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Unformatted text preview: Math 1400 Graphing Functions and Optimization DUE: NAME: I
(7~page Assignment) Exercise Set 3.1
Show all relevant work neatly and legibly in order to receive credit. Please circle or box your ﬁnal answers. A.
1. Refer to the graph of y = f (x) given below to answer the questions. y=f(x) I
a
r
I
I
1
a
I
I
I (A) Identify the intervals over which f(x) is increasing. (0mm, (A26); (B) Identify the intervals over which 1" (x) > O.
(a, t") , ( it, e.) ( in i)
(C) Identify the intervals over which f(x) is decreasing.
(1.9/03) CC)0L) 3 10)) (D) Identify the intervals over which f ’ (x) < 0. as mm) 1 (E) Identify the x coordinates of the points where f'(x) = O . Lye: (F) Identify the x coordinates of the points where f '(x) does not exist. a) 0L) 73» 5.5m k (G) Identify the x coordinates of the points where f (x) has a relative maximum. E) £2 65L (H) Identify the x coordinates of the points where f(x) has a relative minimum. 0‘2 f 97 Math 1400 Graphing Functions and Optimization 2. Match each of the graphs below with a sign chart (a) _ (1'). Sign chart g/ corresponds to this graph. Sign chart CL corresponds to this graph. Sign chart b corresponds to this graph. Sign chart (a) (b)
f’(x) ++++++++EB++++++++ f’(x) m — _ _ _ _ __0++++++++
3 3 (C) (d) f'(x) ++++++++8++++++++ f'(x) —————— ——ND++++++++
.3 _ “M3 (0) _ (i) f'(x) f’(x)++++++++g —————— m 3 1‘ 98 Math 1400 Graphing Functions and Optimization 3. Find the intervais where f (x) is increasing, the intervals where f (x) is decreasing, and the
relative extrema. ‘ (A) f(x)=x2 —8x+3 \ f fly» .5 ,g a “wwwinusmmmmw 4 ( min cum/W3 Wm was LizW): .42. 7 ' So MLWCVL aﬁj’
i, (4”) fig)
mmmommg mow New). ' .
(B) f(x)m—2xsm9x2+60x”8 Xe 'wg" as ﬁx): #2 3/; WD/(X) ‘3 “(0X11 )gx G D 3;. o X: a 1 GO
5:; H (9 (>4 Pi“ 3% ~10) ‘90 mm M («§‘,»2.g><3>'
7:7 ~——r(g‘(><+5") (xi—«7,330 ‘
r I \ I My (10(00
73> >< : «i w xn’L M WW ﬂ ' > "CIC")
#5" Z
(c) f(x)=~x4+50x2 430/9 m 492 3"
fix) a: He LFXK —r WOX "HOB: 0
382? #413 + (O'DX 30 : (9M:
:52 #qxbﬁwzﬂ w me my M
s mm (we) We) no (4“; M) News J mv X} we", 5 g
o ) W “V60 W5 “JG < 0 2 0) /\/\ 99 Math 1400 Graphing Functions and Optimization B.
4. Find the intervals where f(x) is increasing, the intervals where f(x) is decreasing, and sketch the graph. Add horizontal tangents.
(A) f(x) m 2+ 62cwa /  t 5: E: f (>0 r» (a w 2/“ ==  ==
w Wm M) a» w 3 ii ‘ «ii
._ m ' ii.“
WW % a: = no:
Jr “l” + O w “— > an run Inu‘u
~— —————— "am—mg mm =:.=.i== ===a
=a=ﬁ= ~ as»... . __ ﬂIﬂl I.
73(5) *' l l a E = HE
1621,, 'Wx at“ (i H) ﬂ f/(xo: 3X??— G>< M 9
ﬂ 3 (>52; 2* ‘3)
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Sa+tuxmjaax=gwr~z WWO u
Nu. m
allnull luluIn
wt 5 “ﬁg, luaugain!
/”\\k//’ ion. IilliaEII 5. Suppose that f(x) is continuous on (w 00, 00). Use the given information to sketch a graph of
f .
(A) 
f’(x)++++ﬂw—' ———— —NQ++++§B++++ i
g}
..__i 100 Graphing Functions and Optimization Math 1400 IIIEll..
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101 ); is not deﬁned
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11x
f’(x (B) Math 1400 Graphing Functions and Optimization C.
7. Find the critical values, the intervals where f (x) is increasing, the intervals when: f (x) is decreasing, and the relative extrema. Use a graphing calculator to help sketch this graph. (A) y m 3\/(2x2 —8)2 3(9/1X‘2‘w “y IIIIIIIIII
,fgzséﬁ;®:s ,Iﬂllﬂﬁﬂlnl
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g. h..____. m+__.;: ‘3.” i ,A Saw—D M
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m%\€33/”%%waﬂ_ ”’ swam Math 1400 Graphing Functions and Optimization 8. The concentration (in milligrams per cubic centimeter) of a certain medication in a patient’s
. . . . . :2
' body if hours after injection IS given by C(t) x m
+
for when the concentration is increasing and the intervals for when the concentration is decreasing. Also, ﬁnd any/alt relative extrema. , Where 0 St S 4. Determine the intervals 9. The graph in the figure beiow approximates the rate of change of the US. share of total ‘worid
production of hightech electronic devices over a 20—year period, where 8(1) is the US. share (as a percentage) and t is time (in years). The graph is of y = S '(t).
