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# test2 - Math 107 Linear Algebra and Differential Equations...

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Unformatted text preview: Math 107: Linear Algebra and Differential Equations Test #2 Name: Z I HS W( V“ WE] Thursday, November 19, 2009 Lecture section: 107.0 Recitation section: 107R.0 All answers must be justiﬁed. No calcuiator is allowed. 4—16 A=216~ 72-~18 An eigenvalue of A is 2. Find a basis for the corresponding eigenspace. Question 1. [5 points] Let ._ .. 246 46 FfoMA21~ Q45 Ngoo meé 000 Y2 “'3 \$5}:le +X:30 o l W]? 6 f Question 2. [15 points] The night before Thanksgiving, I realized to my horror that I had forgotten to defrost my turkey. So I hurried up and took the frozen turkey from the freezer at —5°C into a room with temperature 20°C. After 30 minutes, the temperature of the turkey is —2°C. (a) Describe a model for this problem, based on Newton’s law of cooling, which states that the rate of change of temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e., the temperature of its surroundings). Include a differential equation, and one or more conditions on the solution. State clearly the unit of each variable. 0110:. TJ— '1’“?M °C 414: H: (l 2% k “w Mun/Jo imﬁaﬂaudi'ﬁon‘ Tabs-J5 oar“ wndﬁ‘r‘an' T636) 9 “2 10 I "5:770)=Q+20 \$C'3—25 t 0=TCt)--— 456k +20 \$640425) Question 3. [20 points] Find an orthonormal basis for the plane U with equation :1: + y + z = O. Hint: You might want to ﬁrst come up with a basis for U , orthonormal or not. And then, starting with that basis, use the technique described in class to construct an orthonormal basis. i v ~ “* -~ A?“ g -— ~ 0 “up” VJ- " b Question 4. [20 points] Consider 4 -—3 A - [2 —1 i (a) Find all the eigenvalues and the corresponding eigenvectors of A. (16+ (A ~11): MP; 3;] a *(‘L—AXN-D-(o =0 (b) A is_similar to a diagonal matrix D, i.e., Find P and D. Question 4. (Continued) (6) Compute A6 Without actually multiplying A with itself. Question 5. [20 points] (3.) Find the homogeneous solution to 652?; dy (b) Find the general solution to 3%(53) + Big—(cc) — 24y(:c) = ~~5e2‘” — 6 sin(42:) 2X g H = A X 6 Plug?“ '- ~2LQ'A'X‘CZX :[BAQZX 3) #1:”;5? , ‘3‘”? “(3‘9 (6+2 = gumm— czcoscux) +éc4qCOS4X ”4‘an‘0‘ -—Z‘(' (CISTAQX +C2C°S {K -_—: 726‘ ~24 (>914le +g4q—7Z‘Q 5" (F’gO’ 62:25:21 W27133W4K+4lo l6: gH—Fgﬁd-gfz 5e”: 3(4Aezy-4— 4Axew) + 6; (A6 4- 2A9 ) a??? Question 6. [15 points] Consider the mass-spring system dy2 dy W _ k; = mdt2+wdt+ y f (8.) Suppose m = 2 kg, w = 4 kg/s and k = 1.5 kg/sz. Is this system overdamped, underdamped; or critically damped? but” 4M k =2 (6 -—(2 —"' qW? abet-damped (b) If m = 2 kg and f = 20 m/s2, ﬁnd the general Solution of the mass—spring system. 2'3” +443, 4- 3-3 =20 .._. int (6H: ((9 g+Cze 36: __ ‘m 1:9 C:- 490/3 gpc Wag ,i "if 6 (gm-:qe +96 1+3— ReSOMILcLQ I}: '5: ...
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test2 - Math 107 Linear Algebra and Differential Equations...

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