6. If f(x) is any function, and Ax = a x, show that the first relation in Exercise 4 is true if and only if the second is true. What is the graphical significance of each relation?

NOTE. A function is said to be continuous at x = a if the first relation in Exercise 4 is true for the given function. It can be proved that the algebraic and transcendental functions studied in this course are continuous for all values of x for which they do not become infinite. If a function becomes infinite as x approaches a, then f(a) has no meaning. Hence x = ɑ is a point of discontinuity. There are other types of points of discontinuity.

7. Prove that (a) any polynomial, (b) any rational function, is continuous for all values of a, except, in (b), for the values for which the function becomes infinite.

Ax÷0

Ax±0 Ax

8. If u is a function of x, what is the value of lim Au? of lim ? of Au2 lim Ax÷0 Ax

lim Au Ax=0

Au
Ax

94. Fundamental Formulas for Differentiation. The rules in this section are useful in differentiating a function without the labor involved in computing the limit considered in Section 93. Theorem 1. The derivative of a constant is zero; that is,

For the graph of y = c is a straight line parallel to the x-axis, whose slope, m = Day = Dc, is zero.

Theorem 2. The derivative of the in

dependent variable is unity, that is,

Dxx 1.

y=c

(2)

For the graph of y = x is the straight line bisecting the first and third quadrants, whose slope, m = Day Dax, is unity.

Theorem 3. The derivative of a constant

ул

times a function is equal to the constant times the derivative of the function. Symbolically, if u is any function of x,

Let

Dxcu

cDxu.

Y = си.

(3)

What is the graphical interpretation of this theorem, in the

light of the Theorem on page 89?

Theorem 4. The derivative of the sum of two functions is equal to the sum of their derivatives. Symbolically, if u and v are any two functions of x

Corollary. The derivative of the sum of several functions is

the sum of their derivatives.

Theorem 5. The derivative of the nth power of a function of x is n times the (n - 1)st power of the function times the derivative of the function with respect to x. Symbolically, if u is any function of x

Dzun

nun-1 Dxu.

(5)

and therefore, passing to the limit as Ax approaches zero,

Dxy = nun-1Dzu.

Corollary. The derivative of the nth power of x is n times the (n − 1)st power of x; that is,

Dxxn

nxn−1

For, by (5), Dxn = nx2-1 Dxx nxn-1, since Dxx

=

(5a)

nx2-1, since Dxx = 1, by (2).

If we think of x2 + 4x - 5 as a function u, the given equation has the form y = u3. Applying (5).

95. Derivative of a Polynomial. Any polynomial has the form (page 133)

Hence,

+ An-1X + an.

+ An-1X + An)

+Dx(an-1x)+D2αn

= Dx(αox") + D2 (α1x2-1) + ·.

by the corollary to Theorem 4, Section 94,

The successive terms of the derivative of a polynomial may be found from the corresponding terms of the polynomial by multiplying the coefficient by the exponent and decreasing the exponent by

one. In applying this rule, the constant term, whose derivative is zero, may be regarded as the coefficient of xo

For example, if then

Y 3x4 4x3 + 3x - 7,
Dzy 12x3 12x2 +3.

1.

96. Corresponding Properties of a Function, its Graph, and its Derivative. The derivative is useful in expressing some of the properties given on page 42, and some other properties also.

Rate of Change. The steepness of the graph at any point is measured by the slope of the tangent line, and the rate of change of the function is measured by the value of the derivative.

Changes of the Function. The graph rises (or falls) to the right if and only if the slope of the tangent line is positive (or negative). Hence

A function increases (or decreases) as x increases if and only if its derivative is positive (or negative).

Maxima and Minima. A line tangent to the graph is horizontal if and only if its slope is zero. Hence,

To find the abscissas of the points at which the tangent line is horizontal, set the derivative equal to zero, and solve the resulting equation, Dxy = 0, for x.

The roots of this equation may be values of x for which the function has a maximum or minimum value.

A

སྐྱ

B

D

FIG. 160.

G

F

E

X

At a maximum point, the function ceases to increase and begins to decrease, and hence (see Changes of the Function above), as x increases, the derivative must change sign from positive to negative.

Similarly, at a minimum point, as x increases, the de

rivative must change sign, from negative to positive.

If, as x increases in the vicinity of a horizontal tangent, the derivative does not change sign, the point of contact is neither a maximum nor a minimum point.

For example, the graph of y = x3 has a horizontal tangent at the origin (page 112), since Day = 3x2 = 0 if x = 0. But as x increases through x = 0, the derivative 3x2 does not change sign, and the origin is therefore not a maximum or minimum point.

Concavity. The curve in the figure is concave downward from A to C. If the tangent line at A rolls along the curve until the moving point of contact reaches C, the slope m decreases, and hence Dam is negative (see Changes of the Function above). Similarly, from C to E the curve is concave upward. The slope of a tangent line rolling from C to E increases, and Dam is positive. Hence

If the graph is concave upward (or downward), Dam is positive (or negative). It is also true that the curve is concave upward (or downward) only if Dam is positive (or negative).

It follows that a point at which the tangent line is horizontal will be a minimum point if Dam is positive, or a maximum point if Dam is negative.

Since m = Day is the rate of change of y, and since m increases or decreases according as Dam is positive or negative, it follows that

The rate of change of a function increases or decreases, as x increases, according as the graph of the function is concave upward or downward.

Points of Inflection. At a point of inflection (Definition, page 139) m ceases to increase and begins to decrease, or vice versa. Hence m has a maximum or minimum value, and therefore Dam = 0 (see Maxima and Minima above). The curve is concave upward on one side of a point of inflection and concave downward on the other (see Concavity).

Not every root of Dam O is a point of inflection. It is necessary that as x increases through the root in question Dam shall change sign (see Maxima and Minima). For example, if y=x4, then m = Day = 4x3 and Dam = 12x2; Dam = 0 if x = 0, but it does not change sign as x increases through zero. Hence the origin is not a point of inflection. It is, in fact, a minimum point, as is apparent from the graph (page 112),