Page 83 - Schaum's Outline of Theory and Problems of Advanced Calculus

74 DERIVATIVES [CHAP. 4

(Notice that the order of differentiation in each succeeding case is two greater. The nature of the
intermediate possibilities is suggested in the next paragraph.)
It is possible that the slope of the tangent line to the graph of f is positive to the left of P 1 , zero at the
point, and again positive to the right. Then P 1 is called a point of inflection.In the simplest case this
point of inflection is characterized by f ðx 1 Þ¼ 0, f ðx 1 Þ¼ 0, and f ðx 1 Þ 6¼ 0.
0
00
000
2. Particle motion
The fundamental theories of modern physics are relativity, electromagnetism, and quantum
mechanics. Yet Newtonian physics must be studied because it is basic to many of the concepts in
these other theories, and because it is most easily applied to many of the circumstances found in every-
day life. The simplest aspect of Newtonian mechanics is called kinematics,or the geometry of motion.
In this model of reality, objects are idealized as points and their paths are represented by curves. In the
simplest (one-dimensional) case, the curve is a straight line, and it is the speeding up and slowing down
of the object that is of importance. The calculus applies to the study in the following way.
If x represents the distance of a particle from the origin and t signifies time, then x ¼ f ðtÞ designates
the position of a particle at time t. Instantaneous velocity (or speed in the one-dimensional case) is
dx f ðt þ tÞ change in distance
represented by ¼ lim (the limiting case of the formula for speed when
dt t!0 t change in time
the motion is constant). Furthermore, the instantaneous change in velocity is called acceleration and
2
d x
represented by .
dt 2
Path, velocity, and acceleration of a particle will be represented in three dimensions in Chapter 7 on
vectors.
3. Newton’s method
It is difficult or impossible to solve algebraic equations of higher degree than two. In fact, it has been
proved that there are no general formulas representing the roots of algebraic equations of degree five and
higher in terms of radicals. However, the graph y ¼ f ðxÞ of an algebraic equation f ðxÞ¼ 0 crosses the x-
axis at each single-valued real root. Thus, by trial and error, consecutive integers can be found between
which a root lies. Newton’s method is a systematic way of using tangents to obtain a better approx-
imation of a specific real root. The procedure is as follows. (See Fig. 4-7.)

Fig. 4-7

Suppose that f has as many derivatives as required. Let r be a real root of f ðxÞ¼ 0, i.e., f ðrÞ¼ 0.
Let x 0 be a value of x near r. For example, the integer preceding or following r. Let f ðx 0 Þ be the slope
0
of the graph of y ¼ f ðxÞ at P 0 ½x 0 ; f ðx 0 ފ. Let Q 1 ðx 1 ; 0Þ be the x-axis intercept of the tangent line at P 0
then

0 f ðx 0 Þ
0
¼ f ðx 0 Þ
x x 0