William Anthony Granville

“ 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. ”

“ 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 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. ”

“ 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. ”

“ 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? ”

“ 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 ”

“ 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. ”

“ 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. ”

“ Find the equation of the evolute in parametric form, plot the curve and the evolute, find the radius of curvature at the point where , and draw the corresponding circle of curvature.
Substituting in above formulas (A) and then in (50) , §117, gives the parametric equations of the evolute. Assuming values of the parameter , we calculate from (B) and (C) ; and tabulate the results as follows: Now plot the curve and its evolute. The point is common to the given curve and its evolute. The given curve (semi cubical parabola) lies entirely to the right and the evolute entirely to the left of . ”

“ 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. ”

“ 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) . ”

“ Assuming the volume of the wood in a tree to be proportional to the cube of its diameter, and that the latter increases uniformly year by year when growing, show that the rate of growth when the diameter is 3 ft. ”

“ 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. ”

“ In order to determine sufficient conditions that shall be a maximum or a minimum, it is necessary to proceed to higher derivatives. To derive sufficient conditions for all cases is beyond the scope of this book. ”

“ But from Algebra we know that the quadratic expression
always has the same sign as (or ) when . Applying this to (F) , and we see that (F) , and therefore also the left-hand member of (E) , has the same sign as ( or ) when
Hence the following rule for finding maximum and minimum values of a function . ”

“ Show that when the radius of the base equals the depth, a steel cylindrical standpipe of a given capacity requires the least amount of material in its construction.
9. Show that the most economical dimensions for a rectangular tank to hold a given volume are a square base and a depth equal to one half the side of the base. ”

“ This transformation is called changing the independent variable from x to t. It is usually better to work out examples by the methods illustrated above rather than by using the formulas deduced.
Illustrative Example 1. Change the independent variable from x to t in the equation.
(C) (D) . Solution. , therefore (E) . Also ; therefore (F) Also Substituting in the last result from (E) , (G) Substituting (D) , (F) , (G) in (C) ,
and reducing, we get
Ans. ”

“ The nth derivative. For certain functions a general expression involving n may be found for the nth derivative. The usual plan is to find a number of the first successive derivatives, as many as may be necessary to discover their law of formation, and then by induction write down the nth derivative. ”

“ A rapidly converging series for computing logarithms will now be deduced.
By the theory of logarithms,
Substituting in (B) the equivalent series for and found in Exs. 6 and 7 §145, we get [8]
which is convergent when is numerically less than unity. Let
and we see that will always be numerically less than unity for all positive values of and . Substituting from (D) into (C) , we get
a series which is convergent for all positive values of and ; and it is always possible to choose and so as to make it converge rapidly.
Placing and in (E) , we get ”