16C08finalsol

# 16C08finalsol - Math 160 Final Exam Name"x S a l M l...

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Unformatted text preview: Math 160 Final Exam 6/9/08 Name: "x. S a l M l 0A5 Signature: Student ID: 0 There are ten (plus cover and bonus) pages to the exam. 0 The exam totals 100 points, plus 10 bonus points. 0 You will have 50 minutes to complete the exam. 0 N0 calculators, notes, or books allowed. 0 Good luck! Your Score 1 :Total E1: 1. Deﬁnitions and Examples: a. (2 points) Write the standard form of the ﬁrst order linear ODE. (an example is not sufﬁcient for full credit) V4Mnyrqw) b. (2 points) Write the deﬁnition of a Geometric series. (an example is not sufﬁcient for full credit) 00 fl gar n50 c. (2 points) Write the deﬁnition of a p-series. (an example is not sufﬁcient for full credit) 90 ! Zr fljl d. (2 points) Write the deﬁnition of a power series centered at c. (an example is not sufﬁcient for full credit) e. (2 points) Let f = f (at). Write the deﬁnition of the Taylor series for f centered at c. (an example is not sufﬁcient for full credit) ‘90 Mk) ' 'n i0<:'J‘Wﬁﬂ{*{) , 0 2. Short Answers a. (5 points) State the ratio test for inﬁnite series. V1900 an 2? W cm ‘ ' A900] n . (5 points) State the p-tczst for inﬁnite series. F>ll Thom 3. Consider the ODE, y’ = 2ty — 2t. a. (5 points) Solve the ODE. b. (5 points) Calculate the solution for the corresponding I VP with initial condition y(1) = 1 + e and its limit as t ——> 00. M: l+€ -:>1+e : W90 :><:/ 1 / A} ~> yup/+97! 4. (10 points) Find and classify all critical points for the function 5. (10 points) Classify and graph the surface given by I ‘ x \ ‘30 f [2" x2+(y—l)2+z=1. Q (7 Z:l~:'[ 41 ‘ H) mph/‘46 elm/I4 WM 6. Determine whether the following sequences converge or diverge. If a sequence converges, ﬁnd its limit. a. (5 points) an = (3n_1)2. )\ ‘ m “x v h M : u N719” " h’70‘7 qhX 4, .- 0/ ,7 J, / :' C( A b. (5 points) an 2 2 + (—1)". W“ 1+ Mm PNE {(S'(/\)""%) M700 / 7. Determine whether the following series converge or diverge. If a series is geometric and converges, ﬁnd its limit. \ Sp / / 05/ a. (5 pOints) 22:1 43% r J F; 7/3 > I DNWypS MST) b. (5 points) Zf=15"—% Px 39(585/ P "K; > / Vp/jg‘ ) 9. (10 points) Of the following two series, one converges and one di— verges. Determine which is which, and explain your reasoning in 2-3 sen- tences . 232021 4 n+5 PAWS 22:1 53% bUQEA gag L1 6? A r u \ / LA ‘ A4”? ~— I K (/1 VWC/‘(J 1L0 Z6 (0/ )9 M i 0,, Clap; 9a m. um ﬁlo/916W, k0hc€/ 267,) (Cm/(0x948 will? ﬁlm, dim 03, ., ‘3 {M09 W00 {6 W: M? EA v70 ‘lciSil-er/ ~{_~/q pm ﬁe (CM/- 10 Woulci \flplcl 2 LP, b,‘ ((1/1 V. ‘“ C? ((m HCfdl\(‘[L‘l \ 10. (10 points) Recall the solution to the ﬁrst-order linear ODE via integrating factors, Prove this result. (Hint: start with an ” educated guess” ll Bonus. (10 points) Mixing tank A initially contains 100 liters of pure water. Mixing tank B initially contains 200 liters of pure water. Salt water w/ﬂ with concentration 2% enters tank A at a rate of d. Mixed solution _ travels from tank A to tank B at the same rate. Mixed solution leaves tank 3 ll 5 B at a rate of d. Write and solve I VPs describing the amount of salt in the tanks as a function of time. \/ \j‘ \lOluMP { l l 00 ((OASlC‘Wl > £01 ) VA: W05 :30“ a fme (g) ﬂied Mel : m0) "3 O n : q/fygaifé : [0/5 M :7 Ugh): «t ; VlO'Vltalw dX/ifé%%*[email protected]'é WEE/cit W M” M , at m i . J I : 00(1) Xﬁ 4’7; X4 5 Wm} + ago J? 3(6) 3 / ﬂy (35%“) :79“? :aCloQ +. M : [0100(1) ,1; W .,'t/0l’ _ I v. v J u :7 XA‘QCOdew / l4 0va —7 5M”1L 536ml” [m ( 1 ...
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16C08finalsol - Math 160 Final Exam Name"x S a l M l...

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