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Math 142 Midterm 1

# Math 142 Midterm 1 - Math-142 Midterml You must Show all of...

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Unformatted text preview: Math-142 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 mew-3‘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: fem-gemi W35 0 1 - 2 ‘ [31" L I \ i :- .— K‘l r”) u-ik 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 :c-axis 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 semi-elrcle? 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 Lab-3 — 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 (1-1: 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— i-ngg 2.33? :3.Fa 3' ‘5 2 a 4?“ F" #M (1-3;; 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 x-axis 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 ...
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Math 142 Midterm 1 - Math-142 Midterml You must Show all of...

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