x,y, 2)(x 1)3(y2)3 ( 1)2(y2)2(z 3)2 (a) Compute the increment Af if (r,y, z) changes from (1,2,3 (b) Compute the differential df for the corresponding change in position. What does (2,3,4) to this say about the point (1, 2,3)? ( 13y2)3 ( 1)2(y 2)2(z 3)2 with C (c) Consider the contour C = a constant. Use implicit differentiation to compute dz/Ox. Your answer should be a function of z. (d) Find the unit...
PLEASE DO LETTER d.)
PLEASE DO LETTER f.)
The plane from e.) is 4(x-2)+6(y-1)+(z-1)=0 or 4x+6y+z=15
15. The temperature on an unevenly heated metal plate positioned in the first quadrant of the xy-plane is given by 25xy + 25 C(x, y) = 7 (x - 1)2 + (y - 1)2 +1° Assume that temperature is measured in degrees Celsius and that x and y are each measured in inches. (Note: At no point in the following questions should you expand...
Question 1. Consider these real-valued functions of two variables TVIn (r2y2) f (x, y)- 9(r,)2 2+4 (a) (i) What is the maximal domain, D, for the functions f and g? Write D in set notation (ii) What is the range of f and g? Is either function onto? iii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z0 0, 20 2, 204 (Note: Use set notation, and draw a single contour...
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