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**Unformatted text preview: **MA 113 —— Calculus I Fall 2010
Exam 1 September 21, 2010 Answer all of the questions 1 — 7 and two of the questions 8 — 10. Please indicate which problem
is not to be graded by crossing through its number in the table below. Additional sheets are available if necessary. No books or notes may be used. Please, turn
off your cell phones and do not wear ear—plugs during the exam. You may use a calculator, but
not one which has symbolic manipulation capabilities. Please: 1. clearly indicate your answer and the reasoning used to arrive at that answer (unsupported
answers may not receive credit ), 2. give exact answers, rather than decimal approximations to the answer (unless otherwise
stated). Each question is followed by space to write your answer. Please write your solutions neatly in
the space below the question. You are not expected to write your solution next to the statement
of the question. Name: —/’%L—*__— Section: Last four digits of student identiﬁcation number: Total I |-‘ r—1 r—- H H I
r—Ai—a
COGS) -- (1) Consider the functions f = -;—6 and g($) = x2 + 2.
(a) Compute f (9(5)) and g( f (5)). Give exact answers. 1 s @ [ {/5331}: ((27): 11131 0 [ 3({(€))= 3&5}; =3(-‘} 7:3; (b) Let h be the composite function h(a:) = (f o Find the domain of h. As usual,
justify your answer by showing your work. ‘ 1%!) = ¥(5{x)) = {{xzw‘zu/
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(a) f(g(5)) = g*, g(f(5)) = _3_.__ V (b) Domain of h is Maj—— (2) (a) Solve the equation 5z+2 = 7. Show all steps of the computation and give the
exact answer. u. 'Tqﬁz‘ma Away mdoeé x-r-Z 2 tags '7, @ Mm x:107€7-2_- (b) Express the quantity 1
10g4($6 + 1) — 10g4(955) ‘l' 510g4($) as a single logarithm. _‘ 71; 14m 6; layoff/CM Mm‘oQ
103‘! (X614) " 103‘, (7x24'é1o3? (X) @ =/959(7;:‘(]+/o§§/§K) _ (x64!) FX/
._ ﬁx (a) Solution is ﬂig— (b) A;— 3 (3) Consider the function
_ 2:3 —— 4 fwd—53:“. (a) Find the domain of f. “ " x .L
@[ l0 $K+(20( “MC/to X:“S~' @Z "W A AM or If» 1x / wéé = (wnéM—aw (b) Find the inverse function f “1 of f. ~ 7: =2X‘LI 0w 0W ; M v ((x/ 5W x IM—x y(5x+/) = 2x41, Mm X {S‘y—Z} = ~y—9. z
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a limit does not exist, but is 00 or —00, then clearly indicate that. I ._.
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thﬁau MA M4¢-Jf¢v0.o( Lani/g an cor/WA / @ xe‘ztua‘z 4m “'01 W191, ® (5) Let f and g be two functions such that the following limits exist limg(a:) = 7, lim [3”‘f(x) — = 13. z—>2 :c—>2 Use the limit laws to compute the following limits. ‘ L‘M (K)
(a) mg?) 2 z<_:z~_§___.__.. = 37 1-;
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L23 . ._—._—. (01017.3:- ﬂu c7M~°ﬁw (6) A particle is moving on a straight line so that after t seconds it is s(t) = 5t + 1 meters to
the right of a reference point. (a) Find the average velocity of the particle over the time interval 1 g t S 2. A _. ( .—
® All )0 1‘9 “S (1/ S U = “‘“—"H 6 = 5 r}:
2 —~( I ==s (b) Find the average velocity over the time interval [1, t], where t > 1. Simplify your
answer. (c) Use your answer in (b) to ﬁnd the instantaneous velocity of the particle after 1 second. A ( F.
