James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

224 Chapter 3 Differentiation Rules
We know that if y − f sxd, then the derivative dyydx can be interpreted as the rate of
change of y with respect to x. In this section we examine some of the applications of this
idea to physics, chemistry, biology, economics, and other sciences.
Let’s recall from Section 2.7 the basic idea behind rates of change. If x changes from
x 1 to x 2 , then the change in x is
and the corresponding change in y is
Dx − x 2 2 x 1
Dy − f sx 2 d 2 f sx 1 d
The difference quotient
Dy
Dx − f sx 2d 2 f sx 1 d
x 2 2 x 1
y
P{⁄, fl}
Q{¤, ‡}
Îx
Îy
0 ⁄ ¤
x
m PQ average rate of change
m=fª(⁄)=instantaneous rate
of change
FIGURE 1
is the average rate of change of y with respect to x over the interval fx 1 , x 2 g and can
be interpreted as the slope of the secant line PQ in Figure 1. Its limit as Dx l 0 is the
derivative f 9sx 1 d, which can therefore be interpreted as the instantaneous rate of change
of y with respect to x or the slope of the tangent line at Psx 1 , f sx 1 dd. Using Leibniz notation,
we write the process in the form
dy
dx − lim Dy
Dx l 0 Dx
Whenever the function y − f sxd has a specific interpretation in one of the sciences, its
derivative will have a specific interpretation as a rate of change. (As we discussed in Section
2.7, the units for dyydx are the units for y divided by the units for x.) We now look
at some of these interpretations in the natural and social sciences.
Physics
If s − f std is the position function of a particle that is moving in a straight line, then DsyDt
represents the average velocity over a time period Dt, and v − dsydt represents the instantaneous
velocity (the rate of change of displacement with respect to time). The instantaneous
rate of change of velocity with respect to time is acceleration: astd − v9std − s99std.
This was discussed in Sections 2.7 and 2.8, but now that we know the differentiation
formulas, we are able to solve problems involving the motion of objects more easily.
ExamplE 1 The position of a particle is given by the equation
s − f std − t 3 2 6t 2 1 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? After 4 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.
(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.
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