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Unformatted text preview: BALL 95:49 MATH 3 EARLY FINAL 1217/2004 Instructor: Frank Béiuerle, Ph.D. No books, graphing utilities or notes allowed. Show your
work. Show your work. Show your work. Your Name: *0 Your TA:
Max Your score
Problem 1: 10
Problem 2: 10
Problem 3: 15
Problem 4: 10
Problem 5: 10
Problem 6: 10
Problem 7: 10
Problem 8: 20
Problem 9 10 Problem 10: 20
Problem 11: 15
Problem 12: 20
Problem 13: 15
Problem 14: 10
Problem 15: 10
Extra Cr 16: 10 TOTAL: 7 205 Relax, good luck, have a great winter break. 1. (10 points) Use the reference angle principle to get the exact value of the following: (a)sin225° : _ gang—5Q h I E ’l‘ T 2, (371T “flew
owshl 2. (10 points) Give a table that gives the exact values AND decimal approximations, rounded
to the nearest hundredth, for sin a,cosa and tana for cr = 0°,cr = 30°, or = 45°,a = 60°
and a: = 90°. Also give the radian measures of all the angles. 33‘” lam M lo ya +9 mle 3. (15 points) Consider the function ﬁx) 2 2 cos[2;z: + Find the following: 1 (b) (2 points) The period of f. (7.2
__ g: h
,7. (a) (1 points) The amplitude of f. (C) (2 points) The phase shift of f .
‘ _._L f
€13: : :ﬂ—iqr (d) (6 points) The graph of f. . l 
(e) (4 polnts) Now graph g(:c) = 2sec(21: + into the same coordinate system above. 4. (10 points) Give the exact values of the following by using (and drawing 1) the appropriate triangles:
. _1_§ h
(a)cos[§1n ( a C09 9 :: + 8
s3 ’5
sine:4 9%? 0059 > O
5 ~ nh—r
«a 621 (we
Mai—e g}: 2 we
5
3
Lf.
(b) sin[arcta.n( : R Q l __L : E
:9 {57 ‘5
W " ,. “
tau9:1 (FOQIWQ 51MB 6’ iS' in M
‘H i , . _
1.4%ch FIKJ' quaclt’mﬂ
6:1
. Hg 7_
I
3 5. (10 points) Graph = arctan(3) + 1. Make sure to label all asymptotes. (Extra Credit, 2 points) Use your graph to graphically ﬁnd/ estimate the solution(s) of the
~ equation arctan(x) + 1 = 1. What are the exact values of all solutions ? X=0 6. (10 points) Find all solutions for the following equations:  1
(a) tans: — 5 x: owclomli I73 {ll/WW3 “Mild” X“: Qf frigidJE— + ‘ i‘lrt 1T cunt (Ali New}. (b) sin2$+sinx —6 = 0 (Shaw—3 )C Sim: —— 2 3 3 O Sin/xx“: 3 NO Yolwi‘lm
Slay: 2 ND folmbm_ 7. (10 points) A vertical ﬂag pole is on the ﬂat roof of a. building. From a point on the ground
50 feet away from the building, the angles of elevation to the top and the bottom of the ﬂag
are 15° and 12° respectively. Find the height of the ﬂag. ti goal: moi x mum = h :7 lthUtﬂulZC‘
BO 1‘ gO‘tcmlSO :: gOEamfzoi—X ‘ x: 90 (tango reams?) x 9,77 13+ 8. (20 points) Consider the function f(3:) = ln($ + 1) (a) (5 points) Show that the function ﬁns) is onetoone. (You can do this algebraically or graphically)
JOMSFS HLT’ ‘ [niil‘d] :: luCLJH)
('r L _
5‘0 [H _6y OCH “C bfi
(M*‘iD“W '
a = b (‘0 it!) M"?[‘3“'N« (b) (5 points) Find f‘1($). v= WW I > ran. em 7[ \f: €‘i (c) (10 points) Graph both f and f‘1($) in the same coordinate system. 9. (10 points) 200kg of a radioactive substance with a halflife 01‘20 years is held in a storage
container. How long must one wait until only 2 kg of the substance is left '? Hint: First use the halflife T to compute k for which you can use the following half—life formula: k = r—l—IlT—Q.
l4 _ .. M1
"LU _lu')—
211206 e W‘E
11
( ———‘lr
a: :— 6 1‘“
_.L ._ __lvxl
lv‘mo *— fg—j {' 10. (20 points) Give all real solutions to the following equations. W : 31
x = CH
(1 3
(b) 21112::I+ln(a:+1) 'L sq e
‘2 :2 +
ZIMX— In CHI) =1 / 7: e X
lnx1m(\<+!)=~i x «we: 0
I .
