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Collections >> The First Century of the First State University >>
Primary document
Title:
James Phillips's Lecture Notes, [18--]: Electronic Edition.
Author: Phillips, James, 1792-1867
Funding from the University Library, University of North Carolina at Chapel Hill supported the electronic publication of this title.
Text transcribed by
Bari Helms
Images scanned by
Bari Helms
Text encoded by
Brian Dietz
First Edition, 2005
Size of electronic edition: ca. 10K
Publisher: The University Library, University of North Carolina at Chapel Hill
Chapel Hill, North Carolina
2005
The electronic edition is a part of the University of North Carolina at Chapel
Hill digital library, Documenting the American South.
Languages used in the text:
English
Revision history:
2005-10-18, Brian Dietz finished TEI/XML encoding.
Source(s):
Title of collection: Cornelia Phillips Spencer Papers (#683), Southern Historical Collection, University of North Carolina at Chapel Hill
Title of document: James Phillips's Lecture Notes, [18--]
Author: [James Phillips]
Description: 1 page, 1 page image
Note:
Call number 683 (Southern Historical Collection, University of North
Carolina at Chapel Hill)
Editorial practices
The text has been encoded using the recommendations for Level 5 of the TEI in Libraries Guidelines.
Originals are in the Southern Historical Collection, University of North Carolina at Chapel Hill.
Original grammar, punctuation, and spelling have been preserved.
Page images can be viewed and compared in parallel with the text.
Any hyphens occurring in line breaks have been removed, and the trailing part of a word has been joined to the preceding line.
All quotation marks, em dashes and ampersand have been transcribed as entity references.
All double right and left quotation marks are encoded as ".
All single right and left quotation marks are encoded as '.
All em dashes are encoded as —.
Indentation in lines has not been preserved.
For more information about transcription and other editorial decisions, see the section Editorial Practices.
The text has been encoded using the recommendations for Level 5 of the TEI in Libraries Guidelines.
Originals are in the Southern Historical Collection, University of North Carolina at Chapel Hill.
Original grammar, punctuation, and spelling have been preserved.
Page images can be viewed and compared in parallel with the text.
Any hyphens occurring in line breaks have been removed, and the trailing part of a word has been joined to the preceding line.
All quotation marks, em dashes and ampersand have been transcribed as entity references.
All double right and left quotation marks are encoded as ".
All single right and left quotation marks are encoded as '.
All em dashes are encoded as —.
Indentation in lines has not been preserved.
For more information about transcription and other editorial decisions, see the section Editorial Practices.
Page 1
ELEMENTS
of the
DIFFERENTIAL and INTEGRAL CALCULUS.
of the
DIFFERENTIAL and INTEGRAL CALCULUS.
DIFFERENTIAL CALCULUS.
Of the differentiation of algebraic quantities.
1. We say that one variable is a function of another when the first is equal to a certain compound analytical expression of the second; for example, y is a function of x in the following equations:
y = (a2 - x2)½, y = x3- 3bx2, y = x4/a , y = l + cx2 , y = (a + bx + cx2 + dx3)m/n.
2. Let us consider a function in its state of augmentation, in consequence of the increase of the variable which it contains. As every function of a variable x can be represented by the ordinate of a curve BMM', let AP = x and PM = y be the ordinates of a point M of this curve, and let us suppose that the abscissa AP receives an increment PP' = h; the ordinate PM will become P'M' = y'. Fig. 1. To obtain the value of this new ordinate, we see that it is necessary to change x into x + h, in the equation of the curve, and the value which this equation will then determine for y will that of y'.
For example, if we had the equation y = mx2, we should obtain y' by changing x into x + h, and y into y', and we would have
y' = m(x + h)2 = mx2 + 2mxh + mh2
3. Let us now take the equation y = x3 . . . . . (1),
and let us suppose that y becomes y' when x becomes x + h; we shall have
and let us suppose that y becomes y' when x becomes x + h; we shall have
if from this equation we subtract equation (1), there will remainy' = (x + h)3 = x3 + 3x2h + 3xh2 + h3:
y1 - y = 3x2h + 3xh2 + h3; and dividing by h[y'/h - y/h] = 3x2 + 3xh + h2 . . . . . (2).
Let us see what this result teaches us: y' - y represents the increment of the function y in consequence of the increment h given to x, since this difference y' - y is that of the new state of magnitude of y, as respects its primitive state.
On the other hand the increment of x being h, it follows that [y'/h - y/h] is the ratio of the increment [...]