MATH 131-503 Fall 2015
3.8
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Liu
3.8 Rates of Change in the Natural Social Sciences
Recall: Suppose y = f (x). If x changes from x1 to x2 , the difference quotient
My
f (x2 ) − f (x1 )
=
Mx
x2 − x1
is called the average rate of change of y with respect to x over the interval [x1 , x2 ] and can be
interpreted as the slope of the secant line between points (x1 , f (x1 )) and (x2 , f (x2 )).
By analogy with velocity, we consider the average rate of change over smaller and smaller intervals
by letting x2 approach x1 and therefore letting M x approach 0. The limit of these average rates of
change is called the (instantaneous) rate of change of y with respect to x at x = x1 , which is
interpreted as the slope of the tangent to the curve y = f (x) at P (x1 , f (x1 )):
instantaneous rate of change =
dy
My
f (x2 ) − f (x1 )
= lim
= lim
dx Mx→0 M x x2 →x1
x 2 − x1
Physics Examples:
1. (p. 238) (a) Find the average rate of change of the area of a circle with respect to its radius r
as r changes from: (i) 2 to 3; (ii) 2 to 2.5.
(b) Find the instantaneous rate of change when r = 2.
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MATH 131-503 Fall 2015
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Liu
2. (p. 228) The position of a particle is given by s = f (t) = t3 − 6t2 + 9t, where t is measured in
seconds and s in meters.
(a) Find the velocity at time t.
(b) What is the velocity after 2 s?
(c) When is the particle at rest?
(d) When is the particle moving forward (that is, in the positive direction)?
(e) Draw a diagram to represent the motion of the particle.
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MATH 131-503 Fall 2015
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Liu
(f) Find the total distance traveled by the particle during the first five seconds.
(g) Find the acceleration at time t and after 4 s.
(h) Graph the position, velocity, and acceleration functions for 0 ≤ t ≤ 5.
(i) When is the particle speeding up? Slowing down?
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MATH 131-503 Fall 2015
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Liu
3. (p. 238) If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after t
seconds is s = 80t − 16t2 .
(a) What is the maximum height reached by the ball?
(b) What is the velocity of the ball when it is 96 ft above the ground on its way up? On its
way down?
4. Graph of the position function of a particle is shown, where t is measured in seconds. When is
each particle speeding up? When is it slowing down?
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MATH 131-503 Fall 2015
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Economics: Suppose C(x) is the total cost that a company incurs in producing x units of a certain
commodity. The function C is called a cost function. The marginal cost, that is, the instantaneous rate of change of cost with respect to the number of items produced, is C 0 .
Chemistry:
1. Consider the reaction A + B → C, where A and B are reactants and C is the product. The
concentration of a product C is denoted by [C]. Over a time interval t1 ≤ t ≤ t2 ,
average rate of reaction of C =
M [C]
[C](t2 ) − [C](t1 )
=
Mt
t2 − t1
M [C]
d[C]
=
Mt→0 M t
dt
instantaneous rate of reaction = lim
2. Compressibility (example 5 on page 232)
Biology Example: (p. 239) The number of yeast cells in a laboratory culture increases rapidly
initially but levels off eventually. The population is modeled by the function
n = f (t) =
a
1 + be−0.7t
where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate
of 12 cells/hour. Find the values of a and b. According to this model, what happens to the yeast
population in the long run?
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