Schaum's Outline of Theory and Problems of Beginning Calculus

290 L'HQPITAL'S RULE; EXPONENTIAL GROWTH AND DECAY [CHAP. 36Supplementary Problems36.9 Find the indicated limits:5x3 - 4x + 3 In (1 + ex) 1 - cos x(a) lim (b) lim (c) limX++Q '+' x+o x1 ex(e) lim -x-ro sin x x+o xx3 - x2 + x - 1(9) limx+l x+lnx-1x+o In (x + 1) x x-0 x3" 2"ex(k) lim -(I) lim -x+o xx+o x2(4 lim (: - J-)1 - sin x(m) lim -x-u/2 x - 4 2In x(s) lim -x-rl tan nxX'(f) lim -x+ + a0 (In XI3tan x(h) lim ( - !) (i) lim -(n)lim xsinx-o+1 - cos2 2x tan x - sin x(U) lim (w) limx-0 x2 x-0 x3x3 - 1(0) lim -x-1 x + 1sin x(r) lim -x-o+ Jxsin 3xe3x I(t) lim - (U) lim -x+o sin 7x x-+o tanx36.10 Sketch the graphs of the following functions, indicating relative extrema, concavity, inflection points, andasymptotes :X(4 Y = 2 (b) y = x2e-x (c) (d) y=x21nx (e) y=e-xsinx36.11 A bacteria culture grows exponentially so that the initial number has doubled in 3 hours. How many timesthe initial number will be present after 9 hours?36.12 The half-life of radium is 1690 years. If 10 per cent of an original quantity of radium remains, how long agowas the radium created?36.13 If radioactive carbon-14 has a half-life of 5750 years, what will remain of 1 gram after 3000 years?36.14 If 20 per cent of a radioactive element disappears in 1 year, compute its half-life.36.15 Fruit flies are being bred in an enclosure that can hold a maximum of 640 flies. If the flies grow exponentiallywith growth constant K = 0.05 per day, how long will it take an initial population of 20 to fill theenclosure? (Recall that In 2 2 0.6931.)36.16 A certain chemical decomposes exponentially. Assume that 200 grams become 50 grams in 1 hour. Howmuch will remain after 3 hours?36.17 If 100 bacteria in a colony reproduce exponentially and in 12 hours there are 300 bacteria, how manybacteria are in the colony after 72 hours?36.18 If the world population in 1990 was 4.5 billion and it is growing exponentially with growth constantK = 4 In 2 when time is measured in years, find the population in the year 2020.36.19 Bacteria in a culture growing exponentially increase from 100 to 400 in the first hour. (a) What will be thepopulation in 1.5 hours? (b) What is the average number present in the first 2 hours?