00001 00001 100 100 00001 x x R x e xe 00001 1001 00001 x x e c

00001 00001 100 100 00001 x x r x e xe 00001 1001

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0.00010.0001'( )100100( 0.0001)xxRxexe=+⋅ −0.0001100(10.0001 )xx e=. c. 0.001'(10)100(10.001)99.800Re=, or $99.80/thousand pair. 58.The demand equation is p(x) = 100e–0.0002x+ 150. Next, p(x) = 100(–0.0002)e–0.0002x= –0.02e–0.0002x. a. To find the rate of change of the price per bottle when x= 1000, we compute p(1000) = –0.02e–0.0002(1000)= –0.02e–0.2–0.0163, or –1.63 cents per bottle. To find the rate of change of the price per bottle when x= 2000, we compute p(2000) = –0.02e–0.0002(2000)= –0.02e–0.4–0.0134, or –1.34 cents per bottle. b. The price per bottle when x= 1000 is given by p(1000) = 100e–0.0002(1000)+ 150 231.87, or $231.87/bottle. The price per bottle when x= 2000 is given by p(2000) = 100e–0.0002(2000)+ 150 217.03, or $217.03/bottle. 59.p= 240133240 13 30 00050 00051+FHGIKJ=+eexx..[()]. p= 720(3 + e–0.0005x)–2(–0.0005e–0.0005x)
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315 5 Exponential and Logarithmic Functions 0.0005(1000)0.0005(1000)2'(1000)720(3 + )2(0.0005)0.36(0.606531)0.0168,or1.68 cents per case.(30.606531)3(1000)240(1)40.36,or $40.36/case.3.606531peep== −≈ −+=60.a. The price at t= 0 is 0018369ee=, or $9/unit. b. 2/362.ttdpeedt=+000628tdpeedt==+=and so the price is increasing at the rate of $8/week. c. The equilibrium price is given by 2/3limlim(1836)18ttttpee→∞→∞==, or $18/unit. 61.a. The price at 0t=is 8 + 4, or 12, dollars per unit. b. 22282.tttdpeetedt= −+2220082817.tttttdpeetedt=== −+= − += −That is, the price is decreasing at the rate of $7/week. c. The equilibrium price is 22lim(84)tttete→∞++= 8 + 0 + 0 , or $8 per unit. 62.P(t) = 20.6(–0.009)e–0.009t= –0.1854e–0.009tP(10) = –0.1694, P(20) = –0.1549, and P(30) = –0.1415, and this tells us that the percentage of the total population relocating was decreasing at the rate of 0.17% in 1970, 0.15% in 1980, and 0.14% in 1990. 63.a. The percent is (3)15.5719,N=or 15.6. b. 0.780.78'( )1.5(0.78)1.17.ttftee==So the required rate is '(3)12.146f=, or 12.1% per year. c. 0.780.78"( )1.17(0.78)0.9126.ttftee==So the required rate is "(3)9.474,f=or 9.5%/yr/yr. 64.a. The number of air passengers in 2000 is (0)666,N=or 666 thousand. The number in 2005 is (5)818.759N=, or 819 thousand. b. 0.04130.0413'( )666(0.0413)27.5058ttftee==. The required rate is '(5)33.8147,f=or 33.8 thousand/yr.
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5 Exponential and Logarithmic Functions 31665.C(t) = 1486e–0.073t+ 500. a. The average energy consumption of the York refrigerator/freezer at the beginning of 1972 is given by C(0) = 1486e–0.073(0)+ 500 = 1486 + 500 = 1986, or 1986 kwh/yr. b. The rate of change of the average energy consumption of the York refrigerator is given by C(t) = 1486e–0.073t(–0.073) = –108.48e–0.073t, c. To see if the York refrigerator/freezer satisfied the 1990 requirement we compute C(18) = 1486e–0.073(18) + 500 = 399.35 + 500 = 899.35, or 899.35 kwh/yr. Since this is less than the 950 kwh/yr, we conclude that York satisfied the requirement. 66.a. 0.4( )0.23;tA tte=0.21122( )0.23( )0.094Ae==; 3.2(8)0.23(8)0.075.Ae==b. 0.40.4'( )0.23[ ( 0.4)]ttA ttee=+= 0.40.23( 0.41)tet+. 0.212'( )0.23(0.8)0.151Ae==; 3.2'(8)0.23( 2.2)0.021.Ae== −67.N(t) = 5.3e0.095t²–0.85t.
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  • Fall '09
  • OLDS,VICKY
  • Calculus, Natural logarithm, Logarithm, lim c

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