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Unformatted text preview: MAC 2311 Spring 2010
Exam 4A A. Sign your scantron sheet in the white area on the back in ink.
B. Write and code in the spaces indicated: 1) Name (last name, ﬁrst initial, middle initial)
2) UF ID number _ p 3) Discussion Section number C. Under “special codes”, code in the test number 4, 1.
1 2 3 a 5 6 7 8 9 0
e 2 3 4 5 6 7 8 9 0 D. At the top right of your answer sheet, for “Test Form Code” encode A.
e B C D E .. E. This test consists of 9 ﬁvepoint and 4 twopoint multiple choice ques—
tions, bonus questions worth 6 points and two sheets (4 pages) of partial
credit questions worth 27 points. The time allowed is 90 minutes. F. WHEN YOU ARE FINISHED: V 1) Before turning in your test check for transcribing errors. Any
mistakes you leave in are there to stay. 2) You must turn in your scantron and tearoff sheets to your discussion
leader or proctor. Be prepared to Show your picture ID with a legible
signature. 3) The answers will be posted on the MAC 2311 homepage after the
exam. Your discussion leader will return your tearoff sheet with your exam score in discussion. Your score will also be posted on Elearning within one
week. ,5. = ‘ a ' ,xgtm ,
d If, ‘9 K1: r' é 2L.Iv / NOTE: Be sureto bubble the answers to questions 1—16 on yourfscairi’tfnéii. Problems 1  9 are worth ﬁve points each. 211 _ 2 an 1
1. Use L’Hospital’s rule to evaluate lim iii—i.
w—>O cos(2:z:) — 1 a. 2 b. 0 c. —— d. —1 e. The limit does not exist. 2. A particle moves in a straight line so that its velocity at time t is t2 — 216
feet per second. Find the total distance traveled by the particle on the
time interval 0 S t S 4. 2
a. ~33— feet b. 12 feet c. 8 feet d. 1—36 feet e. 32 feet 3. Find the area of the largest rectangle which can be inscribed in a right
triangle with legs of length 3cm and 4 cm, if two sides of the rectangle
lie along the legs. Hint: find the equation of the line. 9
a. 3 cm2 b. cm“ c. 4 cm2 d. 5 cm2 e. — cm [\DIOJ 2A ﬁ 3
4. If : f ' 2 dt, then F’(:1:) =
1 :1: 1133/2 d 113—1 '233—4 ':::—2 0'2332—4 ';:;—2 '293—4 5. Approximate the area of the region under f = — on the interval [1, 3]
a: with a Riemann sum using four subintervals of equal Width and letting be the left endpoint of the subinterval [$i_1,$i]. a 143+? b §+2+2 1+§+é
'2 6 5 '2 3 5 C 3 5
4 4
d.213 .3 — —
n p e +3+5 6. Find the absolute maximum and minimum values JV] and m of
f(:1:) = {M2 — 1 on [—1, 3] and use the Comparison Property 3
to ﬁnd lower and upper bounds p and q for / f d2:
—1
3
(that is, p 3/ \3/ :32 —1d$ S q).
—1 a.p=—1andq:4
b.p=—4andq=0
c.p:0andq:8
d.p=0andq:4 e.p=—4andq:8 3A f(0) = —1. Find f taro 13.3?” 0 0—1 d.3 e.—3I——2 ‘ 251:2 + 1 x < 0
8. Find the area of the region bounded by f :
el' :3 2 0
and the x—axis on the interval [—1, ln 2].
5 8 10
. — b 1 . — d — 2
a 3 C 3 3 e 9. The population of a new settlement is growing at the rate dP
E : 50153/2 — 4015 where P is the population of the settlement 15 years after it was founded. If 1000 people moved into the settlement initially,
what is the population 4 years later? a. 1340 b. 2880 c. 1960 d. 1320 e. 3720 4A Problems 10 — 16 are worth 2 points each. _ 2
10. Evaluate: fwdmz
as
6 9 9 6 9
a.l——+—+C’ b.—6lr1l:13———+C' c.a;+———+C
:1: 51:2 :1: 51:2 333
9
cl. :1: — 6111— — + C e. None of these
a:
7T/42 2 _
11,] 2;;de
0 cos2:1:
7r 7r 7r
—1 b. ——l —— .— .
a 2 c 4 d 2 e O 0
12. Evaluate / + 1 alas. You may want to Sketch the region represented
—3
by the integral. 5
a. E b. 2 c. d. 3 e. l N100 6 6
. 13.1f/ dry: lOand/ f(:13)da::6, then /1 3f(:13)d:13: .
1 3 3 a. 16 b. 4 c. —12 d. —16 e. '12 5A Bonusll (Two points each) cos513+513 14. Evaluate lim , .
.T—HT/2 s1ncc + 1 a. g b. 00 c. 0 d. —oo e. Use the graph of the function f sketched below to answer questions 15
and 16. 15. Which of the following is a possible graph of an antiderivative F of f? 16. Suppose that /2 f(:I:) d3: = /6f(a:)da: :2 4 and /4 f(:z:)d:1:‘: —1.
