This preview shows pages 1–12. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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,
., ‘3 {M09 W00 {6 W: M?
EA v70 ‘lciSiler/ ~{_~/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 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.
\/ \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%%*5@'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 ...
View
Full
Document
This note was uploaded on 06/30/2010 for the course CAL 100 taught by Professor Bill during the Spring '10 term at UC Riverside.
 Spring '10
 bill

Click to edit the document details