Phys 3750 Midterm Exam 2 Name____________________ 1 **You may use a pen/pencil and a 3"×5", handwritten note card on the exam. Answer all questions as completely as possible; show all of your work. If you run out of space on a problem, continue on the back of that sheet of paper. Good Luck!** Q1. (5 points each) Short and Simple. (a) Is the function 2xxee−+ even or odd? Why? (b) What is the Gibbs phenomenon? (c) Calculate ()xnxxsinlim0→. (d) Consider the two vectors −=ii21a and =1221ib that live in a four dimensional complex vector space. Calculate the inner product ()ba,. (e) Calculate the integral ()()∫∞∞−−++ dxxxx 1232δ.Phys 3750 Midterm Exam 2 Name____________________ 2 Q2. (5 points) Starting with () ()() ()′′+−++=∫+−ctxctxxdxbcctxactxatxq121, , show that ()xa is the initial displacement. Q3. (10 points) Consider a vector space of functions defined on the interval Lx ≤≤0 . Find the norm of the function ()2xxf = .Phys 3750 Midterm Exam 2 Name____________________ 3 Q4. The function ()()202,actxAetxq−−= is a solution to the wave equation. (a) (10 points) Describe in words, as completely and precisely as possible, the nature of this solution. (b) (10 points) Find the initial conditions ()xa and ()xb that give rise to this solution.Phys 3750 Midterm Exam 2 Name____________________ 4 Q5. Consider the 1D wave equation on the interval Lx≤≤0 with boundary conditions () ()0,, == tLqtxq . The general solution can be written as ()()() ()[]∑∞=+=1sincossin,nnnLntdtbxtxqωωπ. (a) (5 points) What is the relationship between ω and Lnπ? (b) (15 points) Starting with the equation above, find an expression for nd in term of the initial conditions ()xa and ()xb . Carefully show your work.Phys 3750 Midterm Exam 2 Name____________________ 5 Q6. the function ()<≥=−000xxexfAx. (a) (15 points) Calculate the Fourier transform ()kh of ()xf . Express your answer for ()kh in the form () ()kihkhIR+ , where ()khR and ()khI are both real functions. (b) (5 points) Find