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Unformatted text preview: Math 16A — 002, Fall 2009.
Dec. 10, 2009. FINAL EXAM \
NAME(print in CAPITAL letters, ﬁrst name ﬁrst): ____EE_____________________________________ NAME(sign): ___________________________________________________ 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. 'ooqcucnpwto TOTAL 2 1. Compute derivatives of the following two functions. Do not simplify! (a)y=m4.\/l—m3 I. 23 _ QXgW—l— er'i)‘: (4,065) sac/2'0”) 'Li‘ (Xr+ 4), “ gxqﬁtamﬂfx) (>5 +431 i0 3 2. Find the equation of the tangent line to the curve (2:2 + y)3 + my + y4 = 2 at the point (0, 1). 2.
3(Xl'tg.) ' (EXt'ff’) +X'313r + a +4Ug§gﬁ=b l0 2 S3298 4%— X=O 3=1L 3'ﬂ=+4 +qﬂ¥0 otx , 4x ) $ 0»:  4— otx at Lu 4lx
! — " :1, r
a J 4x 4  3. Compute the following limits. — (X) /
(16+h)1/4_2 .CX‘H’O '4; t (X
W“ 10:“; L”? j? )
4 _
W :ebOr—x M ’3/4 a
¢X=46. 404/(x)=11_ty
?’/‘+ 4
Ans/war ' 71F (“93 E: 2
<1» hm W £111....”— W '7 3
M3 23—3 ' OH). + «lam1')
z/PA (””32 * (8X+1) 2
M
“3 CX$) (w). + 44%“ ‘3
(3244“ N gy1 (r. = yams, = mxo
‘1 /Q\ ~~~~~~~ Z
M ,
x~>3 CX~<D (x44. + JAM—ml)
= /Q/‘ X""" = 2 = g
X"? Y4”? + 48X+i 5+ 5 =2. 4. Consider the function
a: 2, a: < —1, f(:c)= am+b, —1$a:§1,
—3x2, m>1. (a) Determine the numbers a and b so that y = f (x) is continuous for all :13. Coot. ix ‘1: 1: —«+b
W. 4* 1'. —3 : obiJO gig—o.) b=4 ) ereL,
x2. x<~1
J
: ~ '55
#00 2x 1) [X {
3><‘) ><>l (b) Assuming the values a and b obtained in
is not differentiable. (at), determine all the values of a: for which y = f(:1:) _2x= —2 «st x=17 W4, —_— l=*6x ﬁx=i? No, to K—
‘0 5. Consider the function f(x) = w = 36(252—3 — 27—2). :3
(a) Determine the domain of y = f (ac),
Compute the left and right limits /
"bet/mm“ X¥O. M I \ X _er: WW W ‘ (2—) o) . Mo \9 ,m 1941'“ 4H», E/l / . _ 31, 8’ = I h.a., JAM $50 = JDM ex = 0 "l 98"“ i”) O' X 9~n
X"°¢ Xrna / X its intercepts, and horizontal and vertical asymptote.
at the vertical asymptote. ll. 0 VJ]; X=O / 1010+ 453,141) 003 ) £10~ ﬁ Z(ﬁ}L)> C , m:u <0 4 ’1 (b) Determine the intervals on which y = f (cc) is increasing and the intervals on which it is
decreasing. Identify all local extrema. (NoLL I 3—4,; .— 3‘ . >
3 ‘3t/(><)=3é(é><“t +2><g>= 42" <H73+X>3
M =0 a» 8W:_j’*;"i3 (AL) Determine the intervals on which y = f(:1:) is concave up and the intervals on which it is
' concave down. Identify all inﬂection points. (Note: £2; = g.) i”(x) = s( (+214 >5: —£ x‘q) / w Sketch the graph of y = f(:1:). Identify all points of importance on the graph. J
(8) Determine the domain and range of the composite function y = f (x)
( o 2] 2
b 0 mean t ) '
3 J
Rom 3} r. I: 0 / °°> .
(f) Determine the domain and range of the composite function y = f (11:2 + 3).
,2
b OMNK '. w x
. p_ on ‘
3:)835 RMK: 3 X +5 KM VA—Qw” 1“ E3/ > J 3 N E's/ad) 4' kﬂvg VAQUM 1M ['%/°> 8 6. You want to build an open box from a square piece of carton, 12 inches by 12 inches, by
cutting out equal squares from each corner and turning up the sides. (a) Find the largest possible volume that such a box can have. '2. .
V: (42"ny I X ) é? 2. ELY _., 2C4’L2Lx)~(—2—)'X + (12.44) cl 4
x [email protected])<‘+X+41‘2X> : (4L‘LX) (ll6X) olx / X \/ L)
O : «2.8 <— ‘ll‘J‘ '8 3 l WeMM g, o 44 41? (M5) )1 (b) Now assume that the height of the box is restricted to be at least 3 inches. What is the
largest volume under this additional restriction? .0l_\/ <0 a! WM X 44 an, (g/ 6) (g) (In)
0b< w Via M ‘HUL QWd Vpam N . v=3e3= 4022 ms.
[\AJlIkJ/‘A x =3) "l’ko‘+ "/ =/T[) > \20) 9 7. Car 1 is headed East (away from point P) and Car 2 is headed South (towards point P). At
one moment, Car 1 is 4 miles from point P traveling at 60 mph, while Car 2 is 3 miles from
point P traveling at 70 mph. (a) At What rate is the distance between the two cars changing at this moment? (Note: be
careful about signs.) a: = (£5 + ﬂaw /
ZPOU' 2X OU’ Zvobl' ‘lo Lo)
m» X 01! j. at l
a}: 7 wt +~r EU?
‘4' MIL—i to + i ~40
tma w 0% S S( ) 2
_ zuco ZIO ,_ (3 (“4/0/
5' :=—" (b) At what rate is the area of the triangle determined by P and the two cars changing at the
same moment? 10 8. A maker of sunglasses estimates the total market demand (at zero price) for its new Zeroglare
sunglasses to be 1,000. The demand drops to zero at the price of $200. Assume that the demand
function is linear between prices of 0 and $200. The costs of starting the production of these
sunglasses are $2,000; after that, each Zeroglare costs $100 to make. (a) Write down the demand, revenue, cost, and proﬁt as functions of m, the number of units
produced. zoo—0
X 2 IV — 2,00 = ._ X = ——’S.)(
4000 O o_.°°0
{'93
O 2.00 1): %x+uo ( ogx 910003/9) Q: 400 x 4— 3<°°°W 7 V
—_‘
R= xC—fo +Leé) = {X 0 P: —:;x"' 4400);  29o (b) Which production level, and What price, of Zeroglare maximizes the manufacturer’s proﬁt? ‘ 2. J‘Oo ..
~ ,—)( +100 ,,=O “A, , ?X= 100 jx'f‘f lS‘O
' i ‘ rl/)\
m W4. r“ mekd »
pot x=uo =_L zro +uo=4so
) ’l) f __ 13:5.
x
4 2.000 V v (
_ — —~ + 400 — '0)
$3 " 5 X x (5)
2‘3 =  .4. + 743:" =0 mm xl=lolooo
Dix g x x=mo
 E Cowman, 
dP as _— 0° 0 > —— J ,i /J )
x1 ' x5 < W X 0 J :9 "Eimi'zLK—E U‘ // 60 x=lco 4i ‘Hur Max, ...
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 Fall '09
 GRAVNER
 Calculus

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