I“ i :1 heiweﬁm
1'“ Z
_ l
o l 2
(C) 623 + 263:]. = 0 Hint: Rewrite 62$ appropriately to factor as a. trinon‘lial éLM:MiLNW
(6: l) : O MAX If . Unfit/{wad x Q = 4 [No Column
0 (d) 111 111111 :2: = Inw:e°=
Imﬁ = e
[fife—5) 11. (ﬁpoints) a) Two numbers add up to 33. Find the maximal value for the product of these two numbers. [2+ my be Mime Lama;  w : 33 is m meme
{PM} : X3 3 M DEW“ ﬂ Wot/xiwu'éw
' > y z 33 ~,<
z ,
, ‘99“; XC3%~\,<) :¥ +35%
WE Ll Cl jocwaxbom 5pm downward with max at m [HM M K“ CoowUuqu at Lo 3'3 ,_ thfxp 6; [m fa'ﬂoULof" L3
b) Two non—negative integers add up to 33. Find the mainmal an the minimal value for
the product of these two numbers. ‘VIAOUY VVchz M mm;qu mm c? fie [amount afﬁuo
YLOIA— M94506 [4wa [I In} Wot/(If Mk—MgaAb
Maw) Kamtﬂ O, r} attcdwcé’ {:0 a Y; 5% gr XlgbﬁgHYT—O 12. (20 points) Let 6’ be an angle such that 7r S 6 g 4 3 Find the following: 37F and sin0: —— (a) (2 points) tanél 2 ill
3)
(b) (2 points) cost? =  E '
5
(c) (2 points) cot 6' = . L‘.
(d) (2 points) secG = __. 3)
 5
(e) (2 omts) (15:36: — .._
p Ll
(f) (3 points) (303(29): 8333 (3» —§i]‘116 2: j — Lt? : _. 2
25 15 25 {9
(g) (3 points) Is cos(§) positive or negative? Explain. ﬁre—ear fll‘éi u’.—.. 2. (h) (4 points) 6‘ =
calculator and your brain here) é: Slhﬁwg) K3“ U3,
Wu ...._H h)
(N 10 Z
I 3‘4 — .'_._ ear w; in degrees and radians (you should use your '9 l5 Nﬂallﬁ m ., ~l q N V
‘6: U + 5"“ ("flv LH97
G’: {800+ 914%) ,4; ‘5 $2»3:c—4 13. (33 points) For the rational function = $2 + 3x _ 4 do the following: (c) Find all ve tic symtotes (if any). "
1Q (MW AQLLOWIWM) ((1) Find t e horizontal asymptote. 1 _
i¥=ll 91w {a} 9:35 :4 {iv (mg/c X (6) Where oes the graph intersect the horizontal agkyrznptote ? 1
i: 7L1+3¥MLL3 )L th‘ Lf
XL +3 V“ Lt 6 @ 14. (10 points) Let a = 7,05 = 40° and b = 9. Find one solution of this triangle. E‘Q' RNA F51 )K‘Ifl
MR ((1016; gm: to
{Eur/l h S’Mé‘
b q Sikﬁ‘z 2:1 si‘mtroo a) f t : Simqu mm 5:; SUD , git: 180° LfoJ’3, :3: 8+5" 8370
USe om §;b{10rcggf‘mf 12> “k” _ ; £1 a i
n. aﬂ‘7‘ gfngf‘t Cl giMaﬁ—wSﬂﬂK
('— 3"th 0° "(‘3’ (0‘8 (Extra. Credit, 5 points) Find t e other solution. 15. (10 points) a) Find an equation of the sine function with the following graph: h AWKMBNCEQ :1
0M mm} .7 UM}, gm {1 :— I
13 LL MD which evict}: gull; : 37:"
=— ..L 2
t z) B ' ‘3—
IT
1
:)C:§'f6
:3 «3 :1“
2 3 5 b) Give a reason why the following Wgraphs. cannot represent a polynomial function with
highest degree term —:r:5.
fund/[UH Midi! 50 mecca cu. “5—7 00 13 16. (Extra. Credit, 10 points) Let a. sequence be deﬁned by (1714.1 :ﬂnl‘230q =1,n=1,2,3,... (a) Find the next three elements (1;, a3, a, of this sequence. 01300 also 3' gem (c) Find an explicit formula for an. CKHZZH‘l (d) What is the value of the sum of the ﬁrst 71 elements of the sequence, namely What is thesum ﬂ +l+ ‘
3=W...+n : Mum
k=1 2‘ 14 ...
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