The differential and integral calculus. [Followed by] Elementary illustrations of the differential and integral calculus

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Page 19 - Any two sides of a triangle are together greater than the third. This difficulty arises from the student depending somewhat too much on ocular demonstration, and not entirely on reasoning, in his preceding course, and can only be overcome by close attention to the reasoning. ' We have the result of all that precedes in the following proposition. If two
Page 53 - added to x, will increase the logarithm. Neither does the ratio of the increment of the function to the increment of its variable furnish any distinct idea of the change which is taking place when the variable has attained or is passing through a given value, for example, when x passes from 100 to 102, the difference between log
Page 59 - We may mention, in illustration of the preceding problem, a method of establishing the principles of the integral calculus, which generally goes by the name of the Method of Indivisibles. A line is considered as the sum of an infinite number of points, a surface of an infinite number of lines, and a solid of an
Page 40 - We have then, The differential coefficient of the differential coefficient is called the second differential coefficient ; the differential coefficient of the second differential coefficient is called the third differential coefficient, and so on. The several differential coefficients of
Page 377 - increase without limit, the tangent perpetually approaches to the asymptote both in direction and position, so that the asymptote may be regarded as a tangent whose point of contact is at an infinite distance. Find then the values of
Page 354 - c), takes all the imaginable varieties which can be given to it by changes in the value of c. What is the curve to which it must always be a tangent? Let x and y be the coordinates of the point of contact, when a is the value of
Page 501 - W, g- being 32'1908 feet. In order to consider the latter, let there be a system which, if it move at all, can only revolve about a fixed axis passing through 0, and perpendicular to the plane of the paper. Any pressure applied to a point of this
Page 28 - or half the length which would have been described by the velocity v continued uniformly from the beginning of the motion. It is usual to measure the accelerating force by the velocity acquired in one second. Let this be g ; then since the same velocity is acquired in every other second, the velocity acquired in
Page 10 - to the one immediately preceding. This limit, as has been observed, is finite in every series which we have occasion to use ; and therefore a value for h can be chosen so small, that for it the series in the last-named formula is convergent ; still more will it be so for every smaller value of
Page 29 - which gives equal in value to the one proposed. These can be brought as near as we please to 1 and 2 by making x sufficiently great, or — sufficiently small ; and, consequently, their ratio can be brought as near as we please to — . We will now prove the following: — That in any series of decreasing

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