t2.f.10-sol - MA 113 “4 Calculus I Fall 2010 Exam 2...

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Unformatted text preview: MA 113 “4 Calculus I Fall 2010 Exam 2 October 19, 2010 Answer all of the questions 1 — 7 and two of the questions 8 — 10. Please indicate which problem is not to be graded by crossing through its number in the table below. Additional sheets are available if necessary. No books or notes may be used. Please, turn off your cell phones and do not wear ear-plugs during the exam. You may use a calculator, but not one which has symbolic manipulation capabilities. Please: 1. clearly indicate your answer and the reasoning used to arrive at that answer ( unsupported answers may not receive credit), 2. give exact answers, rather than decimal approximations to the answer (unless otherwise stated). Each question is followed by space to write your answer. Please write your solutions neatly in the Space below the question. You are not expected to write your solution next to the statement of the question. Name: Section: Last four digits of student identification number: . , sine? (1) Gwen that $121] 6 work and give the exact answers. sin(7t3) : 1, use the limit laws to find the following limits. Show all your (a) i133} 4t3 ' l' l ( ——-—M (7%} ) ‘-—"' '2- th SW‘K ‘1 +3.3 (i) 79’ 3:39 4 90 X (D <0 E _ (b) lirrézrslntr:2 + 83) - cot(3:). I lulu “€le2+€3)((037{) mu S‘D‘X -- ‘ .1— ” l 1) P ‘— 132:0 9C“) ' pukflte) gaulébx) ' 2‘3.l‘ Ca. >60 7‘ ’xxmxfl) gal“) : " Dinl01+€3)‘c®o 21‘3-123 (2) 7 (a) hm sin(7t3) : H0 41:3 (b) lim 2: ln(a:2 + 83) - 0013(33) :flWB— z—#O 2 (2) Use the differentiation rules to find the following derivatives. Show your work. You need not simply your answers. (a) : sin(33:) + 2V5 + 111$. {L HM: C9$(3x\.3+ 2 J1. X l + 316 O J‘ .L 3 : QCmGX) f’x 2+ [X (b) Mm) = 334$: 9 a? ) 2x (x4+q)th(+K3) (gx‘i-QXF K 3 2 “A (M +602 (RM) 2. W 6) (c) M3) = x3e3x2+6x+1l XL” H 3KZ+62<+| Hm: 3X28 " x3e “(Si—Maxim W ('9 3X firm; 1 [NHL 7(3(6><+g)] 9 W -J. (grep—32M CD 18K~ZKS (b) g’(2:) =_me,)2____ (C)h'(m)= 3XL+O<3(6><+6) €3X+6x+1 3 (3) Use the differentiation rules to find the following higher order derivatives. Show your work. (a) Find f”(2:) if f(:c) = era. (b) Find g”(1) if 9(33) = (a: —i— 2) - Sinflx). Give the exact answer. 517x): [Simk'li‘xi +g+ZJ (oi (lid-12; 31,00: (53%Ixnglr -[— CODEEX).IC% ~($}1(K+2)§IAEX) I ’1’, :AE.‘...._~J; :. :5:ch (fx) ‘(%)Z(K+Z)Slil( X) @ 3% = 911mg} *W (1+2) 3 W?) 1 if? :— [JEJ-(W '3" a) (are): €X+QX4 9x3 (mm: %“3(I4E)2 Ag (4) A particle is moving along a straight line. Its position after 15 seconds is given by 3(t) : t3 — 3152 — 9t + 10 meters. (a) Find the velocity, v(t), and the acceleration, .105), of the particle at time t. We): Sim—suites) (D ode: ’U"(tl=— éJc—é (D (b) Find the time interval(s) when the particle moves in the positive direction. hardcle moves in to taste errata-at at avid out, F? *U‘R) > O . CD ‘ . me) = 3(+_,2‘-—21e——3)= 3 Gee) 0e +1) >0 4 wisely qt £> 3 ‘ HQVILL THUR film? l-Wl‘eflfdl 3w (3 Jlbo). (c) Determine the total distance traveled by the particle ging the first six seconds. ®{‘flne Powhle mom 5136Mva dwriwa (<1 , 3) M enema cLuF'i/nj (3 , 6). (2) Total ctr-stance = (3(3)- 302M +l 5(5)“ 5(3” CD { - l27~ 27~27+io ~to| +l 63-3'63‘téfl0—(27eyeyrsl .