1. Let A and B be 3 X 3 real matrices that commute: AB 2 BA. If .35, is a real eigenvalue of A. let V}. be the real subspace of ail eigenvectors having this eigenvalue. {a} Show that if V). is one-dimensional, then ever}.r {nonzero} vector in IE LG, is also an eigenvector of B. possibly.r with a different eigenvalue. {b} ISiva an example showing that if dilIlV)l :n- 1, then some vectors in V). may not be eigenveittors of B. {c} If all the eigenvalues of A are real and distinct {so each has algebraic multiplicity one}, show that there is a basis in which both A and B are diagonal. 2. Consider the Function at} = 1-1. [a] Find lim at\". irlIH' {h} Find e). 3. An ant walks along the surface a = .172 1:2 + 43,; + 14 so that the distance From the ant to the zaxia remains constant at 2 units. (a) 1iWhat is the highest and lowest elevation obtained by the ant during its wallt':rt {b} Suppose the ant- modulates its speed so that it circles the aaxis at a constant rate of 2% radians per hour. 1What is the longest continuous interval of time during which the ants eievatiou is increasing? 4. Find out whether the following series converge absolutely1 converge oondition- all]!1 or diverge. In each case? expfain your reasoning and give the name of the tEst or method you used. 5. Consider the following pharmacohineties mode}: FL drug is ingested and enters the gas trointestinal tract [G1]. lit-em there, it is absorbed into the blood stream {E} The drug is distributed in the i'_lor_l;,.r and slowiy removed from the blood stream by metabolize tian. We assume that absorption of the drug from the GI ta him-d happens at at rate proportional to the amount in the GI with preport-ionality constant (1' = [LTD hour1. Metabolizing of the drug From the blood happens at a rate preportional to the amount in blood 1with proportionaiit'}.r oonstant ,5 = [1.11 hour1. Drug is ingested at. the constant rate ofimgfhonr for 15 minutes [0.25 hero's} starting at t = I]. Let t} denote the amount of drug in the G1 [in mg} after t hours and Mt] the amount of drug in the blood. [a] Denote by Hi} the drug intake: Ht} _ clmgfhour? for I] gt g [1.25 _ a for t s {1.25 6. Consider the sphere a? + y' + (2 -3)? = 1. Find the point P = (a, b, c) on the sphere such that the tangent plane to the sphere at P intersects the ry plane in the line y = 3r. 7. Let V be a vector space and (: V - R be a linear map. If = E V is not in the nullspace of f, show that every r E V can be decomposed uniquely as r = v + cz , where v E V is in the nullspace of / and c E R is a scalar. 8. Consider the three functions a(x), b(x), and c(x) given by: 3+1 T-3n+2 a(x) = > (3n)! b(x) = > (3n + 1)! c(x) = (3n + 2)! 2=0 2=0 2= 0 (a) Determine the radius of convergence for each of function. (b) Give a formula for the oth derivative (")(r) for all ne N\f