@ [ TA('g (‘3 A‘M M g g :2 5 '3‘“; __ é*( _,
6 7/ (D f l G (D (a) Average velocity over [1, 2] is 5 %
(b) Average velocity over [1, t] is 5 %
(c) Instantaneous velocity at time t = 1 is 5 § (7) Using the deﬁnition, ﬁnd the equation of the tangent line to the graph of the function
f($) = :32 —— 555 at x = 2. Write your answer in the form y = ma: + b. q 010/“ 01 M4 9K X'al 1'; ® {’(U=AWW=OM ’4—70 A A—MD A
WMML—w—sm +6
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0 #— Equation 0f the tangent line; MM Work two of the following three problems. Indicate the problem that is not to be graded
by crossing through its number on the front of the exam. (8) (a) Deﬁne what it means for a function f to be continuous at (1. Use complete sentences. <9 i A Ame/Vow 15 A“) (WAMQM aL a (‘f A‘M “alga , Y—J (1 Let
03”” — 16, if cc < 2,
f(:v) = 5, if a: = 2,
x2 — c, if x > 2. As always, justify your answer to the following problems! (b) Find all values for c such that lim f exists. (is—>2 X \
Jag“ c3 -/6 4040‘ xZ—~< art COHAMQO’M. Mama; ,0“ x I MM " Au“ /(x/—. u (c—Z"~/6) = C-SZ—Jé : 7c -/6 «N4 Kai" X“?—
Z @ I: (on ((322 A». (kl-c/ 2 2 —c 2 (1“C-
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TA our-mun! A'Mn‘l: aw 2/; 7c—I6 :Q—c, (fig (Dc: +2.0 [/1 cal C3) [ [ﬂu Aim X4035 1% (:1. C29 Y ~72. (c) For which of the values for 0 found in (b) is the function f continuous at 2? C29 / 2! {Ila (wéhud‘b 0‘ 27 My. AM Mw‘ Mctbﬂ. ﬂ¢ﬂ(6],/QU‘, {99% C22. X-JL s3; 7m r! caa, m ,4»- !W 71*" =1 3‘ $— ﬁﬂlﬂ We 1" '3 MW \<~)Z (WK/mum 0/- 2 . ((1) Find all values for c such that the function f is continuous at 1. _. X
.ﬂﬂ‘q YX/Q ch'M-ccm I Mu And-“w Cg -—/6 1'9 chs‘Mum ,{N was? c_ @[ No“: { r3 (oh/swam cl /4me ﬂumévc (b) _L__ (c) MW (d) (R (9) (a) State the Intermediate Value Theorem. Use complete sentences. 0) CD a
F 3% 0- ,ﬁMO‘VN «K 4‘: COM/{Mum om. Ck (Xm4 Juk/VGX [6/67 mud N I"
Q) My MtM-u‘u rfo‘dé 41(0/ Md 15/5) ( W M "3 Q/um6—d c A, a WM 2qu (4,5) mac. Idol (Kc/aw,
0 0 (b) Explain in detail Why and how you can use this theorem to show that the equation 2m—3\/§5=1 has a solution in the interval (1, 4). @l- (aw-‘0‘“ K4 Mo‘rw «((X/= ZX—gr’: fir/“q law/ml: (:0! M4 {19¢ M0414: on. Cm/tﬂam an Mair dM/Aa( {N r
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l “Iv/«Mu. 490m C 4;“ (“9/ out}. #o/ (={(G/ a 2 “Br: ‘ (D t: c fa ML 00297114 oWﬁ‘w. 10 (10) (a) State the deﬁnition of the derivative of a function at a point a. Use complete sentences. ‘ _. \ —- {Q} t . i(¥)~f(q/
A ‘ a ' 44 to A’AM [(—9qu! J ( ‘9 ’ U‘dt“ AN“
C; A (/u alt/n0 60c v; ,{ a A—qo A or y. H}, ‘0‘)“ x—q / ” mm'oac! M—c 4‘4“)! xxx/06 . r q 6 (— 3“ 1‘2 lavage: I" Wﬁw Mat/W. fwo Aha/é. (b) Using the deﬁnition, determine the derivative of the function 'f <0
W): 96+ 1””—
7 if$>0.
atx=—1 andax=0ifit exists.
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- = -M .——————- \K ' crud [cm ¥ K) I‘M
3(ng— lﬂx} Kf20— le L kqof X—w-l Ll ﬂow {0:} at»; MY will. [Av-u M4 au¢—.n‘a£w( Iv‘pacé ave 01c X—ao <2) L— ”cucc fio «ﬂ “VII-kaum 0" 01/40» 15’? “’7 WW“"°‘"‘ “'4 0‘ f’(-1)= M4 f’(0) = W5 11 ...

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