This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 16A (LEC 003), Fall 2007. Dec. 12, 2007.
FINAL EXAM
/
NAME(print in CAPITAL letters, ﬁrst name ﬁrst): _________l;:E:{______________.____.____
N AME(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. OONOBU‘rhCJONI—I TOTAL 2 1. Compute derivatives of the following two functions. Do not simplify! (a)y=x3\/1—x4
l 7. f q‘ 3 q A 4x3) to
%I: "ﬁx A—x +>< .JiC4X) ’C" 2 a. ' l O 3 2. Find the equation of the tangent line to the curve (2:3 + y)4 + xyz — y = 0 at the point (0, 1). A d 
4C2‘*d—fz> +11. ”ﬁ‘0
0! A
8+1 + 4'33 91?“)
gﬁva 3f=—3<—— SnotL ‘
.93.
81=‘3><
f
ﬂz’gx 4" 4 3. Compute the following limits.
1/4 _
(a) lim (16 + h) 2 h—’0 h
76620 = ></1/q / £/(X)’J¢TX*3/q
4/4 4/4
/ _ , (4am) —— 45
av W 2&2: —~ (a)‘1im :1:— 32+4 = A)
$—>4 :1: — X4 q 4. Consider the function
—3:c , an < 1
f(a:)= aa:+b, —1_<_a:§1,
m2, :1: > 1. (a) Determine the numbers a and b so that y = f (x) is continuous for all x. CM.aul*i '. ‘3 = ~a+b LP
25: »2 J la=~4
=1b ) QBZ. 3 (b) Assuming the values a and b obtained in (a), determine all the values of :L' for which y = f (51:)
is not differentiable. _0L_ _ L 59— .
(Dy/H. at ~Ar’. (AK A 2X) 6%) M )(="1 "HME‘PC
f (2K‘1> =. Z Mi Mi (jdbuvvovm ){ 3,
h 3," ’14 w: . ‘l I 'TLQ J4. “we 0 M. ,1 «JAR/Q
.d. ()3) :LX / wlrx =i é—o 6 My) 3
ON, 44 abél ml A gnu—
.— ARM 3: 41%) 'lF ”Lil/f WWW «scatcw.
«A x=~4 2 _
5. Consider the function f(x) = 902:3 3) = 9(a:_1 — 3113—3). (a) Determine the domain of y = f (:3), its intercepts, and horizontal and vertical asymptotes.
$0M MM ‘. X # O A
31446402441: C 533/ 03/ (DE 0) A V:
’ ._ . :: — (>0 1
VolJV. Aijmpialt. X'O ) QM $(X) ) QM @543 190+ Ham. aﬂmpkk" JOAN $(7‘lto. / 3=o . x~‘> D ‘ (b) Determine the intervals on which y = f (x) is increasing and the intervals on which it is
decreasing. ((1) Determine the intervals on which y = f (as) is concave up and the intervals on which it is
concave down. 5 = 4?X—§ (XI—lg) =0 7 (e) Sketch the graph of y = f(:(:). (Use m = 3\/§ R5 4, f(3\/§) z 1.8, f(—3\/§) R: —1.8.)
Identify all points of importance on the graph. (f) Let g(x) = M57. Determine the domain and range of the composite function f (g($)), that is,
the function f (ﬂ) ﬁmuvxn X>O ,"'°' (04") 2
M1612 & $122; 2m, (—m,2].3 ( O 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. MAX/{Mitt 2
2.
3y”: 2(412x>(~z)x +(4zLx)
X :(«2—1x) (—l—lx + 42~L><>
= C42LX>(42_ 6.x) 5' 631—0 all x=l€~t olx ‘ ’
x V
'0 O
12% m 4
Z '12—? 1’ —ll~9. x 'l ure. 5
g o «22 (m3) . (b) Now assume that the height of the box is restricted to be between 4 and 5 inches. What is
the largest volume under this additional restriction? 33 ms <0 mm x '14 {.4 (77g) Md so V $8)
dx .
go M ‘Hu. ,QW'E VfaAM 4; VOLMA X: 1+)
‘H/tkt H I
l0 9 7. A point P is moving on a parabola y — m2 .At one point in time, the x—coordinate of the
point P 18 observed to be 1 and the distance from P to (0,1) is increasing at the rate 4. Find the following two rates at this time.
(a) The rate at which the a: coordinate of P is changing. y 3:)?" PM) x1)
(4)03 x :D = did—amen MW ? MA @, 4) “mm d, ,M_._M,_,...W.MW»...~. . . W‘ 5‘ = \ﬂ 1.x; + (ximz FLD. = (2.x +2<x1+32><> 33%
‘VC 2 W2+( 41?
P \j 4“ “=12 33:4; f
4 , .02: Ax 
H ‘2 ‘2 M 3‘4. (b) The fate at which the distance from P to (1,0) is changing. = (x24): + X“
cl
ﬁg :2 _.‘.____.__ (2(x—1)+ka3> 31): 5
0H 1 \, (x431 +x‘+
oi><
9.; NA =4. ———‘:
P (3 X / 0U” 2" j 5
am 4. , =
1 1.. 4 4 :2 10 8. A bicycle maker estimates the total market demand (at zero price) for its new DirtsurferTM
mountain bike to be 10,000. The demand drops to zero at the price of $2,000. Assume that the
demand function is linear between prices of 0 and $2000. The costs of starting the production ,
of Dirtsurfers are $50,000; after that each bike costs $1,000 to make. (a) Write down the demand, revenue, cost, and proﬁt functions. X a, O ,_ 2,000 "'0 (X 1olooo>
(0,000 «I O ’ ‘0’ 00°
0 2,000 u ‘ c: " i. + 2000 q
DWOL 4 . /l> 5 X
 _. .. .4. 1 azooo X 2 ‘ yo
R’X? (X
C =, 4000 X + $70,000 2 P=R~ C, 1: ~31__XL + 4000K ' ”ism/000 .2.  (b) Which production level, and what price, of Dirtsurfer maximizes the manufacturer’s proﬁt? “31:2:‘2fx +4600 :0 a at x =p—{nooo :U’Doé
? «‘3 ex «Mew 4M6“ 0:: Y ”Mi“ Maw" L} jhkobva, Wad/1F) Xeli‘DO d" VHM WMM ,1M 3:. 20 0 Ianco 4 :oo . __._'—— (c) Which production level will maximize the proﬁt per bike, that is, I5 = 5?
CE
BOO
wcaLXl‘MR'. F‘F= " A‘X +1000  $9 QSKSIOQ’O
\T X 2
.. r)
O! —— _L + “59000 7: O «Si’ XlBJO/DOO
.— :..: 5 9.
Li dx X  , x =J’oo
1— —~
fig __ _ 9 2,413,000 < 0 5:0 'P h meowe chum) ...
View
Full Document
 Spring '10
 NA

Click to edit the document details