This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math142 Midterml. You must Show all of your work to receive full credit. You have 50 minutes in which to complete the exam. No calculators; The number of points for each question is given in the
margin. Name: .  ,....' u an, 1004,  . (a? 1. Find the area bounded by the curves y = m, y = i, :1: 2 1 aide; e. I H MN“ I . 1— E = e g mew3‘s) we“? ' 6 AL—
I \
2. Given that f(2) = 8 and f’(2) = for a certain function f, determine (f‘1)'(8). @ £53: 0WD: 4. Scientists working in the Isle of Man ﬁnd a large, ancient ceramic pot at the bottom of a mine shaft 100m deep. They wish to lift the pot out of the shaft using rope and a ® winch. Unfortunately, the Winch they have will only do ZOQOOOJ of work before failing. _ The scientists want to use the strongest rope possible (as this will reduce the chance of the rope snapping). If the pot weighs 1000N, what is the maximum that the roEe can
weigh per meter that will allow the Wincrﬁh raise the pot to ground level? (Hint: Let the weight per meter of the rope be c. Then ﬁnd the work needed to lift
the ot and there e. Then solve to ﬁnd c.
P P ) Ft‘oaﬂN. 1‘0an WM Er M) C 0
3 wt) to a
i gnu
View 0 :ﬁtau“ I
:'.":':>';
E (53 100 M“
7’ 6 Page 2 5. Find the volume of the solid obtained by rotaing the region bounded by y = —:n2 + 1 and y z 0 about the line a; = 2. _
 \f: femgemi W35 0 1  2 ‘
[31" L I \ i : .—
K‘l r”) uik zri: 0+2] \5
6. The base of S is the region bounded by the line. g + 321 = 1 and the :L' and y axes. Cross
sections of S paralell to the x axis are semi—circles. j
I I—’
@ (a) Draw the base of S. g 2)
3 * =2 3
/ k aﬁ'ﬁu
l 0am Di?!wa L:
(b) What is the diameter of the semi—circle paralell to the :caxis at “height” 3;?
2? wartime W DioiHQtUé.
a: (yam >5 =D E334
V /\
q a
‘32» [5: . . . . 1 v
/ /’1 (c) What IS the area of this semielrcle? r Tf C r 7 g‘ i a,
i”? \b 2 L '{ﬁ'irgjﬂj‘l’kk ‘1
'3 4 t 2
“v 2, n , 2 2,9 ,I’a‘bi'i‘q
“6‘ o . g, 2 1‘ T a 1 I x
., 34: 1‘ : a '3; .I «3 1
1%; “[3bj E? 1 g 1c.) I 3 j 7 a
"lilo (d) What is the volume of 3? TT Lab3 — 12? *r 3 4”] ' 1:04, \ i ’ “Ems “ital +31] ' \I=TU LZ' 2x ML : ;E X" ﬁlo ‘5 (wigs) 3.543237%;zm My) 2 Y” 7 0’
‘ Pa. e 3 (11: x5 u 2—» g
a x; g ,3; I Bonus Questions
1. Find the average value of ﬁx) 2 V1 —— m2 on the interval [0,1]. i b 1 I I’ I’ \J’
_—H :2.
—— \ ,e 3 — ~\x
W jfaxmx 21> I’D NW2. ax Joy) c
0" o
l
I ' 2— [Z—  71
{in ‘3‘“ '_3 x 72) o
0
i f» . . s 2%
. i'rlclx'i EY'XJ‘ E 3L3) .L: 25%
D ‘5 0 q
I
. r—
ingg 2.33? :3.Fa
3' ‘5 2 a 4?“
F"
#M (13;; 343
G.
if (m) nﬁ
2. Find
[4  . , 1 z 2
j [log2 $)(21) + $1n2lm )da:
(Hint: think about the product rule and the fundamental theorem of calculus.)
\ "1
J‘O‘lz" (1*) *‘f‘: WV”
x111 .,
UT: peg.
‘ x l I ﬂaw"? ‘3»: U!
:jiadjng.'1”xj dx i" « 5x1) m m
x111? 31¢." «ML
2,
4"“ ,  f .
>< ’J llﬁdw—jiwiii U“) M
A n'. L .  1
L11 > la},
I ,_ 2.
%.1X 4} CM 4 U I i. L U ck) : g’jilﬂg’ ._L a”
2)) "W a»: if x. wt )3 W ? é J QKdJL
& \nrv
—’ 1
Earémr xtﬂérihc 3
3 2' 1 21']: Z, +L
%LZ+3 ' 5X?" EEXDX b‘
X 7‘; To ' ﬂ. Math 142  Midterm 1 ~ Formulae Sheet 0 The areas betWeen the curves 3; = f(a:) and y = (m) and between .7; = a and x = b is _
A = INC) “ 9(xlldir 0 Let S be a solid that lies between a: = a and :c z b. If the cross—sectional area of S in
the plane through a: and perpendicular to the xﬁaxis is A(a:) then the volume of S is V j: A($)d:r. o The volume of the solid obtained by rotating the region under the curve 3} = f (m) from
a to b about a vertical axis is b A
V = / 27r(radius of a shell at m)(height of a shell at whim 0 Work z force >< distance. 0 Units of force are N = kg.m/32 (metric) and lb (imperial). 
0 Units of work are W = N m (metric) and f H!) (imperial). o Acceleration due to gravity: 9 2 9.8m/32. 0 Suppose that an object moves along the xaxis from a to b and that a force of f
acts on the object. Then the work done in moving the object from a to b is W = f(:c)dx
' d 1
EEUOE“ m) = :1: in a
0
d ...
View
Full Document
 Spring '06
 staff
 Math

Click to edit the document details