0 4 2 If F (0) = 2 for some antiderivative F of f, then use the Fundamental
Theorem to ﬁnd a. 11 b. 5 C. 9 d. 7 e. 10 6A MAC 2311 Exam 4A Spring 2010 Sect # Name
UF ID Signature
SHOW ALL WORK TO RECEIVE FULL CREDIT. 1. A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the page are to be 1— inches, and the margins on the left and right are to be one inch. What should the dimensions of the page be so that the least amount of paper is used? Let :1: and y be
the Width and height of the print area as shown. Function to be maximized Constraint
:1: = in.
y = in.
Dimensions of the page: in. by in. Use the Second Derivative Test to conﬁrm your results. 7A 2. If f(:1:) = I. Determine the following. Leave function values in terms of e. If none,
write “none” . (a) domain of f vertical asymptote: a: = (b) Evaluate the limits, using L’Hospital’s Rule if necessary: 1) lim f(a:) = ' (IF—>00 2) 1n_n ms) = {B'—)OO (c) horizontal asymptote(s): y = (CI) number lines indicating intervals of positive and negative values for f’ and f”: fl m f” (e) relative(local) maximum at a: =
maximum value:
relative(local) minimum at a: = minimum value:
(f) inﬂection point(s): (g) intercept(s): 8A' 93—1 ea: 2. (continued) II. Sketch the graph of y = from part I. Clearly indicate all important features of the graph. 1 2
Note: —2 R: 0.14 and —3 Rs 0.10.
e e 3. Use L’Hospital’s Rule to evaluate lim(1 + 3tan x)2/”". z—rO 3
4. Find a Riemann Sum which approximates / (332+2) dm using 3 subinter—
0 . vals of equal Width and letting x2“ = midpoint of the subinterval [xi4, 9A 5. Consider the area under the graph of f = $2 + 2 on [0, 3]. (a) Set up a Riemann sum which approximates the area using n subinter—' vals of equal Width with = right endpoint of the subinterval [n1, 231]. 1:1 (b) Find the exact area under the graph of f = $2 + 2 on [0,3] by
evaluating the limit of the Riemann sum as n —> 00. n  “(71+ 1) n 2 “(71+ 1)(2n+ 1) n 3 “(71+ 1) 2
N13: =——————— =———__———__—.. = __
oe £2 2 ,;z I 6 ,;z 2 Area = 10A MAC 2311 Spring 2010
Exam 4B A. Sign your scantron sheet in the white area on the back in ink.
B. Write and code in the spaces indicated: 1) Name (last name, ﬁrst initial, middle initial)
2) UF ID number 3) Discussion Section number C. Under “special codes”, code in the test number 4, 2.
1 2 ' 3 e 5 6 7 8 9 O
1 e 3 4 5 6 7, 8 9 O D. At the top right of your answer sheet, for “Test Form Code” encode B.
A a C D E E. This test consists of 9 ﬁvepoint and 4 twopoint multiple choice ques—
tions, bonus questions worth 6 points and two sheets (4 pages) of partial
credit questions worth 27 points. The time allowed is 90 minutes. F. WHEN YOU ARE FINISHED: 1) Before turning in your test check for transcribing errors. Any
mistakes you leave in are there to stay. 2) You must turn in your scantron and tearoff sheets to your discussion
leader or proctor. Be prepared to show your picture ID with a legible
signature. 3) The answers will be posted on the MAC 2311 homepage after the
exam. Your discussion leader will return your tearoff sheet with your exam score in discussion. Your score will also be posted on Elearning within one
week. NOTE: Be sure to bubble the answers to questions 1—16 on your scantron. Problems 1 — 9 are worth ﬁve points each. 3 — 1
1. Use L’Hospital’s rule to evaluate lim M".
z—>0 €33: — 362; + 2 a. The limit does not exist. b. —— c. 3 d. —— e. 0 2. Find the area of the largest rectangle Which can be inscribed in a right
triangle With legs of length 4 cm and 3 cm, if two sides of the rectangle
lie along the legs. Hint: ﬁnd the equation of the line. a. 4 cm2 b. 5 cm c. 5 cm2‘ d. — cm e. 3 cm x/5 t2 I
3.IfF(:c)=/1 2+3cit,then}7‘(:c):
a. (6+1 b ﬂ c. ﬂ (1. wig/2 e. m
2z+6 33+3 2513+6 2$2+6 $+3 2B 4. Find the absolute maximum and minimum values A! and m of = \3/ $2 — 1 on [—1,3] and use the Comparison Property 3
to ﬁnd lower and upper bounds p and q for / d1:
—1
3
(that is, p 3/ \3/5132 — 1dcc S q).
