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Unformatted text preview: Math 123 Calculus I Name: §OLUT70N§ “ BLUE. Exam I Fall 2007 RWJ
No calculators allowed, one page of notes allowed Instructions:
0 Read questions carefully 0 Help me award you (partial) credit by showing your work (except on problems 4
and 5) 0 Note point values on questions — I 05 points are listed, 1 00 points is possible
0 Check your answers if you ﬁnish early
0 The exam ends promptly at 2:52
0 Good luck! 1. / 15 2. / 28 3. / 20 4 / 10 5 . / 8 6. / 9 7. / 15 /105 l. (15 pts.) The position of an object at time x is given by f(x)=\/x+l a. Find the position of the object at time x = 0. Also ﬁnd the position of the object at time x28. nC/o)=J\—=[y
«9/804?— =(3) b. What is the average velocity of the object over the time interval [0, 8]? 41/9} We) 3:: l c
’ ‘ Q '0 87,0 0. Give an expression for the average velocit over the time interval [x, x + h].
’P/X*L)'L(x)  J 7(4'L\*I  x44 ’ (1. Using your answer from part c (and not any “shortcut” methods) determine the
instantaneous velocity of the object at time x. \jxvrh’rf ’ MI mel JrJ'xH _______________________. In J‘kafl l'\j7<+l
Qwhqtoun) ; \ ______’.__————————— In 4—5:) {\l'wa] 4—5;?) 4A ﬁ/xru) ﬁ/x) __ A J , ﬂ; k (m MEI) @‘ 2. (28 pts.) Compute the derivatives of the following. Don ’t waste time simplifying. a. 3x7 —6x3 +5x—10 " lgx?’* 5 b. (2x6 +5x4 +3x2 —x+7)sinx (/276‘4' Zoxat’éx—(D 9&1: +—
(ZxékE'qu— hm'xr7> WX 13x5—4x3+3x+1
x3+2x—7 (45%+— /’2_ 'x" #3) {763+1x—7) ——(/37(5— f‘x3+3x%t)(3x"+z)
_//_—__—_’__________________—_
(W3 f'Zx—7)L C. d. tan[(5x2 + 3x +1)“2] .vI/l %z[(5xzf3X+l)btj  2%(5’X1437ﬁ4'l) (/ox43) 3. (20 pts.) Find Don ’t waste time simpliﬁzing.
X x3 sin x a. y=
x2+x+3 (aklﬁixk kgmvc)('x7’+‘x+})— 6(3ﬁx)(7.7cﬂ) 4% (>9 ,L x*3)L b. y = (6x3 + 5x+8)10(3x2 + 2x+1)8 4%: ; /0(6x34’5ka)q(/9x"/'5) [3x7'4—1x4rl): 7"
[Aﬁrhkﬂw 5’(3«14—1x4—1)7 (4x44) dy 0. xy2 = sin(xy) (your expression for d— will be in terms of both x and y)
x
,4— (‘PC 1 ’— ; EI'LC 1047
47c ’7 ) x U) 7" Ir X g— (o‘)  mew)  :1: (my)
Ax Jx Ax ' 1207 _. xm[¢7) 4. (10 pts.) For each of the following indicate Whether the statement is true or false (no
work necessary): a. If f is continuous at x, then f is differentiable at x. True oircle one)? b. If f is not continuous at x, then f is not differentiable at x. or False (circle one)? c. For any function f, lim f (x) = f (0). True o(circle one)?
d. If f and g are differentiable, then (fg)' = f ' g'. True ocircle one)? e. If f and g are differentiable, then (f + g) '= f '+ g' @ or False (circle one)? 5. (8 pts.) Fillin the blank: a. If f '(x) exists, then we say that f is '1‘ M'b £6 at x. b. If 1imf(x) = f(c), then we say that fis mﬁhuws at c. X—)C 6. (9 pts.) Suppose f is a continuous ﬂinction on [1,5]. Also suppose f(1) =—4
f(5) = 6 a. f necessarily has a zero (or root) between what two x values? [+5 b. If the bisection method is used, at what x value do we now evaluate f? 0. Suppose f is negative at the x value you gave in part b. Where do you now evaluate f? 7. (15 pts.) For the following limit problems:
0 If the limit exists, give me the (ﬁnite) value of the limit 0 If a limit does not exist, answer using 00 or —oo when appropriate, otherwise
write “Does Not Exist” a.lim "2—3 z A, x.) . A “J‘— — .L
H3" ‘9 >993 Q‘JX’V’” xa} Qua) é x2+2x—l _ (1)1440) —I [*4 b. lim
H1 x + 4 H MD" ...
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 Fall '06
 JOHNSON
 Calculus, Velocity, pts, Continuous function

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