William Anthony Granville

“ Here is zero or unity according as is even or odd, and although does not become infinite as increases without limit, it does not tend to a limit, but oscillates. It is evident that if all the terms of a series have the same sign, the series cannot oscillate.
Since the sum of a converging series is a perfectly definite number, while such a thing as the sum of a nonconvergent series does not exist, it follows at once that it is absolutely essential in any given problem involving infinite series to determine whether or not the series is convergent. ”

“ The first derivative gives us the slope of the curve at any point; the second derivative determines the intervals within which the curve is concave upward or concave downward, and the points of inflection separate these intervals; the maximum points are the high points and the minimum points are the low points on the curve. ”

“ Our notion of velocity tells us at once that (C) does not give us the actual velocity at the end of two seconds; for even if we take very small, say or of a second, (C) still gives only the average velocity during the corresponding small interval of time. But what we do mean by the velocity at the end of two seconds is the limit of the average velocity when diminishes towards zero ”

“ For instance, if we say that shall be the arc of smallest numerical value whose tangent is , that is, shall take on only values between and , then we are limited to the branch passing through the origin, and the condition for continuity is satisfied. (9) Similarly,
is found to be a many-valued function. Confining ourselves to one branch of the graph of
we see that as approaches zero from the left, approaches the limit and as approaches zero from the right, approaches the limit Hence the function is discontinuous when Its value for can be assigned at pleasure. ”

“ Let the base CD (= x) increase to 10 inches (the diameter) ; then the altitude will decrease to zero and the area will become zero. Now let the base decrease to zero; then the altitude will increase to 10 inches and the area will again become zero. It is therefore intuitionally evident that there exists a greatest rectangle. By a careful study of the figure we might suspect that when the rectangle becomes a square its area would be the greatest, but this would at best be mere guesswork. A better way would evidently be to plot the graph of the function (1) and note its behavior. ”

“ Assuming that the intensity of light varies directly as the sine of the angle under which it strikes an illuminated surface, and inversely as the square of its distance from the surface, how high should the light he hung in order that the best possible light shall fall on a walk along the circumference of the plot? ”

“ The derivative of the quotient of a function by a constant is equal to the derivative of the function divided by the constant.
All explicit algebraic functions of one independent variable may be differentiated by following the rules we have deduced so far. ”

“ Curvature of a circle. If the point P with its tangent be supposed to move along the curve to P', the total curvature (= ) would measure the total change in direction, or rotation, of the tangent; or, what is the same thing, the total change in direction of the arc itself. Denoting by s the length of the arc of the curve measured from some fixed point (as A) to P, and by the length of the arc P P', then the ratio
measures the average change in direction per unit length of arc. [1] Since, from the figure,
or
it is evident that this ratio is constant everywhere on the circle. ”

“ Maximum and minimum values of a function. A maximum value of a function is one that is greater than any values immediately preceding or following.
A minimum value of a function is one that is less than any values immediately preceding or following. ”

“ A variable is said to vary continuously through an interval , when starts with the value and increases until it takes on the value in such a manner as to assume the value of every number between and in the order of their magnitudes. ”

“ By experiment it has been found that a body falling freely from rest in a vacuum near the earth's surface follows approximately the law , where s = space (height) in feet, t = time in seconds. Find the velocity and acceleration (a) at any instant; (b) at end of the first second; (c) at end of the fifth second.
which tells us that the acceleration of a falling body is constant; in other words, the velocity increases 32.2 ft. per sec. every second it keeps on falling. ”

“ If the given product contains factors, then since each infinitesimal may be assumed less than the th root of , the product can be made less than itself.
(4) If is a variable which approaches a limit different from zero, then the quotient of an infinitesimal by is also an infinitesimal. For if limit , and is any number numerically less than , then, by definition of a limit, will ultimately become and remain numerically greater than . Hence the quotient , where is an infinitesimal, will ultimately become and remain numerically less than , and is therefore by (2) an infinitesimal. ”

“ To determine the direction of curving or direction of bending in the vicinity of a point of inflection, test for values of x, first a trifle less and then a trifle greater than the abscissa at that point. ”

