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Unformatted text preview: MA 1021 A ’06 Final Exam Name Instructions This test is closed book. Calculators are not allowed. \ Part I  Basic Skills A01 Heinricher, A G3LC) A04 Tashjian, G (8:00) A07 Masamune, J (1 :00) A10 Abraham, J (10:00) A13 Tashjian, G (12:00) Please Circle your Section A02 de Oliveira, G (3:00) A03 Servatius, B (8:00)
A05 Onofrei, D (9:00) A06 Fehribach, J (9:00)
A08 de Oliveira, G (3:00) A09 Malone, J. J. (10:00)
All Lui, R (2:00) A12 Masamune, J (3:00) A14 Abraham, J (11:00) Part I  Basic Skills Work the following problems and write your answers in the space provided.
Use the scratch paper provided for your work. You need not simplify your
answers. d 4
1. Find—y 'fy:x3 +3x2——3+5
dx x l2. d
2. Find—J: ify=x3lnx
dx dy (x2+1)
3. F' d — 'f =
I“ a1 y e (em 4. Find an equation for the tangent line to the 4 .
curve y = x — x3 at the pomt (1,0) Ans. z: x‘l dy 3+x
5. Find — if y = 2 I
dx l+x (DU+1.1)" (3+X)(7>0
«W
Ans. ll'l'x‘f“
d 11
6. Find—yify=(x3—3x2+5) I
dx 0
u (xvnus) (htex)
Ans.
d
7. Find l ify = sin3 (x)
dx 3 Slh1(¥) ((05.3!) Ans. Part II ‘ Work all of the following problems. Show your work in the space provided. You
need not simplify your answers, but remember that on this part of the exam your
work and your explanations are graded, not just the ﬁnal answers 8. Evaluate each limit or show it does not exist. . x2—9
a.mn7;————
x+3x+x—12 __ 1m (x+3)(x3) _ I"... “'3
! >973 (“ﬁts—3") i “73 7"” .. lm “'9‘
_ ng 136933 3+3 4. —— —.
' lim X + [trim If 7
X93 . ‘16‘93
.'5wa+25
b. 11m—
x—)0 x
.. hm ('5' x+2§ )(H H2? llm 257(54'25)
290 x 5+ may ’9") "(5+\/¥+15')
: llm Fl
x—eo Sh/xtzs‘ —\ ——. .4 __
9M _ lo ":5 him .._.I
54*0 .— __________,_.______..———— —
“m 5+ ‘ ’lhm 6 him 9.. For f (x) = x2  5x , ﬁnd f'(x) by using the limit deﬁnition of
derivative. Hm.) Mgr/ﬂ
[um
97%) '5 kﬁo k Dcﬁmhe» 0“ a. d‘ﬂV‘A hut, Iv \\
p
7‘
+ (7 I \r\ \l
N
x l
V] 10. Find an equation for the tangent line to the curve x 2 + xy + y2 = 19
at the point (2, 3) x
l 7 v4.9
:1 .z .— if...
dx may 11. A ladder 13 feet long that was leaning against a vertical wall begins to
slip. Its top slides down the wall while its bottom moves along the
ground at a constant speed of 3 ft/s. How fast is the top of the ladder
moving when it is 5 feet above the ground? 1: i6“? 1: (7‘ l‘5 iﬂCfﬂU‘ljB (1‘1 "x Q at m1 01" if; t (‘2‘ :: HZ‘: pi'/5l'c‘¢: [5 éecr‘i‘T‘SEﬁ) \/ team ’5 j 11, A LTERNATWE 5 fat—U TtoN A ladder 13 feet long that was leaning against a vertical wall begins to
slip. Its top slides down the wall While its bottom moves along the
ground at a constant speed of 3 ft/s. How fast is the top of the ladder
moving when it is 5 feet above the ground? 12. In this problem, you will analyze the curve given by f(x)= ~x3+3x2 oo< x < +00 Mt
(“MU—*2) E “6:0 x=2_ a. Find all intervals where f(x) is increasing F’H‘; .=  3X (“306*4) M
(Home) (+1 () — '2. (b. 7) m (7 t
In .Lnln\ z..\ (g\ .—
LL’ 1 W \ J k I 'WX) linerease; 6n (017) b. Find all intervals where f(x) is decreasing FCK) dlﬂeak’; 6n (n 00,5) M (2) you) c. Find all intervals where f(x) is concave up Wm: — (Mr : emu—n ~=> W I WM) 0 p’ﬁt) 4. 0 6.00) l)
C‘ﬂ‘”) l5 Concave up on @0030 (1. Find all intervals Where f (x) is concave down in?!) [S Concave clown on (Iii—db) e. Find all local maxima and minima of f (x) crih'catl pomi‘y 1K": 0 Md W137— 4" ﬁéow’) W3
1:0 15 ad'th Gab, l), a. festbn “AuL [5 [onto‘we up ' ﬂu jedcfl'l Mll/Aﬁéejcjé I“; a, Inca] Mlnlmdm. (0,0 1, +0“), 4» N55”. wlmm .
is (maxr9 downr b7 ﬂu Seamd dean/15c 7435i; 70; Z 15 a, [on] maximum. (2, LI)
f. Find any inﬂection points of f (x)
0m possible Fatiml '7 when: FII/k):0
9 x r: I FY!) I} (onhnuo'r fin at neighborhood amt/I‘d x3!)
and (Wat) is POSIM/c +0 “4: fer”! Jf Y7]! “6‘91”: l
1117 HM r5,th of X61; 'hau; Xv! 1'! am lﬂFbcM/l poi44' (1,90 Sketch a graph of f (x) III... . IIIHHII
IIIIl I. III”...
IllII. I' III II.
UI‘. E... % f_ i I _ no»  II I IIIIIIII
M II. I III. III
nwI m III I IIIIII
@ (0,5)  III I Ill{III 7 (HI‘1") ,/ “
t
1,4/ 13. An open top rectangular box with a square base is to built with wooden sides,
and a metal bottom. It will have a volume of 125 cubic inches. Metal costs six
cents per square inch, while wood costs two cents per square inch. Find the
dimensions of the box which will minimize the cost of materials for the box. ‘2 . 3
'
Volumer. V‘J )5 l, ., l'Zf In Casi:67; éxl'l 2(ley) I
1:. l1;
>< x 9} x1 1 175 __ 2 Moo
c602. éx + B’x(.;;), éx +
O4 X < +00
C/{X‘55 11x“ loco 31.14:; Zena MW)
)9“ [2x7 [600
F
x“: I20
[7/
WV
$ecgnd dmvafxirc (.5 pojtﬁw X ._: 19'
4%! x [h AOMﬁIhI' he?!“ cont‘ve {/71 Id {ﬁg . cdmfpemd; fb 4L
ﬁtobal mmmwm “ft 3'0 1‘5 14. Find the derivative of the following functions a. f(x)=ex+xe+(e)(x)
P/xk— €¥+ 676944“ 6 b. f(x):xsinx
In y =, in xsmx“; (S'hxiﬂh x) L 4.2. .. (cmaww @‘M‘Cii
‘/ d)‘  anX
4:1 —; [(6% mam + 11"]7i
A): >‘ c. f(x) = sin(esx) Man—— [Cos (8‘)] E e”) [if ****** End Ofexam ****** ...
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This note was uploaded on 10/13/2009 for the course MA 1021 taught by Professor Tashjian during the Spring '08 term at WPI.
 Spring '08
 TASHJIAN
 Calculus

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