L H PITAL S RULE; EXPONENTIAL GROWTH AND DECAY in .NET framework

Generator QR in .NET framework L H PITAL S RULE; EXPONENTIAL GROWTH AND DECAY

36.9 Find the indicated limits: 5x 3 4x + 3 (a) lim x + 2x 2 1 1 1 (d) lim sin x x 0 x x3 x2 + x 1 (g) lim x 1 x + ln x 1 (j) (m) (p)

ln(1 + ex ) x + 1 + x 1 ex (e) lim x x 0 1 1 (h) lim ln(x + 1) x x 0 3x 2x (k) lim x x 0 (b) lim (n) (q)

(c) (f ) (i) (l) (o) (r) (u)

1 cos x x x 0 x2 lim x + (ln x)3 tan x lim x 0 x ex lim 2 x 0 x x3 1 lim x 1 x + 1 sin x lim x x 0+ e3x 1 lim x 0 tan x lim

36.10 Sketch the graphs of the following functions, indicating relative extrema, concavity, in ection points, and asymptotes: x (a) y = x (b) y = x 2 e x (c) y = x 3 e x (d) y = x 2 ln x (e) y = e x sin x e 36.11 A bacteria culture grows exponentially so that the initial number has doubled in 3 hours. How many times the 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 ago was 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 ies are being bred in an enclosure that can hold a maximum of 640 ies. If the ies grow exponentially with growth constant K = 0.05 per day, how long will it take an initial population of 20 to ll the enclosure (Recall that ln 2 0.6931.) 36.16 A certain chemical decomposes exponentially. Assume that 200 grams become 50 grams in 1 hour. How much will remain after 3 hours 36.17 If 100 bacteria in a colony reproduce exponentially and in 12 hours there are 300 bacteria, how many bacteria 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 constant K = 1 ln 2 when time 5 is measured in years, nd the population in the year 2020. 36.19 Bacteria in a culture growing exponentially increase from 100 to 400 in the rst hour. (a) What will be the population in 1.5 hours (b) What is the average number present in the rst 2 hours 36.20 Prove Cauchy s extended mean-value theorem: If f and g are differentiable in (a, b) and continuous on [a, b], and if g (x) = 0 for all x in (a, b), then there exists a point c in (a, b) such that f (c) f (b) f (a) = g(b) g(a) g (c)

[Hint: Apply the generalized Rolle s theorem (Problem 17.19) to h(x) = (f (b) f (a))g(x) (g(b) g(a))f (x) Note that g(b) = g(a) since, otherwise, the generalized Rolle s theorem would imply that g (x) = 0 for some x in (a, b).] 36.21 Prove L H pital s rule. [Hint: Consider the case limx a+ (f (x)/g(x)), where limx a+ f (x) = limx a+ g(x) = 0. We may assume that f (a) = g(a) = 0. Then, by Problem 36.20, f (x)/g(x) = (f (x) f (a))/(g(x) g(a)) = f (x )/g (x ) for some x between a and x. Therefore, as x a+ , x a+ . Hence, if limx a+ (f (x)/g (x)) = L, then limx a+ (f (x)/g(x)) = L. The other cases can be handled in the same way or can be reduced to this case.]