Transcription of Final exam, Math 240: Calculus III
1 Final exam, math 240: Calculus IIIA pril 29, 2005No books, calculators or papers may be used, other thana hand-written note card at most 5 7 in this web version, answers are at the end of the examination consists of eight (8) long-answer questions and four (4) multiple - choice problem is worth ten points. Partial credits will be given only for long-answer questions,when a substantial part of a problem has been worked out. Merely displaying some formulas is notsufficient ground for receiving partial credits. Your name, printed: Your Penn ID (last 4 of the middle 8 digits): Your signature: Your lecture section (circle one):ChaiCaldararu123456789-12 TotalPart I.
2 Long-answer Compute det(A3), whereAis the matrixA= 1 2 31 4 91 8 27 .2. LetCbe the oriented curveC= (x, y) : 4x2+ 9y2= 36, x 0, y 0 from (3,0) to (0,2). Compute the line integralZC(x+ 1)dy+y dx .3. LetDbe the cubeD= (x, y, z) R3 0 x, y, z 1 ,and letS= Dbe the boundary surface ofD, oriented by the unit normal vector field~nonSpointing away fromD. Compute the oriented surface integralZZS(x2~i+xyz~j+z3~k) ~n dS .4. LetSbe the surfaceS= (x, y, z) R3 x2+y2+z2= 1, z 0 ,the upper half of the unit sphere centered at the origin, oriented by the unit normal vectorfield~n=x~i+y~j+z~konS. Compute the surface integralZ ZS(x~i y~j+z~k) ~n dS.
3 5. LetCbe the boundary of the rectangle with vertices (3,2), ( 5,2), ( 5, 7) and (3, 7),oriented counter-clockwise. Compute the line integralICy dx x dyx2+ Lety(t) be a function which satisfies the differential equationy (t) + (1 +t)y (t) + (1 +t+t2)y(t) = 0y(0) = 1,y (0) = 0. Determine the values ofy (0) andy (0).7. Suppose that a vector-valued function~x(t) satisfies the following system of ordinarydifferential equations~x (t) A ~x(t) =~0,whereA= 1 20 1 ,and~x(0) = 12 . Determine the function~x(t) Find one ( particular ) solution of the system of differential equations dxdt+y=tdydt 2x= other words, find a pair of real-valued functions (x(t), y(t)) satisfying the above systemof equations.
4 (There are many such solutions.)[Hint: Try to replace the above system by a single differential equation, then try to find aparticular solution of that equation.]Part II begins on the next page2 Part II. multiple choice Consider the following matricesA1= 0 11 0 , A2= 1 11 3 , A3= 1 3 40 1 50 0 2 , A4= 2 1 11 2 11 1 2 Which ones can be diagonalized over the real numbers? (In other words, there exists aninvertible matrixPwith coefficients in real numbers such thatP 1 Ai Pis a diagonalmatrix.) , , ,A2,A3andA410. LetAbe a symmetric 4 4 matrix with real entries. Consider the following have four distinct There exists an invertible matrixCwith real entries such thatC A C 1is a The four roots of the characteristic polynomial ofAare all real a symmetric ones among the above statements are true?
5 A. I, II, III II, III, IV I, III, IV III and IV II and III II and IV I, II, III, IV are all None of the Suppose that a functionx(t) satisfies the differential equationt2d2xdt 2tdxdt+ 2x(t) = 0,andx(1) = 3,dxdt(1) = 1. What is the value ofx(2)?A. 0B. 1C. 3D. 4E. 2F. 1G. 5H. None of the Suppose that a functiony(t) satisfies the differential equationy (t) + 2y (t) +y(t) =e 2t,andy(0) =y (0) = 0. What is the value of the Laplace transformL{y(t)}(s) ofy(t) ats= 1? None of the :1. 17282. 35. 2 (0) = 1, y (0) = 0 7.~x(t) = et+ 2tet2et 32t 4,y=t+32[Take the derivative of the first equation; then use the second].
6 9. OnlyA3andA4can be diagonalized Only II, III, and IV are (t) = 5t 2t2, x(2) =
