Problem 1 Use the Tables of Laplace transforms, along with the operational theorems, find the Laplace transforms of the following functions: (a) (b) (c) Find if (4+4+5=13 marks) Problem 2 (a) Find and...

Assignment, Semester 2, 2012 Instructions: Assessment criteria: Your project will be assessed on: your interpretation of the problem application of the appropriate concepts taught in your course the clarity of your explanations logical reasoning and valid conclusions in your solutions the presentation of the report the accuracy and relevance of your graphs Problem 1 Use the Tables of Laplace transforms, along with the operational theorems, find the Laplace transforms of the following functions: (a) (b) (c) Find if (4+4+5=13 marks) Problem 2 (a) Find and verify that. (b) Using Laplace transforms solve the following initial value problem: (5+5=10 marks) Problem 3 Obtain the Fourier series expansion of the periodic function of period defined below (8 marks) Problem 4 A periodic function is of period 8 defined by. Obtain the half-range cosine series for the function in this range. (7 marks) Problem 5 Find the Fourier sine transform of and hence find the Fourier cosine transform of t. (6 mark) Problem 6 (a) Show that is a conservative force field Find its scalar potential. Find the work done in moving an object in this field from to (b) Evaluatewhere, and S is the portion of the plane included in the first octant. (6+7=13 marks) Problem 7 Evaluate where and is the region bounded by the planes and (7 mark) Problem 8 Verify Green’s theorem for where C is the boundary in the first quadrant enclosed by the semi-circle and its diameter. (6 mark)