# This gives the following graph b the graph of the

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This gives the following graph.(b) The graph of the functionh(x) =ex/2isobtained by scaling the graph of thefunctionf(x) =exhorizontally by afactor of 2.This gives the following graph.15
Exercise Booklet 3Solution to Exercise 35(a) Taking the natural logarithm of bothsides gives3x= 70ln(3x) = ln 70xln 3 = ln 70x=ln 70ln 3= 3.87 (to 3 s.f.).(An alternative method is to take thelogarithm to base 3 of each side, whichgivesx= log370 = 3.87 (to 3 s.f.).)(b) Since 81 = 92= 34, you can immediatelysolve the equation to obtainx= 4.(The approach in part (a) would givex=ln 81ln 3= 4.)Solution to Exercise 36ln(12xy3)= ln12+ lnx+ ln(y3)(a)(by the first law)= ln(21)+ lnx+ ln(y3)=ln 2 + lnx+ 3 lny(by the third law)= lnx+ 3 lnyln 2ln3x212e2x(b)= ln(3x212)ln(e2x)(by the second law)= ln(3(x24))2x= ln(3(x2)(x+ 2))2x= ln 3 + ln(x2) + ln(x+ 2)2x(by the first law)Solution to Exercise 37(a) Letf(t) =aekt, whereaandkareconstants.We know thatf(6) = 500 andf(8) = 5000, soae6k= 500(1)ae8k= 5000.(2)Dividing equation (2) by equation (1)givesae8kae6k=5000500e8k6k= 10e2k= 102k= ln 10k=12ln 10 = 1.1512925465. . .Equation (1) can be written asa(e2t)3= 500,and substitutinge2k= 10 (from themanipulation above) into this equationgivesa×103= 500a= 0.5.Therefore the functionfis given byf(t) = 0.5e0.5(ln 10)t(2t12);that is, approximately byf(t) = 0.5e1.151t(2t12).(b) The expected number of bacteria pergram after 12 hours isf(12) = 0.5e0.5 ln 10×12= 0.5e6 ln 10= 0.5(eln 10)6= 0.5×106= 500 000.(c) Every hour the number of bacteria ismultiplied bye(1/2) ln 10=(eln 10)1/2= 101/2=10= 3.162 (to 4 s.f.).16
Solutions to exercisesSolution to Exercise 38(a) This inequality is true, as it is obtainedby multiplying both sides of the originalinequalityx >2 by the positivenumber 4.(b) This inequality is true, as it is obtainedby subtracting 2 from each side of theoriginal inequalityx >2.(c) This inequality is true, as it is obtainedby multiplying both sides of the originalinequality by the positive number12,(d) This inequality is false. Multiplying theoriginal inequality by1 givesx <2,and it follows that the inequalityx >2 is false.(e) This inequality is false. Multiplying theoriginal inequality by12givesx2<1,and it follows that the inequalityx2>1 is false.(f)It is not possible to say whether theinequalityax >2ais true or false, as wedo not know whetherais positive ornegative.Solution to Exercise 395x+ 2<2x+ 11(a)3x+ 2<113x <9x <3.The solution set is (−∞,3).5x22x11(b)3x2≤ −113x≤ −9x≤ −3.The solution set is (−∞,3].x23>1x2(c)x2>33x25x2>5x >2.The solution set is (2,).Solution to Exercise 40(a) Rearranging the inequality givesx2x >6x2x6>0(x3)(x+ 2)>0.The graph off(x) =x2x+ 6 isu-shaped, withx-intercepts 3 and2.So the solution set of the inequality is(−∞,2)(3,).(Alternatively, instead of considering thegraph, you can find the solution set byconstructing a table of signs, as below.)x(−∞,2)2(2,3)3(3,)x30+x+ 20+++(x3)×(x+ 2)+00+(b) Rearranging the inequality gives2x2<2x2>1.Asx2is always positive, the solution set

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Term
Summer
Professor
NoProfessor
Tags
Order theory, Complex number
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