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CHAPTER 4 • •
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Applications of Differentiation
59. The cost of manufacturing x items is given by C(x) = 2. One way of numerically approximating a derivative is by
38. Suppose that a population grows according to the logis- 48. Suppose that the total cost of moving a barge a distance ′
2
p
′
p at speed v is C(v) = avp + b , representing energy ex-
tic equation p (t) = 2p(t)[7 − 2p(t)]. Find the population at 0.02x + 20x + 1800. Find the marginal cost function. Com- computing the slope of a secant line. For example, f (a) ≈
f(b) − f(a)
which the population growth rate is a maximum. pare the marginal cost at x = 20 v to the actual cost of produc- , if b is close enough to a. In this exercise, we will
pended plus time. (a) Find v to minimize C(v). (b) Trav-
ing the 20th item. b − a
eling against a current of speed v , the cost becomes
C
39. It can be shown that solutions of the logistic equation have 60. For the cost v function in exercise 59, find the value of x that develop an analogous approximation to the second deriva-
p
2
+ b
. Find v to minimize C(v). (Sug-
the form p(t) = B , for constants B, A and k. Find C(v) = ap v − c average cost C(x) = C(x)∕x. tive. Instead of finding the secant line through two points
v − v
minimizes the
1 + Ae −kt c on the curve, we find the parabola through three points
the rate of change of the population and find the limiting gested by Tim Pennings.) on the curve. The second derivative of this approximat-
population, that is, lim p(t). ing parabola will serve as an approximation of the second
t→∞
derivative of the curve. The first step is messy, so we rec-
40. In exercise 39, suppose you are studying the growth of a ommend using a CAS if one is available. Find a function of
EXPLORATORY EXERCISES
population and your data indicate an inflection point at EXPLORATORY EXERCISES the form g(x) = ax + bx + c such that g(x ) = y , g(x ) = y
2
1
2
1
p = 120. Use this value to determine the constant B. In your and g(x ) = y . Since g (x) = 2a, you actually only need to 2
′′
1. Let n(t) be the number of photons in a laser field. One
3
3
study, the initial population is p(0) = 40. Use this value to 1. A simple model for the spread of fatal diseases such as find the constant a. The so-called second difference approx-
2
′
AIDS divides people into the categories of susceptible (but a
determine the constant A. If your current measurement is model of the laser action is n (t) = an(t) − b[n(t)] , where imation to f (x) is the value of g (x) = 2a using the three
′′
′′
′
and b are positive constants. If n(0) = a∕b, what is n (0)?
p(12) = 160, use this value to determine the constant k. not exposed), exposed (but not infected) and infected. The points x = x −Δx [y = f(x )], x = x [y = f(x )] and x =
Based on this calculation, would n(t) increase, decrease 1 1 1 2 2 2 3
proportions of people in each category at time t are denoted or
S(t), E(t) and I(t), respectively. The general equations for this
′
neither? If n(0) > a∕b, is n (0) positive or negative? Based x +Δx [y = f(x )]. Find the second difference for f(x) =
3
3
model are calculation, would n(t) increase, decrease or nei-
on this √ x + 4 at x = 0 with Δx = 0.5, Δx = 0.1 and Δx = 0.01.
′′
′
′
S (t) = mI(t) − bS(t)I(t), negative? Based on
APPLICATIONS ther? If n(0) < a∕b, is n (0) positive or Compare to the exact value of the second derivative, f (0).
this calculation, would n(t) increase, decrease or neither? e − e −x
′
x
E (t) = bS(t)I(t) − aE(t),
41. The function f(t) = a∕(1 + 3e −bt ) has been used to model Putting this information together, conjecture the limit of 3. Forthehyperbolictangentfunctiontanh(x) = e + e −x ,show
x
′
I (t) = aE(t) − mI(t), under the assumption
the spread of a rumor. Suppose that a = 70 and b = 0.2. n(t) as t → ∞. Repeat this analysis that d tanh x > 0.Concludethattanh(x)hasaninversefunc-
dx
Compute f(2), the percentage of the population that has that a < 0. tion and find the derivative of the inverse function.