(A) Write a brief description of the graph of y m S(t). Include a discussion of any/ail
extrema. (B) Sketch a possible graph of y = so). .3.
a
g
i
1.
i
a
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i
I. 103 Math 1400 Graphing Functions and Optimization DUE: ' NAME:
‘ (7—page Assignment)
Exercise Set 3.2 Show all relevant work neatly and legibly in order to receive credit. Please circle or box your
ﬁnal answers. 
A. 1. Refer to the graph of y = f(x) given below to answer the questions. (A) Identify the intervais over which f (x) is concave upward. @c 19)) c. w ,. (ace) ) (e) p); (t, 3/) (B) Identify the intervals over which "(0) > 0 . (C) Identify the intervals over which f '(x) is increasing. (Se/m1, m (/0 (D) Identify the intervals over which f (x) is concave downward. (t, a (w) J (m Li) (B) Iéentify the intervals over which f "(0) < 0. M D) (F) Identify the intervals over whiCh f ('(x) is decreasing. (G) Identify their coordinates of inﬂection points. 112 _ Math 1400 _ Graphing Functions and Optimization 2. Match 621011111116 graphs beiow with the appropriate conditions (a) — (d). ' corresponds to this graph. Condition 10 corresponds to this graph. Condition (a) f'(x) > 0 and f”(x)< 0 (b) f'(x)< 0 and f”(x)< 0 (c) f’(x)< 0 and f"(x) > 0 (d) f’(x)> 0 and f"(x)> 0 \ . 113
“33 Math 1400 Graphing Functions and Optimization 3. Find the second derivative of each of the ftmctions.
(A) m 2x3  7x2 +’5x — 8 I (B) = (x2 W9)4 _ . 1 _ M, . ' I Z (3
1%)“; (a X M I 4X. + $ 73’64) :— 4647’qu "(ZX):9X(X"i>
/ Llij  I/ “2.. I' ea?"
7050:! if ~10 (>0: ﬁxaafMXGCX @Qx)  : Wiqﬂ eg‘x'zfxiﬁf" w 4. Find the intervais where the ﬁinction is concave up and the intervals where the function is
concave down. Find any/all inﬂection points. 1:“ (9X “I 8/ i ' . ' aim/Mn (m i7 .  ' .. (B) f(x)= x4 «~2x3 —24x+12
fin :2 M ~—~ We 2%
fer/va 1:: (AYE: [COX 1”“ if“ Miramo —~<> 4x(3><q>)ao es? X 90 W X '5; O i Math 1400 Graphing Functions and Optimization 3 w L9 P, f.
B. r .
5. Find ali relative maxima and minima using the second~derivative test Whenever it applies. If
the secondmderivative test fails, use the ﬁrst~derivative test. \ H #3 3 (A) f(x)= 3x4 —8x3 m90x2 —150 m; r41 x? MWWOX rawPk: s; ‘ff/(Xy ":3 3 (0 38’ “W PM! ,4¢¢dcz".!~ Mg)? 0 :> M. Win mm my» LOW Ker: (B) 1120:“? PM a: t #84???” 7:; (a, gag; @ My: P43 X1?” My) : {Mfg 3.1925 E15 Math 1400 Graphing Functions and Optimization 6. Suppose that f (x) is continuous on (— 00, 00). Use the given information to sketch a graph of IIIll
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f (x) W .
aa:::%5a:
"x m m — — ——0+ + ~i« ‘ '
“’“mmfrx*ﬂi?‘“ =::=::::
(B)
f(*4)=0: f(?~)=3a f(0)=02f(2)=2= IllIi..
f (4) m 0, f (5) = 3 ; ' IIIIII.
f '(~ 2)=0, f ’(2)=0; IliiﬂII
f’(x)> 0 on (—00 —2)and (2 so) Illlﬁllll
f ((x))< 0 on («2, 0 and (0, 2) f"4=0; ' ‘ i ' .." t
f "(O)< 0 on (— 00, 0) and (4, 00); fiwgmi M eﬂiy ::::=:=::=
i..i,.§w~5~w—j~w I IIIIll...
4,. m . i
J? W. we m “t” “H 9 w 4x
7. Find the; inﬂection points of tho function: f(x) = 1+ 2x .
.f/( x eat(f s + 22$ '—~ («3475) a Q. m *4
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SO ha’i‘i V20 Jﬂj/“CzuHomv/gomfs_ _ Math 1400 I Graphing Functions and Optimization DUE: NAME:
(3page Assignment)
Exercise Set 3.3
Show all relevant work" neatly and leginy in order to receive credit. Please circle or box your
_ ﬁnal answers.
