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Unformatted text preview: MACZ311 MGl Algebra/Trig Review Name: Sotu’l“ DV‘5
Fall 2011 Section: (circleyours) MWF 9:00 MWF 10:00 . Directions: Use all available resources, including the math lab, Tcague’s ofﬁce hours, and fellow
students. To justify answers and earn partial credit, show work in the spaces provided on this document. MCI is due at the start of class on Tuesday 8/30/11, and the ‘flate watch” starts at the
startofclass deadline. Papers submitted less than 24 hours late are penalized one letter grade (10%), and papers are not acceptable more than 24 hours late. E. An open box with square base x meters on each side must have volume 40 1213 (cubic meters). a) Find a function formula for S(x) , the singlevariable function that expresses
surface as a function of base dimension x. a”; Voiume V = X‘X'ln “Ni surface Ate“ = lPx‘L‘ Jr x'x 7.. qo=xzk (AzltK EfﬁeX h= L‘°/><‘ «(.0 x:
Answer: S(x)= T + b) Suppose material for the sides of this box costs $3 per square meter, and material for the bottom
costs $5 per square meter. Find a function formula‘for C (x) , the singlevariable function that expresses total cost of materials as a function of be e dimension x. f L if
To'l‘o.‘ Mo'l'eria‘ Cast" a: ‘3 3° Ctrea :5 5.4595 + ($. arcane :2 am use 5 “a: 31§Q+ 50‘?"
Answer: C(x)= x + X 2. Recall that the standard equation fora circle of radius r centered at (hit) is given by 7_ L
A 2 2 2 . l , 1+6... 0) == '3
(lﬁf’l) +(y—k) =7‘ . Full Cure e (X‘O')
xi + Y1. = C]
a) Write the equation for the full circle centered at the origin with radius 3. Y1 : Cf..)('z‘ 7— '2.—
Answer: X. +2 zcl / Y": Lt oi”X2'
b) Let y = s(x) = “09 —— x2 . Solve the eguation in part a} for y to Show that the graph of s is the non—positive semi—circle centered at the origin with radius 3. Sketch s and state its domain and
or ice mamiﬁ‘fev ‘ : Domain: [’3 3] of {xi ’3éxé’ag “.4 H.
Rangeimw l ‘3‘; 3 5 a} ‘ 'i ‘ a m AgntS(CLtO.lce+0 LL59. '
(1+; Mtge“ or seiuhu‘ilder no‘l‘od'io‘ﬂ 'unkhr oil no
i 0) Complete the formula for the piecewise—deﬁned function shown. The left and right pieces are linear and the middle piece is the semi—circle of radius 2 centered at the origin.
‘L. .__ 7— ' 3. Consider r(x) = xﬁ 3 . a) Rewrite r in radical form. Answer: r x. = ='" z 2. b) Each of the non—zero xvalues below produces a rational number output. Complete the table
without using technology, expressing function values in exact form. Sketch the graph.
Ijo_te; It will be virtually impossible to make a handdrawn sketch 1‘0 scale, so don’t try to do
this... just sketch the graph without establishing ﬁxed scales on either axis. 6) State the domain and range of 1'. domain: ix! x ¢ 0‘ I range: l, S i 9 > o} _ Q—oo) 0) U (oft0°) Coltoo) a) b) Conclusion: lim f = lim f ( x): C) d) x~iniarmfh onur‘ wiun Yto
__, 3x+t
x 0:3X'l’l x=~ Consider f = 3X
x 3x+1. $00 = + for xii). X.
Simplify the formula for f by carrying out the. division. Answer: f z 3 “i” As x tends toward +co and —oo , the function values tend toward a speciﬁc numerical value,
called a “limit”. Determine this limit by using technology to examine numerical tables and the
graph of f . The result should give information about an asymptote in the graph. 3 , so the graph has l1°YlEOniEal asymptote {'1 3 . ( horizontal or vertical ‘2 ) .T*>+D'J x—)—DD ( equation ) As x tends toward 0 from the left
(negative) side of 0, the function values
again get inﬁnitely large. numerically.
Determine whether the function values
tend toward +00 or woo . AS x tends toward 0 from the right
(positive) side of 0, the function values
get inﬁnitely large numerically.
Determine whether the function values
tend toward +00 or —oe. 00 Answer: iim f (x) 2 "1'50 x—> 0+ Answer: lim f = .14) 0" asymptote x: O . ( equation ) Conclusion: The graph has VQY‘Jfl all (horizontal or vertical ? ) Sketch the graph of f . If the graph is asy ptotic to a line other than a coordinate axis, sketch this asymptote using a dashed line, and 1a e he asymptote with its linear equation. Also label
the x—intercept. .L
5 f(x+h)—f(x) _ 5. Deﬁnition: For function f , the difference quotient at input x is given by I
2 a) Let f = 3 m x2 . Setup the difference quotient and simplify it completely. _Fgr ﬁg; ‘Rmc’i‘im 'F) £(Mk) "‘ PM = infﬁx
\r\ b) Let f = . Set up the difference quotient at x = 2 and Show that it simpliﬁes to — 2: h Forﬂ‘gmcifion 'F anci x27.)
