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KOÇ UNIVERSITYMATH 102 - CALCULUS
Midterm I April 13, 2010Duration of Exam: 90 minutes
TRUCTIONS: No calculators may be used on the test. No books, no notes, and noallowed.You must always explain your answers and show your work to receivedit. Use the back of these pages if necessary. Print (use CAPITAL LETTERS)n your name, and indicate your seetion below.
name: ---------------------------
ature: --------------------------------
ion (Check One):
Seetion 1: Sultan ErdoğanM-W (14:00)Seetion 2: Benjamin Smith M-W (17:00)Seetion 3: Selda Küçükçifçi T-Th (11:00)Section 4: Selda Küçükçifçi T-Th (14:00)Seetion 5: Sultan Erdoğan M-W(12:30)
OBLEM i POINTS SCORE1
100
142 243 84 185 206 16
OTAL
x-ıı. Let f (x) = 2 2 3x + x-
(a) (6 points) Find the domain and the range of f (x).
X2+2 )(-3::= (X+6) (X -I)::=: O -+hen X=-3> Oc x= 1.
'bomaı(\ o~ ftx) tR -f-3/1~
~ if- x-=t i ) f-(x)=t0 ..
fHm) lııYI x-1 (fr()x;rf
- (14X--l'1 x"2+2X-.3 (x+s)rx;G-x-lI
RD'f' of- f( x) fR-i 0,1/4 3(b) (8 points) Let g(x) be the function that corresponds to the graph obtained by first
reftecting the graph of f(x) about the z-axis, then shifting the reftected graph 3 units right
and finally shifting ıunit up the graph obtained after the horizontal shift. Determine g(x).
9i (X);- - X-1)(1+2X-3
x-I
(X-I) (Xt3)
9J-(X)=~
X-4(X-4) X
3rd (ç-tq>: S~ff1i(\-3 .{un(t- Up ') j
93(X)= _ X-4-
(><-4) X+1
2. (24 points) Evaluate the following limits, if they exist. if the limit is infinity, specify
whether it is positive or negative infinity. (Do not use l'Hospital's Rule):
(a) lim sinx -= II,," Sf(\)( (\~ )X-40 Jx + 3 _ J3 _ 2x .v i • '( X+3 +-V3~2xr
)<-\0~j;Z-~2><-~
3X
" \(+3 '+'1'3-2 vıO}= -L f Ur)') Si" X ~ ilr3 X~O -;ç mX-Lo
~,1_2R-=2(33
(b) lim x2
- 2x L ;/if X 2- )3 4 = Irv. , -
X-40 X + X .11
X-7 o vX;7--{ X+i )X 1..
=-llrvıX-Io
= 11tV) -ix-to xı X-2
X-t-I~---2
ıo1 ı(c) lim x - (Hint: Use definition ofderivative.)
X-41 X - ı
\\~X-H
(d) L~ f(x), if 4 aretan x ~ f(x) ~ 7rln(e/x) for z > O.
)Irvı Lt orcta n x = Ll. -ır z: 1f)(-4 1 "1
ır tn ( etx) = LT. -1 :=: Tr
k the following function continuous fort nd b that ma es3. (8 points) Find the constan s a a
{~:~~:o~;~ıf (x) = x2 - ı, ı< x
x-ı
all real numbers:
ond
on (i It?O) / -f('X)=- x~-1)(-(
bo.,./ we M.ed +o chWC. x=-0 Qnd X=~ orıı~.
\ir() .f-( X) - 2
io -o =0
)(40-
lhtn i \b~()(tiM ~()() == bx.,o+
2
4. (18 points) Consider the function f(x) that satisfies the given eonditions:
limx-+o- f(x) = 00, limx-+o+f(x) = -00,
limx-+ı- f(x) = -00, limx-+ı+ f(x) = 00,
limx-+±oof(x) = O ,
f'(x) < O on (1/2,1) U (1, (0),
f'(x) > O on (-00, O) U (0,1/2),
f"(x) < O on (0,1),
f"(x) > O on (-00,0) U (1,00).
(a) Find the horizontal and vertieal asymptotes of f(x).
+bi20ntol aS8mp1vtG 8:::0'X-=D and
(b) Determine where f is inereasing or deereasing.
(c) Determine where f is eoneave upward or eoneave downward.
f-U >0 0(\ (-')O (O) U ( LıOC)
ftı Co on
+kre. -f- is ccnca.e) upword
fulL..) f fs (O flGQVe dOU)(\uJ;:)rd
o o f the following function foo ) FO d the derıvatıve o5. (20 poınts ın
t1eXX -C)eX 1 heX(x ıı..f. i(x) == O + 4X +-. :. =- 4 X + oj __
X~ X~
(b) f(x) = )ı- ...rx -=- ( ~ _ \/7 ) 1/2..
f i ( x) = ~ i i . 0_-12 VI-'fi7 2Vi'
-1
-2x-to(2k-r-l) (X-I)
(d) f(x) = (~r8::-(7))( ~
In.:J :::- x. In -~ z: _ :xInxx
~. ~(::: - ( (Y\X -+,lt)- ) = - j -Inx
lhtn ) y / =- Id. ( - f - ınx)
b(x-I)1 - LI (lx-H)
(2)(+ i) (X-ı)
. ti to find dy if6. (16 points) (a) Use implicit differentıa ıon dx
sin(x2 + y) = -eY cos(x2).
~i ( LcS(X2+!) ) +e~cos(L(ı ») cc 2.)( e!oJ sin ( !(.) _ 2xcos (!(l+y)
~/::::. LxeY.s{n(x2) -2)(Go.slxl+0)
(os (xl+y) +eY co s (Xı.)
. - x - 3 at the point (2, -1).(b) Find an equation of the tangent lıne to the curve y - x2 _ 3
'1. CXL.-3) -( X-3) . 2x
(x'2-3)'L= t:) =-m .
ot (2,-1)