“ If the cost of the sides is $1 a square yard, and of the bottom $2 a square yard, what are the dimensions when the cost is a minimum? What is the minimum cost?
Ans. Side of base = 4 yd., height = 4 yd., cost $96. 12. A rectangular tract of land is to be bought for the purpose of laying out a quarter-mile track with straightaway sides and semicircular ends. In addition a strip 35 yards wide along each straightaway is to be bought for grand stands, training quarters, etc. If the land costs $200 a.n acre, what will be the maximum cost of the land required? ”

“ Since the differential of a function is its derivative multiplied by the differential of the independent variable, it follows at once that the formulas for finding differentials are the same as those for finding derivatives given in § 33, if we multiply each one by dx. ”

“ Similarly, the distance a boy can run may be considered as depending on the time. Or, we may say that the area of a square is a function of the length of a side, and the volume of a sphere is a function of its diameter. ”

“ This illustrates Rolle's Theorem:
If f (x) vanishes when x = a and x = b, and f (x) and f' (x) are continuous for all values of x from x = a to x = b, then f' (x) will be zero for at least one value of x between a and b.
This theorem is obviously true, because as x increases from a to b, cannot always increase or always decrease as x increases, since and . Hence for at least one value of x between a and b, must cease to increase and begin to decrease, or else cease to decrease and begin to increase; and for that particular value of x the first derivative must be zero (§ 81) . ”

“ Since the formulas deduced in the Differential Calculus generally involve derivatives of y with respect to x, such formulas as (A) and (B) are especially useful when the parametric equations of a curve are given. ”

“ Infinitesimals. In the Differential Calculus we are usually concerned with the derivative, that is, with the ratio of the differentials dy and dx. In some applications it is also useful to consider dx as an infinitesimal (see §15) , that is, as a variable whose values remain numerically small, and which, at some stage of the investigation, approaches the limit zero. ”

“ A man is walking at the rate of 5 miles per hour towards the foot of a tower 60 ft. high. At what rate is he approaching the top when he is 80 ft. from the foot of the tower?
Example 2 illustration. 2. A point moves on the parabola in such a way that when x = 6, the abscissa is increasing at the rate of 2 ft. per second. At what rates are the ordinate and length of arc increasing at the same instant? ”

“ By continuing the process as illustrated in I, II, and III, it is seen that if the first derivative of which does not vanish for x = a is of even order (= n) , then (47) is a maximum if a negative number; (48) is a minimum if a positive number. [3] If the first derivative of which does not vanish for x = a is of odd order, then will be neither a maximum nor a minimum. ”

“ The statements made concerning the remainder and the convergence of Taylor's Series apply with equal force to Maclaurin's Series, the latter being merely a special case of the former.
The student should not fail to note the importance of such an expansion as (65) . In all practical computations results correct to a certain number of decimal places are sought, and since the process in question replaces a function perhaps difficult to calculate by an ordinary polynomial with constant coefficients, it is very useful in simplifying such computations. ”

“ The fundamental problem of the Differential Calculus is to establish a measure of this change in the function with mathematical precision. It was while investigating problems of this sort, dealing with continuously varying quantities, that Newton [1] was led to the discovery of the fundamental principles of the Calculus, the most scientific and powerful tool of the modern mathematician.
26. Increments. The increment of a variable in changing from one numerical value to another is the difference found by subtracting the first value from the second. ”

“ Yet we may show that the harmonic series is not convergent by the following comparison:
We notice that every term of (G) is equal to or less than the corresponding term of (F) , so that the sum of any number of the first terms of (F) will be greater than the sum of the corresponding terms of (G) . But since the sum of the terms grouped in each bracket in (G) equals , the sum of (G) may be made as large as we please by taking terms enough. The sum (G) increases indefinitely as the number of terms increases without limit ”

“ Dividing by the increment of the variable, , we get
(D)
.
The limit of this ratio when approaches the limit zero is, from our definition, the derivative and is denoted by the symbol . Therefore
(E)
defines the derivative of [or ] with respect to .
From (D) we also get:
.
The process of finding the derivative of a function is called differentiation.
It should be carefully noted that the derivative is the limit of the ratio, not the ratio of the limits. The latter ratio would assume the form , which is indeterminate (§14, p12) . ”

“ Construct the series
When , series (H) is a geometric series with the common ration less than unity, and is therefore convergent. But the sum of (G) is less than the sum of (H) , as shown in the above inequalities; therefore (G) is also convergent.
When series (G) becomes the harmonic series which we saw was divergent, and neither of the above tests apply. ”