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Unformatted text preview: MACllGS Lecture Worksheet 4 i. 50 ( ujilﬁﬂ § Sections 4.1, 4.2, 4.4, 4.64.8 Fall, 2011 Decide whether each function is a polynomial. If so, state its degree and leading coefﬁcient, a. 31) 3":4YS—V +6 Poly} cla35'3 [1:53. 6) y=3x4 Poly; okajtlsf): Q23
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b) y 3 217 r Palmdeﬂz.’ QT”. f) y t+7 (Fab, dﬁl ) 6t. Z.
c) y=3f4 moi—oc‘aoly g) y
d) y=(2x—1)2:(2xrl(zx~—I) h
=qu‘ahixni—l ) y
{>947} 2.1mm: a) Asx—>+w,f(x)—9 “‘90
b) Asx—>—oo,f(x)—> 4‘00 c) f is an a) degree polynomial, and its leading coefﬁcient is % 06%;! V6; . d) The local minimum value is "' ’ 0/3 , and . . 12*
the local max1mum value 18 Z? . Time, e) True 01' False? Function f has no global (or absolute) extrema. ~00 t) Over what xaXis interval(s) is f deereasing? 5 “2'3 U [2) +99 True or False? ¢+M057Ln
a) Ea, ng Every 4thdegree polynomial has} 3 local extreme points. (Local extrema are sometimes called “turning points”.) b) 17f“ a“ Every 3rddegree polynomial has at least one zero (Jr—intercept).
Egan/x. Guitava (g, olel A03 r6161. {gab ave _\ f Lawn 5. image) avian in Off” 0 S" / ‘8} cl i We c'ii‘cm 5‘) 5‘6 “ll;ka jr‘eplt Meow" Cite053 )4 .7. 5‘ al Laxa 3 cit hcﬁx‘  E 4. ﬁnite portion of the graph of polynomial f (x) is shown. Use the graph to answer all parts of this question. w 25 maximum value is 2.253
w I'D b) The global (or absolute) minimum value is 3 , a) The local minimum value is ‘“ “X3 , and the local and the global (or absolute) maximum value is 1 V?) . 5. Consider f (x) a 4x — g x3 . a) The graph of f is shown. Judging from the graph,
is f even, odd, or neither? 554C! Shah!» of“? gin, gammar bay " b) Use the algebraic deﬁnitions on p.262 of the text to conﬁrm your decision in part a). Show the work. (ax; 1" 6%)) 6. The graph of f(x) = 2253 H 4x2 H1016 +12 is shown. Write f in factored form. ad 0“; Z
W :— «(Xe VT'JXXWWe m _
M) a l (v (»3)1(x~=I)(>< ~23) 5: ZlKMbXXIéD 3) Answer: The'factored form of f is ﬁx): ZOHZXWDQME 7. The factored form of polynomial f is f(x) = 0.23:2 (x + 4)(2x m1) . ﬂ OZX'Lan'Z“ 45M # At) a) f isadegree polynomial.
*1“— 0:‘r‘><"”+ 1.9x m 0,? b) Find the xintercepts of p(x) algebraically (without the calculator). Express each answer as an ordered pair. 7..” gm x”? l
X P 0.1 i 2.x” I :1: 0
$250 ?X‘:..
X230 X 2&0) (93,0) (“é/Oi c) Use a calculator to View this graph, and sketch the graph. Label the x—intercepts. Also label the
local extrema with both coordinates, rounding coordinates to two decimal places, Fwy} Lice. a wimiow) weéxeq
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mfég)’ «5:3 8. Write the factored form for the lowestdegree polynomial f (x) passing through (0,9) with xintercepts at x = ﬁ3 , 3, and 6. lionx: CL (wthOée’a) (Ksé)
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Wgﬁeﬁ a main (0;?!) 9. An open box is made from a 11inch by 8.5—inch piece of cardboard by cutting x—inch by x—inch
squares from the corners and folding up the ﬂaps. (See ﬁgure.) Volume is the product of length width and height. a) Write a formula that gives volume Vas a function of cutsize x.
For what realistic domain does this construction technique make
sense? ' V(x)= X IPZK 9.5%» 2.x
Domain: MP Em.th
b) Use the calculator to ﬁnd the approximate cut size x that maximizes volume. Sketch the portion of the graph over the realistic domain, and label the local maximum point with both coordinates.