— 7: 27+8’l:\08' (d) Find the interval(s) where the particle is speeding up (that Iiieans, the time t where a(t) and Mt) have the same Sign). (b) Particle travels in positive direction during (C) Total distance is l O 8‘ (d) Speeding up during w . 5 meters (5) Consider the curve given by the equation 33y3 + 12.152 + :92 z 51. Assume this equation can be used to define y as a function of 3: (ie. y = f (39)) near (2, 1). Find the equation of the tangent line to this curve at the point (2, 1). Show your work and write your answer in the form 3; = ms: + b. Equation of the tangent line (6) As usual, show your work in answering the following questions. ,,. was“ xl-I / (af)F1ndf(2)1ff(:c)——1(Thkl) / r ( XL‘) -£(XJ= <va (K310) “Ram” ;F(x) 2 x’m @ { % 2X 4K3 0“ km) _ K1“ _‘ X4+j 1. 2X(x‘r+1)~4‘3(¢l) (xfiamw 3 <0 [2): 92 .fi “~22: % fat/1 2%»: W 85‘ (b) Find 91%) if 9(39) = tan(sin m). ’ z \ A? (gx @ i M 2 39(2( SAX) C031 ( _ 39;) = sesmw meg) _ l _ F 6‘ <2} gr -—— E: (or ’32: 3921—27?” (7) In this problem use the fact that the mass of a radioactive substance changes over time according to the formula m(t) = mac“, where m0 and k are constants. Show all your work and give exact answers. A radioactive substance known to be 600 years old is 75% decayed (that means, 25% is left). (a) What is the half-life of the substance, that is, after how many years was half of the original amount mg decayed? @ ’Yllo €166“): 95% mo : "limo J/ [ : Mt) I” O 6mg 1-“? (b) What percentage of the substance is left after 1200 years? {Jaime ~ {aim/«>100 Webb) = me = moe @ L .._ m ~—QQVI4 - o E l l 3 if}: 3;... mo ‘ a “W l W e (was... X (a) The halfnlife is Lyears (b) After 1200 years there is ___ J— percent of mg left. 0’. 29% 8 Work two of the following three problems. Indicate the problem that is not to be graded by crossing through its number on the front of the exam. (8) A particle moves on the curve y : 2\/§$3 so that its position after t seconds is given by PU) I (m(t),y(t)) = ($(t),2\/§m(t)3), measured in centimeters. As the particle moves through (1, 2fi), 55(15) is increasing at the rate of 5 centimeters/second. How fast is the distance from the particle to the origin changing at this instant? Recall that if d is the distance from the point P = (m, y) to the origin, then d = (£2 + 312.) The Millw— ST’WE (lit): xlt'cH‘ngEl 6 :f @ :K(f)l+ $249 D&aaamfiruustq (Mite Mt) 21%(6 059:) + $61“) S_’)<((’c) 4%) OVGE} _: x“) + Q4- «(i—l? cite) M fie PDuLf (1225)) we have ')((_+)=1, @ dim: VFHH‘ '1‘}?! :3 dx__ C ‘U?’ 5”“ g Heme AIR)": 3 I '5’ = _—3— (“/5 @ X%a <~G§ (9) (a) State the chain rule. Include all the necessary assumptions to make the rule valid. 1% fils figamiftal/le OKJL‘ X Midi di-REMQJ—mme (X) 144%“ fig f5 le‘geved‘r‘alégg cuE 1_ ngwer "£05"? (X) ‘2}“(3W -3’<x) 2 ( b) Suppose f and g are differentiable functions such that f(3) = 2, f’(3) = 4, 9(1) = 3, and g’ (1) = 6. Find each of the following: (i) h’(3) where h(a:) = ln([f(:c)]4). (ii) £’(1) Where 1(33) = flat“ - 0) Wm: M =4 @ Qi) fix): fixigm (figw/ E 69 (10) Consider the function f(:15') = tan(:r:) with domain (—g, g) and let 9(3):) be the inverse function of f a) Give the domain and range of g(;r). @‘Demm Hf 3: (we/we) 3 @“nge «f a=<€t in. (13) Use implicit differentiation and the formula seczy = 1 + tan2y to find ShOW that g’(:c) = You need to show all your work. Since “PMWX ) Implicfii Cii‘wr‘aifem "Tn/THU) @ E J H ‘5 I! 11 ...
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