where m, b and a are positive constants. Notice that each
′
heard the rumor after 2 hours. Compute f (2) and describe equa-tion gives the rate of change of one of the categories.
what it represents. Compute lim f(t) and describe what it Each rate of change has both a positive and negative term.
t→∞
represents. Explain why the positive term represents people who are
entering the cate-gory and the negative term represents
42. After an injection, the concentration of the medicine in a people who are leaving the category. In the first equa-
muscle is given by a function of time, f(t). Suppose that t is tion, the term mI(t) represents people who have died from
measured in hours and f(t) = e −0.02t − e −0.42t . Determine the the disease (the constant m is the reciprocal of the life
time when the maximum concentration of it occurs.
expectancy of someone with the disease). This term is
43. Suppose that the size of the pupil of an animal is given by slightly artificial: the assumption is that the popu-lation
f(x) (mm), where x is the intensity of the light on the pupil. If is constant, so that when one person dies, a baby is born
who is not exposed or infected. The dynamics of the dis-
160x −0.4 + 90 ease are such that susceptible (healthy) people get infected
f(x) = ,
4x −0.4 + 15 by contact with infected people. Explain why the number
of contacts between susceptible people and infected peo-
show that f(x) is a decreasing function. Interpret this result
in terms of the response of the pupil to light. ple is dependent on S(t) and I(t). The term bS(t)I(t), then,
represents susceptible people who have been exposed by
44. Suppose that the body temperature 1 hour after receiving contact with infected people. Explain why this same term
1 2
x mg of medicine is given by T(x) = 102 − x (1 − x∕9) for shows up as a positive in the second equation. Explain the
6
′
0 ≤ x ≤ 6. The absolute value of the derivative, |T (x)|, is rest of the remaining two equations in this fashion. (Hint:
defined as the sensitivity of the body to the dosage. Find The constant a represents the reciprocal of the average
the dosage that maximizes sensitivity. latency period. In the case of AIDS, this would be how
long it takes an HIV-positive person to actually develop
45. A fish swims at velocity v upstream from point A to point AIDS.)
B, against a current of speed c. Explain why we must have
v > c. The energy consumed by the fish is given by E = 2. Without knowing how to solve differential equations, we
kv 2 , for some constant k > 1. Show that E has one critical can deduce some important properties of the solutions of
v − c differential equations. Consider the equation for an au-
number. Does it represent a maximum or a minimum? tocatalytic reaction x (t) = x(t)[1 − x(t)]. Suppose x(0) lies
′
′
between 0 and 1. Show that x (0) is positive, by determin-
46. The power required for a bird to fly at speed v is propor-
1 ing the possible values of x(0)[1 − x(0)]. Explain why this
3
tional to P = v + cv , for some positive constant c. Find v indicates that the value of x(t) will increase from x(0) and
to minimize the power. will continue to increase as long as 0 < x(t) < 1. Explain
why if x(0) < 1 and x(t) > 1 for some t > 0, then it must
47. A commuter exits her neighborhood by driving y miles at
Copyright © McGraw-Hill Education r mph. Assume that the neighborhood has a fixed size, so 1). Therefore, we can conjecture that lim x(t) = 1. Similarly,
be true that x(t) = 1 for some t > 0. However, if x(t) = 1,
r mph, then turning onto a central road to drive x miles at
′
then x (t) = 0 and the solution x(t) stays constant (equal to
1
2
that xy = c for some number c. (a) Find x and y to minimize
t→∞
show that if x(0) > 1, then x(t) decreases and we could again
the time spent driving. (b) Show that there is equal time
conjecture that lim x(t) = 1. Changing equations, suppose
driving at r mph and at r mph. This is a design principle
t→∞
2
1
′
for neighborhoods and airports.
that x (t) =−0.05x(t) + 2. This is a model of an experiment
315