Consider the function f : R R given by f(x,y) = 1 4x + y...

     

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Consider the function f : R² → R given by f(x,y) = 1 √4x² + y² + 1 (a) Draw a contour map for the function f(x, y). The contour map should be clearly labelled and should help to determine the nature of the graph of the function. (2 marks) (b) The contour corresponding to f(x,y) = defines a curve in R². Use implicit differentiation to determine at the point (1,2) on the curve. Briefly describe the geometric meaning of the number that you have just calculated, with reference to your contour map from part (a). (2 marks) dy dx (c) What is the rate of change of f(x,y) at (1,2) in the direction (3,4)? Describe how you can predict the sign of this rate of change without doing any calculation. (2 marks) (d) In what direction is the rate of change of f(x,y) at (1, 2) the most negative? Express your answer as a unit vector. (2 marks) (e) Determine all points (a, b) that satisfy both of the conditions: ▪ f(a, b) = 0.2, and ▪ at (a, b), the function f(x,y) has the most positive rate of change in the direction (2,1). (2 marks) Consider the function f : R² → R given by f(x,y) = 1 √4x² + y² + 1 (a) Draw a contour map for the function f(x, y). The contour map should be clearly labelled and should help to determine the nature of the graph of the function. (2 marks) (b) The contour corresponding to f(x,y) = defines a curve in R². Use implicit differentiation to determine at the point (1,2) on the curve. Briefly describe the geometric meaning of the number that you have just calculated, with reference to your contour map from part (a). (2 marks) dy dx (c) What is the rate of change of f(x,y) at (1,2) in the direction (3,4)? Describe how you can predict the sign of this rate of change without doing any calculation. (2 marks) (d) In what direction is the rate of change of f(x,y) at (1, 2) the most negative? Express your answer as a unit vector. (2 marks) (e) Determine all points (a, b) that satisfy both of the conditions: ▪ f(a, b) = 0.2, and ▪ at (a, b), the function f(x,y) has the most positive rate of change in the direction (2,1). (2 marks)