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Unformatted text preview: Math 16A (LEC 004), Fall 2008.
December 9, 2008. FINAL EXAM , ' < : NAME(print in CAPITAL letters, ﬁrst name ﬁrst): ______ _L__}:_: _______________________________ __
NAME(sign): ________________________________________________ __ ID#: ___________________________________ __ Instructions: Each of the ﬁrst four problems is worth 20 points, while problems 5 to 8 are
each worth 30 points. Read each question carefully and answer it in the space provided. YOU
MUST SHOW ALL YOUR WORK TO RECEIVE FULL CREDIT. Clarity of your solutions
may be a factor in determining credit. Calculators, books or notes are not allowed. Make sure that you have a total of 10 pages (including this one) with 8 problems. Read
through the entire exam before beginning to work. concavipwwp—r TOTAL 2 _ «5—7
(80y 3m_r1 J/p—
3/ = ‘: <><~ LE1..:3L: ..... g“ M ' 3
(2x  432 (b) y = m4  cos(29:) 39 Maw) — WWW '9 l O 10 3 2. Find the equation of the tangent line t0 the curve (as + y)4 + $23; — y = O at the point (0, 1). 3 d 101 d,
ACM—gﬂ (4+ £53 +lxﬂ+xg£7 £50 12 paws m x=QJ3=4i 0* __
An;
4 +de
ﬂaai 4—day; Cf
ow 3
4
93’ 3“: “ix
L/ 4 3. Compute the following limits. . (1+h)—4_1
(Mira h V140 h ¢ ~l : P Lek co = x” M x = 4., EM 4 («ti :er i J "A WWW ~ ‘4
/ f “I 4. Consider the function :34, m<0,
f(m)= am2+b, US$31,
.r3+x, x>1. (a) Determine the numbers a and b so that y = f is continuous for all :r. W.Ot* O; O = MM
Cowtyakl Q+b=2 } “‘2‘ (b) Assuming the values of a and b obtained in (a), determine all the values of x for which
y = f is not differentiable. . A... H’ = ers
AJV'O' o‘x (X) art 0 ﬁx:o
i [llXL} l3;Ll' K
01%
Aki' 2‘. Li qu
dx< X) bow“ 4} MFX=1L
3L (X?+x) = P5XZ—‘l’iL r
dx
m jvuvujwm "“ 4O —— 2
5. Consider the function = 86:62 ) = 8:13“1 — 16:3‘2. (a) Determine the domain of y = f (3:), its intercepts, and horizontal and vertical asymptotes.
Compute the left and right limits at vertical asymptotes. 'Doma/va: X750, Iwwmt’“ ‘(2’ O) e 410
Hag, Watx):,O/\M 1~71 =0, x—am “a”
e x~L)
U a 1 X=O ' pm“ ) Q)“ £327. raw
' ) xeo+ X1) an“
[ww.>o = W (b) Determine the intervals on which y := f is increasing and the intervals on which it is
decreasing. Identify all local extrema. ‘i/(x) = ' “(1+ 374% 7"89530‘4“) g
Q/HVLJLa/L W5. " ><20/x=(+ (d) Determine the intervals on which y =: f is concave up and the intervals on which it is
concave down. Identify all inflection points. (Note: 3 ~32 = 6 I 16.) W00 = No >63 4‘32 sf” = May‘wa 7> Qr‘mr‘; \(30/ X;6 J (e) Sketch the graph of y = f Identify all points of importance on the graph. Y (f) Determine the domain and range of the composite function y = f DOMMNL X>l)\'.l. Ell”); .’ to, 9,] 1 (g) Determine the domain and range of the composite function y = f (:02 + 1). DOMN‘M Z all X 8 6. A construction contractor needs to build a small rectangular storage box with a square base,
ﬂat roof and volume 1,600 cubic feet. Every square foot of side walls costs $10 (that is, the cost
of side walls is the area multiplied by $10), and every square foot of the roof costs $4. Assume
that the base does not need to be built, so it costs nothing. (a) Determine the dimensions of the box that will minimize the building costs. Justify all your
conclusions! h = 4,500 )3 X1“: A1600 X2 /\ (:00 7‘
__ 40 x i 3:3: + ‘+ X
00° 1.
X
P___’________,_.
Oi C 6 Lt ‘30 0 ’ 3 _ V = 10 \
______ __ 2 +8x A x 8000 L‘: \2
d K x2. / j \
' 4 soc ‘
_V h: _z——— « Lt
(0 lo) Cw/ w) \‘\ ____._. qoo =.
0‘0—  + A"
dx T . //
(b) Now assume that the height of the box is restricted to be between 5 and 10 feet. With this
additional restriction, what dimensions of the box minimize cost? s in 5 4o '
'3' 490° = 2' ._ ’XI W‘ I 2"; r X __. $1:— 3 O ) x ' 4,320 \2
'L
Whom \n :40 ) X — 4.4%? :1b0 )X : 4450' «$424k 9 7. Water flows into a conical reservoir (with radius of the base 5 m, and height 10 m) as shown
in the picture. At one point, the water level in the reservoir is measured to be 4 m, and the rate
of inﬂow of water is measured to be 2 m3/sec. Find the following two rates. (The volume of a 1
cone is —7r'r2h, where 7" is the radius of the circular base of the cone and h is its height.) (a) The rate at which the water level in the reservoir is increasing. 2 A
r: 4 '2: in / 4 z ‘Drhg i
V=§3TTM:4Z 4 ’th 4 _._._ ,k_:3rh I
our‘nsvz can ‘* Pauli AMln=Lb 5;:2vio
I
ah — 2. a .4.— CM )
ou— LNT Di": ML
WWRYGY) (b) The rate at which the area of the water surface in the reservoir is increasing. (Surface refers
to the top circular surface of the water that is exposed to air.) 1/; v : 2L! — L db
9% «~ h Li) M w e)“ 2
0‘A ; iii—r 2. (l (ML/m) I 0 7,0 10 8. Your hardware company makes a ﬂash drive called Bolt. The total demand, at zero price, for
Bolt is 6000 units per year. At the current price of $60 per unit, you sell 3000 Bolts. Your ﬁxed
operating costs are $20,000 per year, and each Bolt costs you $20 to make. As usual, assume that the demand function is linear.
(a) Write down the demand, revenue, cost, and proﬁt functions. x éo—O _
6900 o ‘ Cx 6000) 3000 £0
: " ~36 Cx—eooo)/ (b) Which price should you charge for Bolt charge to maximize your proﬁt? How many Bolts
would you then sell per year? QUE _ a, A _ r 
dxv Ex+loo ’50)><—74'00l Y’W
A5 P 1.5 at quAAaJQL #Mc’km“ MW MGR/“Vt S, 09%c/Jux) P it“ WOL‘KUMMM ﬁx=Lroo w (0) Which production level would maximize the proﬁt per unit sold, that is, I3 = F=—’ix +A‘oo ~7/°/°°°
m x
2L]: ._ __ 1‘. ‘t 323‘: E X? : 50 . w/ooo = 4,000/000
olx ’ w "8’
 y ‘= 4})00
‘I‘
0i? 1/0 coo "’ W
\—  v M GOMCW v\
axl’ ’Z‘Uﬁ’ <O\3 g0 l) W;
owed Sr? X'= mooo win;ng P I lo ...
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This note was uploaded on 10/22/2010 for the course MAT 16A taught by Professor Gravner during the Fall '09 term at UC Davis.
 Fall '09
 GRAVNER
 Calculus

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