Math 51, Winter 2007 Final Exam March 19, 2007FINAL EXAM• This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted.• In order to receive full credit, please show all of your work and justify your answers. You do not needto simplify your answers unless specifically instructed to do so.• Please sign the following:“On my honor, I have neither given nor received any aid on this examina-tion. I have furthermore abided by all other aspects of the honor code withrespect to this examination.”Name:Signature:1 10 pts 9 8 pts2 8 pts 10 5 pts3 5 pts 11 5 pts4 7 pts 12 12 pts5 10 pts 13 10 pts6 17 pts 14 10 pts7 10 pts 15 10 pts8 15 pts Total 142 ptsCircle your TA’s nameLan Oren Josh Peter Chad Leo Rob Nikola JianPage 1 of 16Math 51, Winter 2007 Final Exam March 19, 2007(1) (10 points) Find bases of the null space and the column space of the matrixA =1 2 0 1 21 2 0 2 31 2 0 3 41 2 0 4 5.Page 2 of 16Math 51, Winter 2007 Final Exam March 19, 2007(2) (8 points) What condition(s) must b1, b2, b3and b4satisfy so that the following system has a solution?x − 3y = b13x + y = b2x + 7y = b32x + 4y = b4Page 3 of 16Math 51, Winter 2007 Final Exam March 19, 2007(3) (5 points) Let−→x ,−→y , and−→z be vectors in Rnwhose magnitudes are 1, 2, and 3 respectively. Suppose that−→x is parallel to (and in the same direction as)−→y , and−→x is perpendicular to−→z . Find the constant(s)c such that−→x +−→y +−→z and−→x + c−→y +−→z are perpendicular.(4) (7 points) A matrix A and its reduced row echelon form are shown below:A =1 ? 5 92 ? 6 103 ? 7 114 ? 8 13and rref(A) =1 0 0 10 1 0 10 0 1 10 0 0 0.What is the second column of A?Page 4 of 16Math 51, Winter 2007 Final Exam March 19, 2007(5) (10 points) A box containing pennies, nicke ls and dimes contains 13 coins altogether, with a total valueof 83 cents. How many coins of each type are in the box?Page 5 of 16Math 51, Winter 2007 Final Exam March 19, 2007(6) (17 points) LetV = span−2110,−1101, u =1111,−→v1=1110,−→v2=1021.(a) Show that−→v1and−→v2belong to the orthogonal complement V⊥of V .(b) Is {−→v1,−→v2} a basis of V⊥? Explain why or why not.Page 6 of 16Math 51, Winter 2007 Final Exam March 19, 2007(c) Find an orthonormal basis of V⊥.(d) Find the orthogonal projection of u on V.Page 7 of 16Math 51, Winter 2007 Final Exam March 19, 2007(7) (10 points) Let T : R3−→ R3be projection onto the plane P that passes through−→0 and is orthogonalto the line spanned by109.(a) Find an eigenbasis for T .(b) Write down a matrix in standard coordinates which represents T . You can express your matrix asa product of matrices and inverses of matrices.Page 8 of 16Math 51, Winter 2007 Final Exam March 19, 2007(8) (15 points) Globo-tech Marketing monitors the dollars spent each year by its customers on apples andoranges. With a(k) representing the number of dollars spent (in millions) on apples in year k, and o(k)the number of dollars spent (in millions) on oranges in year k, they determine thata(k + 1) =210a(k) +410o(k)o(k + 1) =810a(k) +610o(k)We shall write−→vk=a(k)o(k).(a) Find a matrix A so that A−→vk=−→vk+1. Notice that this will imply Ak−→v0=−→vk.(b) Find the eigenvalues of A, and for e ach eigenvalue find a basis for the corresponding eigenspace.Page 9 of 16Math 51, Winter 2007 Final Exam March 19, 2007(c) Express21as a linear combination of the eigenvectors you just computed.(d) Suppose that−→v0=21. Using your answers from above, what is a good estimate for thenumber of dollars (in millions) s pent on apples in year 100? What about dollars (in millions) spenton oranges in year 100?Page 10 of 16Math 51, Winter 2007 Final Exam March 19, 2007(9) (8 points) Show that if A is an n × n matrix then there exist scalars c0, · · · , cn—not all zero—so thatdet(c0In+ c1A + c2A2+ · · · + cnAn) = 0.(Hint: For a vector−→v , what can you say about linear dependence of the collection−→v , A−→v , · · · , An−→v ?Why might this help you?)Page 11 of 16Math 51, Winter 2007 Final Exam March 19, 2007(10) (5 points) Does there exist a constant c such thatf(x, y) =((x+y )2x2+y2if (x, y) 6= (0, 0)c if (x, y) = (0, 0)is continuous? Why or why not?(11) (5 points) Let S be the surface in R3defined byx2+y24− z2= 1.What is the tangent plane to this surface at the point (1, 2, 1)?Page 12 of 16Math 51, Winter 2007 Final Exam March 19, 2007(12) (12 points) Consider the function f (x, y) = x2/y4.(a) Carefully draw the level curve passing through the point (1, −1). On this graph, draw the gradientof the function f at (1, −1).(b) Compute the directional derivative of f at the point (1, −1) in the direction−→u ="4535#.(c) Suppose that f(x, y) gives the height of a mountain above (x, y), and suppose further that you arestuck on the mountain at position (1, −1, f(1, −1)). In what direction∆x∆yshould you takeyour first step if you want to descend the mountain as quickly as possible?Page 13 of 16Math 51, Winter 2007 Final Exam March 19, 2007(13) (10 points) Consider the functionf(x, y, z) =pln (e2xyz3)(a) Write down the first order Taylor polynomial centered at the point (2, 1, 1).(b) Find the approximate value of the numberpln(e4.01(.98).(1.03)3).Page 14 of 16Math 51, Winter 2007 Final Exam March 19, 2007(14) (10 points) Find all critical points of the function 2x3+ 6xy + 3y2and describe their nature.Page 15 of 16Math 51, Winter 2007 Final Exam March 19, 2007(15) (10 points) Use calculus to find the point on the circle (x − 1)2+ (y − 2)2= 1 which is nearest to theorigin.Page 16 of