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17. If the demand function f is differentiable, prove that 13:38 to the mixture. This is directly proportional to the amount
p
′
[pf(p)] < 0 if and only if f (p) < −1. (That is, revenue of converted acid in the region where 0 < x < 1.)
′
f(p)
decreases if and only if demand is elastic.)
18. The term income elasticity of demand is defined as the per-
centage change in quantity purchased divided by the per-
centage change in real income. If I represents income and
Q(I) is demand as a function of income, derive a formula for pH
the income elasticity of demand.
19. If the concentration of a chemical changes according to the
′
equation x (t) = 2x(t)[4 − x(t)], (a) find the concentration x(t)
for which the reaction rate is a maximum, (b) find the lim- ml of base added
iting concentration.
20. If the concentration of a chemical changes according to the 26. In the titration of a weak acid and strong base, the pH is
′
equation x (t) = 0.5x(t)[5 − x(t)], (a) find the concentration given by c + ln x
x(t) for which the reaction rate is a maximum, (b) find the 1 − x , where f is the fraction (0 < x < 1) of
limiting concentration. converted acid. What happens to the rate of change of pH
as x approaches 1?
21. Mathematicians often study equations of the form rx
′
x (t) = rx(t)[1 − x(t)], instead of the more complicated 27. The rate R of an enzymatic reaction is given by R = k + x ,
′
x (t) = cx(t)[K − x(t)], justifying the simplification with the where k is the Michaelis constant and x is the substrate con-
statement that the second equation “reduces to” the first centration. Determine whether there is a maximum rate of
′
equation. Starting with y (t) = cy(t)[K − y(t)], substitute the reaction.
y(t) = Kx(t) and show that the equation reduces to the form
′
x (t) = rx(t)[1 − x(t)]. How does the constant r relate to the 28. In an adiabatic chemical process, there is no net change in
constants c and K? heat, so pressure and volume are related by an equation of
the form PV 1.4 = c, for some positive constant c. Find and
22. Suppose a chemical reaction follows the equation dV
′
x (t) = cx(t)[K − x(t)]. Suppose that at time t = 4 the con- interpret dP .
′
centration is x(4) = 2 and the reaction rate is x (4) = 3. At
time t = 6, suppose that the concentration is x(6) = 4 and In exercises 29–32, the mass of the first x meters of a thin rod
′
the reaction rate is x (6) = 4. Find the values of c and K for is given by the function m(x) on the indicated interval. Find the
this chemical reaction. linear mass density function for the rod. Based on what you
23. In a general second-order chemical reaction, chemicals A and B find, briefly describe the composition of the rod.
(the reactants) combine to form chemical C (the product). If 29. m(x) = 4x − sin x grams for 0 ≤ x ≤ 6
the initial concentrations of the reactants A and B are a and 3
b, respectively, then the concentration x(t) of the product 30. m(x) = (x − 1) + 6x grams for 0 ≤ x ≤ 2
′
satisfies the equation x (t) = [a − x(t)][b − x(t)]. What is the 31. m(x) = 4x grams for 0 ≤ x ≤ 2
rate of change of the product when x(t) = a? At this value, 32. m(x) = 4x grams for 0 ≤ x ≤ 2
2
is the concentration of the product increasing, decreasing ............................................................
or staying the same? Assuming that a < b and there is no
product present when the reaction starts, explain why the
maximum concentration of product is x(t) = a. 33. Suppose that the charge in an electrical circuit is
Q(t) = e −2t (cos 3t − 2 sin 3t) coulombs. Find the current.
24. It can be shown that a solution of the equation
′
x (t) = [a − x(t)][b − x(t)] is given by 34. Suppose that the charge in an electrical circuit is
t
a[1 − e −(b−a)t ] Q(t) = e (3 cos 2t + sin 2t) coulombs. Find the current.
x(t) = .
1 − (a∕b)e −(b−a)t 35. Suppose that the charge at a particular location in an elec-
Find x(0), the initial concentration of chemical and lim x(t), trical circuit is Q(t) = e −3t cos 2t + 4 sin 3t coulombs. What
t→∞ happens to this function as t → ∞? Explain why the term
the limiting concentration of chemical (assume a < b). e −3t cos 2t is called a transient term and 4 sin 3t is known as
Graph x(t) on the interval [0, ∞) and describe in words how the steady-state or asymptotic value of the charge function.
the concentration of chemical changes over time.
Find the transient and steady-state values of the current
25. In example 9.5, you found the significance of one inflection function.
point of a titration curve. A second inflection point, called 36. As in exercise 35, find the steady-state and transient values
the equivalence point, corresponds to x = 1. In the gener- of the current function if the charge function is given by
alized titration curve shown on the following page, identify Q(t) = e −2t (cos t − 2 sin t) + te −3t
on the graph both inflection points and briefly explain why + 2 cos 4t.
chemists prefer to measure the equivalence point and not 37. Suppose that a population grows according to the logis- Copyright © McGraw-Hill Education
′
the inflection point of example 9.5. (Note: The horizontal tic equation p (t) = 4p(t)[5 − p(t)]. Find the population at
axis of a titration curve indicates the amount of base added which the population growth rate is a maximum.

314 | Lesson 4-9 | Rates of Change in Economics and the Sciences