Round answers to the nearest tenth, and include appropriate units. When 3:: l ““‘\"‘?v’5 , volume is maximized at V5 :l edible» \wa; . 2x+6 x — 3
represent asymptotes, and label each asymptote with its equation. Also label the X and yintercepts as
ordered pairs. \I'Q Whitetail W have, Knit? .3; Q
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y‘g‘ Mitt? “i hijL—PoWaf ﬂier/m. x yr‘gg a] nas$ '1}, my Fm we; vm amend» aur w:
gﬁ J 5 X m w A %m$mhmmhkﬂ .m 10. Consider the rational function given by f (x) a . Sketch the graph. Use dashed lines to yes; 35; x Z r —s—7—5_:;:3 2 4 s 6 r s. 910:
’ I
yr: 2». ' U)
x “ﬁnite Mmpﬁ” y‘alth __g
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X11“? ii. 12. The surface area of a sphere is directly proportional to the square of its radius. Said another way, The
surface area of a sphere is varies directly with the square of its radius. A sphere of radius 3 units will have surface area 367: square units. a) Find the constant of proportionality, and write an equation that gives surface area of a sphere, A, as
a function of its radius, r. ““‘=Wﬂz~~.~ m' A a: K r2" g: Afﬁrm.
arm a tats)?” 3W _. .. __ _
BettaQK e» K «aw arr h) Find the surface area of a sphere of radius 6 inches, expressing the answer using appropriate units.
2) W ' .' ' ,
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c) If radius of the sphere is tripled, surface area is multiplied by C? . Said another way, if radius of qyf‘i iwao the sphere is tripled, surface area increases by a factor of g . R
am a... Pt: l’l“!’t’(3¥‘)2‘:: Hefnal‘nl“ Ci, 4th 1:: a «'3
rﬁ’! The intensity of Eight on an object is inversely proportional to the square of the distance from the light
source to the object. At a distance of 6 meters from a particular light, the intensity of the light is 50
watts. _ a) Find the constant of proportionality, and write an equation that gives intensity, I, as a
function of the x, the distance from the object to this light source. 1:: KXWZ" or” Ian»: K$50~3éﬁzl900 xi
5"" "i ii?  w are.
kl equation: “ w 0 ‘ 2(1“ 30": ~ 3t, b) Find the intensity at a distance of 10 meters. Answer: W ck
t?) 00 7 pg (3:;
j: : WW?“ WMWW” i 8 medic:
l0 ' too 13. The properties of exponents and radicals are presented on p.350 and reviewed again on p.357. Use
these properties to evaluate each expression: a) 49% :3 £th ‘ b) (—64% “ﬁn(pa 0) 10,000” :2: 5‘ mood .7 .. (awmm 7:; is . it; at? ' I
13. a) Sketch the graphs of y 2 xé and y = 1% .9 Range [.0 1 + (19) Range @499 ~%~ (:43 z b) For integer n 2 2 , the function y = 35% is a root function. For 11 even, the domain is M and the range is I
For 7? odd, the domain is M and the range is (We 0'4) {~ 00 I 14. The graph shows a power function of the form
y = kt." where p is a non—zero integer and k is a
non—zero constant. Tme or false? a) M k is positive.
13) i: is e p is positive. 6) T‘l‘ ULE This is an even function. ywmﬂe 53vneﬂé’i‘f‘);
t) False The domain is (—00,100). Mwﬁ be, m K xhej even
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15. Suppose y is inversely proportionai to the cube of x, and y = 6 when x : m1. at M «mow s a) Write an equation that expresses y as a function of x. Mgmiw wer 7:: w W;
ﬁ '5 _ MK )4 a: “'5 . WK his...
7 Q (0 G“); 2.
b) Findywhenx=3. Answer: y: '" gas"n 3% 2.7M ‘7 0) Sketch the graph, making sure your artwork shows the
asymptotic behavior of the graph. VIZ wwwggmm w d) Finish each statement: i) The domain of this function is (“00; O) U (0 (Tm). ii) The range of this function is (’00, 02 U ( 0 +410). iii) Asx—>+oo,yh+ O .
iv) As x ——> —oo , y —> O
V) This is an (even or odd) function because it has (yaxis or origin) 0 M a} )5, symmetry. ...
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 Fall '08
 Algebra, local minimum value

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