KOÇ UNIVERSITY MATH 102 - CALCULUS Midterm I April 13,...

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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)