—1 a.p=—4andq=0 b.p=0andq:4
c.p:0andq=8
d.p:—4andq:8 e.p:—1aiidq:4 5. A particle moves in a straight line so that its velocity at time t is t2 — 215
feet per second. Find the total distance traveled by the particle on the
time interval 0 S t g 6. 1 2
a. g feet b. 24 feet c. 36 feet d. 9? feet e. 2—8 feet 6. Suppose that f is a function so that f”(.'c) : — sings, f’(7r) : 2 and
f(0) : —1. Find f a.——2 b.3 c.— d.—1 e0
2 2 3B 2 7. Approximate the area of the region under f : — on the interval [1, a: with a Riemann sum using four subintervals of equal width and letting be the left endpoint of the subinterval [33211, 4 4 5 4 1‘5 2 .3 — — b1 — — .— — —
a +3+5 +3+5 c2+6+5
3 2 2
d.— — —~ .213
2+3+5 e n 8. The population of a new settlement is growing at the rate dP
—— = 50753/2 e 4075 where P is the population of the settlement 75 years dt
after it was founded. If 2000 people moved into the settlement initially, what is the population 4 years later? a. 2320 b. 4720 c. 3880 d. 2960 e. 2340 2:62 + 1 a: < 0
9. Find the area of the region bounded by f :
~ ex :1: > 0 and the asaxis on the interval [—1,ln2]. 8
a. — b. — C. 2 d.
3 03101 e.1 4B Problems 10 — 16 are worth 2 points each. _32'
10. Evaluate: /($—2)d$=
m
6 9 6 9 9
a.$+—2——3+C b.1——+—2+C C.$—6lnl$I———+C
m m  m a: m 9
d. —6ln — — + C e. None of these
cc 0
11. Evaluate / $ + 1 elm. You may want to sketch the region represented
—3
by the integral. 3 5
a.2 b.3 c.— (1.1 e.—
2 2
12_/”/4E§Eié£:_1dm:
0 cos ct:
a E b —E C. 0 d ——1 e —1
2 4 8 8 1
13. lf/ dm : 12 and/ da: : 8, then/ 2f($) dz: : .
1 4 4 a. 4 b. 16 C. 8 d. —16 e. —8 5B Bonusll (Two points each) os
14. Evaluate liin :r—nr/Q 811153 + 1 a. E b. 0 c. g d. 00 e. ~00 Use the graph of the function f sketched below to answer questions 15
and 16. 15. Which of the following is a possible graph of an antiderivative F of f? 2 6 4
16. Suppose that / f(:z;) d1; =/ d1; :2 4 and / f(:c) dzc : —1.
0 4 2 If F (0) : 3 for some antiderivative F of f, then use the Fundamental
Theorem to ﬁnd F ‘ a. 9 b. 12 c. 9 d. 7 e'. 10 6B MAC 2311 Exam 4B Spring 2010 Sect # Name
UF ID Signature
SHOW ALL WORK TO RECEIVE FULL CREDIT. 1. A rectangular page is to contain 48 square inches of print. The margins
at the top and bottom of the page are to be 2 inches, and the margins 1
on the left and right are to be 1 — inches. What should the dimensions of the page be so that the least amount of paper is used? Let a: and y
be the Width and height of the print area as shown. Function to be maximized Constraint
3: = in.
y = in.
Dimensions of the page: in. by in. Use the Second Derivative Test to conﬁrm your results. 7B 2 If me) = 1 — CC :6 — 2 3 _ IT l i, A»
then f’(:c) = — . l, 3’ €11 , em ‘1 4*"
I. Determine the following. Leave function values in terms of e. If noné;
write “none”. (a) domain of f vertical asymptote: cc = (b) Evaluate the limits, using L’Hospital’s Rule if necessary: 1) lim f(:c)= I—l—OO 2) lim f(x) = :L‘—}OO (c) horizontal asymptote(s): y = (d) number lines indicating intervals of positive and negative values for f’ and f”: fl m fl! (e) relative(local) maximum at cc 2
maximum, value:
relative(local) minimum at cc 2 minimum value:
(f) inﬂection point(s): (g) intercept(s): 8B 1—3: 2. (continued) II. Sketch the graph of y = , using the mfg  62‘.
from part I. Clearly indicate all important features of the graph. Note: —2 £6 ——0.14 and ——3 m 73.10. 6 p e 3. Use L’Hospital’s Rule to evaluate lim(1 + 4tan 3:)3”. 2—)0 3
4. Find a Riemann Sum which approximates / (3324—5) do: using 3 subinter—
o vals of equal width and letting = midpoint of the subinterval [$i_1, QB 3"?
f
if 5 a] i ‘L
If
.
~§§‘¢/ v 5. Consider the area under the graph of f = $2 + 5 on [0, 3 (a) Set up a Riemann sum which approximates the area using n subinter—
vals of equal Width with = right endpoint of the subinterval [rm4, 'n.
i=1 (b) Find the exact area under the graph of f = $2 + 5 on [0, 3] by
evaluating the limit of the Riemann sum as n —> oo. 71 Note: : Ego—2w, 27:2 = i3 : [7101+ 1)]2 Area = 10B ...
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