3 8
4(Z+\\\”f(23 __ 2+1“ 1 k 7 in I 6. Without using technology, ﬁﬁd the exact value of each expression or write “undeﬁned”. Express  answers for pans 10—11) as exact radian measures. ‘ , .
 ‘ Ver‘l‘lt‘ao
J: “ﬁdeﬁﬂed ( as: m‘ﬁ‘o‘l‘e} “ I .  " . i. 37: ___I See Un‘c‘l c‘rrcle
cos[?]—  Z ‘ ‘ 1) 31116;}— or an)?“ C) .tan[27r]: 5}“ 2%) = :2 .— 3  j) 36%”): uhAegv‘eA. (Ver'l‘ltaD ) ﬂ
2 a) sin[2§£]: _ I h), tan[ l’drde. oil 2% —3_ cos (1%) — i 7 “awhjte
a = 3 mm “72. * 1; or, a]
q; . :—
V‘ 1) cos'1 —1 = 2"— \o mu 211  .ml—
e) ese[2—ﬂ)= *; =3 "L": 2 .3 56’ cos 2“.
3 , 6142"?!) 5/2. [3' 3 . _ and 3315 'm [out] fl eot Ezr— =' (05673) I {9; m) tan—3(1): E laemuse +&“% 3 land
3 '— I ' 11. TI" .' ' '1: _ E 15 \v‘ E /2 , 2_ 2. u 243: t 2
g) COS(7r)= ....I . a I 11) sec 1[__~3 )2 Eil lnzcause. secigs — TE
' ' ‘ . I h .‘ 11— See (mll cw: le or’ jraeln (meg is m [OJ1T1’av‘d‘t'é ,7:
7. a) Sketch at least two periods of the ' b) Sketch the function y = 23in‘E x. Label the funCtion y = 008616) ' Label enough “Ck ' endpoints with exact x~ and ycoordinates, marks“ eaCh ads to indicate their _ and label enough tick marks on each axis to r Scales" . _ “211 111” indicate their scales.
(,0; (Ex) has Period — *. Here) 3— _ 8. Solve each equation, ﬁrst over [0,271) , and then over (—00,+00) . Express answers in exact radian
measure form. When solving over [0, 27:?) you will, of course, get a ﬁnite number of solutions. The solutions over (—0o,+00) include all solutions over [0,27r) and all angles coterrninal with these, so ‘F
n *7 this will be an inﬁnite set. / “5e dqulale omjle Ede
a) Zoom—1:0 ' b) sin(2x):cosx I
ZCOSX :l 25““x cosx :: (05"
055K : é , Passride "In QI, 0N 25inxcosXCOSX 4": g
x = If 5.71 omcl all (ole'fminal M‘le C°5K (zsw‘yﬂ i)” [5 o
3 , S , ' ' ,0 25'“X”
. _ cosx '— _ $— __ y
_ I $11: ‘hxi‘: 7. ﬂ
X‘= > ..
2 7— X ’ 6 ) 5:
Answers: .‘TT . 11 11— 5". 3h
Over [0,27r): /3  ’ 3 Over [0,275): "a; J 3.— , 7: ’ E
If 5"” L“ 5.1: :1:
Over (—co,+o:>): + in“) T + awni Over (—00,+00): to + 2h“) (a + 2W") 2. + NF
h anq tutejer‘ ‘ I h any .mtejer . 9. Consider f (x) : xefx. a) Simplify the function formula to express f without the negative exponent, and state the domain. ‘ ' ‘ K
Answer: f(x)= .25; domain: (~00JHo) he cause 6 35 0 list”. a“ x '
e .
b) As it tends toward +00 , the function y m e" grows much faster than y = x . Consequently, the
function values for f tend toward 0 asxtends toward +00. That is, liin f (x) = O .‘Cﬁ‘Hﬂ 10. Consider f (x) mi.
1 v
a) Statethedomariri Answer: W U (“+09 . b) As x tends toward +00 , the function y = x grows much faster than y 2 In x . Consequently, the
function values for f tend toward i (D as x tends toward +00. That is, lim f = 'l' 90 xa+0o
I c) The function f'(x):lnx”1 (In that satisfy f’(x) m 0. Show the work. is the derivative function for f . Find the exact values of x, if any, (\“KY— ' [\nK 7" i j: Answer: x= I e
eltﬁx _____ I
X =6 ...
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 Spring '11
 Teague
 Calculus

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