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Unformatted text preview: MAT 21B—A, Spring 2007, Prof. Opmeel'
Wednesday Apr. 25, 2007. MIDTERM EXAM 1 NAME (print in CAPITAL LETTERS, last name ﬁrst): _______ ID#: __________________________ _..'_ Instructions: 0 Read each question carefully and answer in the space provided. If the space provided
is not enough, please continue on the opposite (blank) side and CLEARLY INDICATE this. 0 YOU MUST GIVE CLEAR AND REASONABLY COMPLETE ANSWERS TO CEIVE FULL CREDIT. Answers like ’yes’ or ’no’ or ’2’ will be awarded no credit.
Proper notation and (mathematical) readability of your answers play a role in deter mining credit.
. Calculators, books, notes or similar things are not allowed. 0 The numbers between brackets refer to the number of points (out of 100) for that
exercise. U H 1.[24] Find all antiderivatives of the following functions (i.e. ﬁnd f f dos). Indicate the substitution that you make (if you make one).
(a)l6l f (x) = 26’ — 38'“ S 1! (ze‘o 30 (b)[6l f (x) = 44? 3 4+6? Ax
x5 = SUNSr )5?“ 63*
4 {1+ ixdh + C
r; a 1.[continued] Find all antiderivatives of the following functions (i.e. ﬁnd f f dz). Indicate the substitution that you make (if you make one).
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WONT)" dw Z‘de
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’ 2 S u“ du
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»\ (d)[6] f(a:) = 6'”st + cosx SIX 71%: cos x> dx V 1 x3
Xngx +1 Saudi:
Al _ m: X3
' Au: Kldx
is Jr'SCOSXdK
3
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Q“ A. g‘mx +Q 2.[16] Evaluate the following integrals.
Indicate the substitution that you make (if you make one). (a)[8] f; (3a: — 2—3) dz: 4
3° ($X~ ﬁxﬁdx H .“al."‘l“ ’0 3.[18]
(a) [9] Set—up the area between the sicaxis and the graph of the function x2 — 4a: — 5 over the interval [7, 7] in terms of integrals. Do not evaluate. Do not leave absolute value signs in
your ﬁnal answer. “A: xtttxv's'
(32 (x, 6)CK&‘) : 0
X36! 3‘3" (b)[9] Setup the area of the region enclosed by the curves y = 9:2 — 2 and y = 3x + 2 in
terms of integrals. Do not evaluate. Do not leave absolute value signs in your ﬁnal answer. Xi; = 3M 7
Xl— 5x v“\ ‘0 4.[18]
(a) [6] Using Sigma notation, express the uppersum and the lowersum of f (2:) = 25:: obtained by dividing the interval [0, 3] into n equal subintervals. Do not evaluate the sums. we ’aI—‘i: .3.
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O'Fn n '2 3 ll 4. [continued] (b) [6] Suppose that the uppersum and lowersum for a certain function f obtained by dividing
a certain interval [(1, b] into n equal subintervals are, respectively,
n k k2 n—l k k2 L: ———+——.
n2 n3, [£112 n3 1 ~ (x
\\m (%*“3)~Z(%*:zl]
W ~ 0 “:0 Iv!
\Ldn n “3 1
.5. \c‘ ’ _\_ Efk L E l1
vElam 1 ~53 l1 + “3 2; n1 g” (‘1 La)
on ‘)
\‘m \ _ “(nH) + __L . nCﬂKan) ‘L Lnﬂ )Cnl _\_(_(\—O(nl(.2n
“mo ‘57 '2. n3 (a n1 2 n" l.
_\_ . \ + _l_ . 2. _l_.  _L . 2.
z b 2 e (c)[6] Does the result in part (b) Show that the function f is integrable over [a, b]? Be sure
to justify your answer. “We vvhoo
l‘ \n \
“\80, \mn < 970(de < \W‘ U
“an a “A”
/\
L____________§_T
‘m M m
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%0 El \3 wﬁrﬂ (Qb\€ Ode‘ [@001 . 5.[8] Show that the value of fol sin 2:2 dx cannot possibly be 2.
\ ‘ ‘
3 %\n x Au: 3 \ dx
6 O 
= x\.
f \
\
gsznx‘ax s; \ < 2. 6.[8] Suppose that f is differentiable on [a, b]. Is it true that f is the derivative of some
function on [a, b]? Yes i “Q ‘ 6%emo‘bu 0“ lad/)3, “\QWR ‘Q \S Conﬁnuogs 31“? V3 ::VI\\)S ‘\Y\\"€ afoul/A & ~ \ e (Kydx t ed (x\ I
t . .5) 29(x3 ickqxb. 7.[8] Calculate % 11”}? l‘ﬂ’n‘t dt. Hint: use the fundamental theorem of calculus and the chain
rule, do not ﬁnd an explicit antiderivative. \l ...
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