AP Calculus(Mr. Surowski)

Homework from Chapter 4

Oct. 22 (4.1): 1–19 , 22 , 24, 25, 29.

(A) Find the extreme values of the functionf(x) = 3x + cos x on the interval [0, 2π].

Oct. 24 (4.2): 1–9, 12 , 15–19, 25–34, 39, 40, 43.

(A) The surge function looks likef(x) = ate−bt (a, b > 0 are constants).(a) Graph y = f(x), x ≥ 0. (Use graphing calculator if necessary.)(b) Find lim

x→+∞f(x).

(c) Find the relative extrema of f(d) Determine where f is increasing and where f is decreasing.

Oct. 27 (4.3): 1– 11 , 12–14, 16, 18, 19, 23, 24, 28,31 (work in groups of one each), 33, 36, 41, 42, 45, 46

(A) The logistic function looks like

f(x) =L

1 + Ce−kt(L, C, k > 0 are constants).

(a) Find where f is increasing, decreasing, concave up, and concave down.(b) Show that f ′′(x) = 0 where f(x) = C/2.(c) The number L is called the carrying capacity of thelogistic function. Why do you think this is? (Look at lim

x→+∞f(x).)

(d) The point of inflection is called the point of diminishing returns.Why do you think this is?

AP Calculus: Homework from Chapter 4—Continued

Oct. 29 (4.4): 1–3, 5, 8, 9, 12(a), 13–15, 17, 19(a), 20 , 37.

(A) A rumor spreads among a group of 1,000 people. The numberof people, N(t), who have heard the rumor by time t (hours)is often modeled by a logistic function of the form

N(t) =1000

1 + 999e−.69t

(a) Approximately how long will it take for half of all the people to haveheard the rumor?(b) Approximately when is the rumor spreading the fastest?

(B) If t measures years since 1990, one popular world-population model is

P (t) =40

1 + 11e−0.08t(P is in billions).

(a) What does this model predict for the maximum sustainable populationof the world?(b) According to this model, when will the population be growing the fastest?

(C) The concentration C of a drug (measured in ng/ml) as a functionof t (hours) is often modeled by a surge function. Assume that the model isC(t) = 12.4te−0.2t.How many hours after administering this drug will the concentrationreach its greatest?

Oct. 31 (4.5): 1–6, 7 (note that this is just the first two

terms of the binomial expansion), 8(a), (b), (d), 10, 19, 20 –22,25, 27, 28, 32–34, 39–41, 43–45.

Nov. 4 (4.6): 1–5, 8, 11, 13, 14, 18, 20 , 21, 24, 27, 30.Nov. 5 Review for Chapter 4 testNov. 7 Chapter 4 test