Answer the following questions. 1. Compute all first and second order partial derivatives of the function...

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Answer the following questions. 1. Compute all first and second order partial derivatives of the function f(x, y) = In(x + y). 2. A function f(x, y) is an eigenfunction of the Laplacian if there exists a scalar ER such that fax +fyy = f. Determine the value of A so that f(x, y) = sin(4x+5y) is an eigenfunction of the Laplacian. 3. Let f(x, y) = Q-(2,-3). e42-2, and let P = (-1,2). Determine the directional derivative of f(x, y) as one moves from P to 4. Suppose your professor is in a classroom and is standing at the point (1,2,0). A group of students, who shall remain nameless, begin to talk such that the loudness of the noise they create is given by the function 25 L(x, y, z) = measured in decibels. In what direction should the professor move in order for loudness of the students to decrease the fastest? 5. Determine an equation of the tangent plane to the function f(x, y) = tan(2x + 3y) at the point (0,0). Answer the following questions. 1. Compute all first and second order partial derivatives of the function f(x, y) = In(x + y). 2. A function f(x, y) is an eigenfunction of the Laplacian if there exists a scalar ER such that fax +fyy = f. Determine the value of A so that f(x, y) = sin(4x+5y) is an eigenfunction of the Laplacian. 3. Let f(x, y) = Q-(2,-3). e42-2, and let P = (-1,2). Determine the directional derivative of f(x, y) as one moves from P to 4. Suppose your professor is in a classroom and is standing at the point (1,2,0). A group of students, who shall remain nameless, begin to talk such that the loudness of the noise they create is given by the function 25 L(x, y, z) = measured in decibels. In what direction should the professor move in order for loudness of the students to decrease the fastest? 5. Determine an equation of the tangent plane to the function f(x, y) = tan(2x + 3y) at the point (0,0).