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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 pseries. (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 ptczst 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 23 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,
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10 Woulci \flplcl 2 LP, b,‘ ((1/1 V. ‘“ C? ((m HCfdl\(‘[L‘l \ 10. (10 points) Recall the solution to the ﬁrstorder 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.
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