A. 1. Refer to the graph of y m f (2:) given below. Find the absolute minimum and the absolute x
maximum over each of the indicated intervals. ‘ a f 2 ____ _. / v) K, Le V‘ F7 ‘ “
(A) [0, 5] (B) [2, 8] Absolute Max: 3; 6;) Absolute Max: j l O)
\ Absolute Min: (2 I 1) Absolute Min: G/
(c) [5, 9] (D) {4, 8] Absolute Max: Cl! 02 Absolute Max: All Absolute Min: 7 {_ [2 ) Absolute Min: 1 "jg, (2) 322 Math 1400 Graphing Functions and Optimization 2. Find the absolute minimum value of f m 4 + 3x + ~21 , forléc > O>if it exists. Sketch the
. x graphoff “ff)®::f§ .Wﬂzx 23*??? Pixel (1M: ~ iota iVL alw'smwm I“... .. .
ih  lull IIIIIII
_~ III IIIIIII
Wm Wm xss lullIll... w WM” lullIll... “* lullIII...
suywewmﬂwa IIIIIIIIIII ; , lullIll... 010 m Mam, own/4K. x93 IHIIIIIIIII
Wimssiwm. Innllllllll
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M~Oi+ Rn IIIlllllllm
*Wmswm“ Illlllllllll ' '3 “ — '25?
go) alas. mm act/bong Wax:3t £8); 4+3ig+ ﬁg“: QR
3. Answer the following questions: 5, ‘ mm @6 3 2 L
A) What is the difference between a relative extremum and an a so ute éxtre um? " " 56 (m M‘W is (X (“Mag mg; Wiring, MW". ' B) Can. a relative extremum be an absolute extremum‘? 763 C) Is a relative extremum necessarily an absolute extremum? No 123 4 {H&>:$3X4*Qéyé“wk¢X%“JSD
‘ 11790::‘135 ~ waf—wm '
142% 0x~79<65+;) (/0 0,4342% <0]: i ‘ 245,0
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. &Mkwah@ .‘. Ho Mag. max; x49. i=7
a, o 7 alas. Wm gag (mama). Math 1400 ‘  ' Graphing Functions and Optimization DUE: ' ‘ _  NAME:
' '  ' 7(5~page Assignment)
Exercise Set 3.4 Show all relevant work neatly and legibbz in order to receive credit. Please Circle orbox your
ﬁnal answers. You must‘use a derivative test zrst derivative test or the second derivative test) to
veriﬁ) that the answer is fine desired exiremum. A. .
1. The financial analysis department of a large company determined that the cost of producing
x number of its product to be C(x) = 15,000 + 20x . The department also determined that the associated pricedemand function is p(x) : 100— 0.0053: , Where p is the price in dollars.
(A) Find the revenue function. KOO “:2 7960 “3C eOooHoioogan 31/0076 W 0.0o§7<z (B) Find the proﬁt function. POX) ﬂ KOO WCOO
‘ */0o%m0.00§xZ>”(/5000+%9<)
:2" —~ 0‘ 00998“ +80% —/§DOO
(C) Determine the production level for the number of items (interval) that win yield a profit.
Solve, ~— Ocoog pear[Lon w/fam> 0 TM (1.0ro m$ M513 X: "'80 i EDZ"¢(WQ'GGSD(;(W ,1(wo.oo$‘) Tag) Me: [90 W 35:. [5870 _
“3'0 «map MW m (we 2 15870) (D) Determine the number of items, x, that will yield the maximum proﬁt and the maximum
proﬁtindollars. ‘ ' We.Pm“1&T.W IP/(ij 
(“D/()0 ,Iwonix +343 a ‘/
rm ea ex 184% WW")
0 w gm ‘ {j 1:: W>3305300 130 Math 1400 Graphing Functions and Optimization B.
4. Recreational swimming lake is treated periodically to control harmful bacteria growth. Suppose that t days after a treatment, the concentration of bacteria per cubic centimeter is given
by 000 = 30:2 — 240: + 500, where 0 S r s 8. arc/e) :co t. w am
(A) At what rate are the harmful bactena growxng 3 days after treatment?
(:75) :— Zgoa) Waite : M090
%QUW& £0“ UJLLHO’VL all (éo/Cmg ‘
' a1? (B) How many days aft r a treatment will the concentration be min Oct) so ~=> east (C) What is the minimum concentration? (1(4) 2: 30 Callerarm (a!) +w‘o
'3 9/0 /cm3 5. A fence is to he built to enclose a rectangular area. The fence along the two sides facing north
and south is to be made of material that costs $3 per foot. The fence along the two sides facing east and west is to be made of material that costs $6 per foot. (A) If the area is 20,000 square feet, ﬁnd the dimensions of ﬂae rectangle that will allow the
most economical fence to be built. (B) If $5,000 is available for the fencing, ﬁnd the dimensions ofthe rectangle that will enclose
the most area. l32 ...
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 Spring '08
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