Approximations by the Differential Calculus

Excerpts

• Approximations by the Differential Calculus

  These last are in the proportion of $h$ to $k$, and hence results a proposition of the utmost importance in every practical application of mathematics, viz., that if two different, but small, errors be committed in the valuation of any quantity, the errors arising therefrom at the end of any process, in which both the supposed values of $x$ are successively adopted, are very nearly in the proportion of the errors committed at the beginning.

• Approximations by the Differential Calculus

  For example, let there be a right-angled triangle, whose base is $3$, and whose other side should be $4$, so that the hypothenuse should be $\sqrt{3^{2} + 4^{2}}$ or $5$.

• Approximations by the Differential Calculus

  The errors of the hypothenuse are then $.0008$ and $.0016$ nearly; and these last are in the proportion of $.001$ and $.002$.

• Approximations by the Differential Calculus

  It also follows, that if $x$ increase by successive equal steps, any function of $x$ will, for a few steps, increase so nearly in the same manner, that the supposition of such an increase will not be materially wrong.

• Approximations by the Differential Calculus

  The sun’s longitude is a function of the time; that is, the number of years and days from a given epoch being given, and called $x$, the sun’s longitude can be found by an algebraical expression which may be called $\phi x$.

• Approximations by the Differential Calculus

  And even for this interval, though it can hardly be called *small* in an astronomical point of view, the increments or decrements will be found so nearly the same for four or five days together, as to enable the student to form an idea how much more near they would be to equality, if the interval had been less, say one hour instead of twenty-four.

Equations

• Approximations by the Differential Calculus

  \phi x + \phi' x\, dx

  If x is changed into x + dx with dx very small, the function phi x changes to phi x plus phi' x times dx, very nearly.

• Approximations by the Differential Calculus

  \phi x + \phi' x\, h

  With the value x replaced by x + h, where h is a small error, phi(x + h) is very nearly phi x plus phi' x times h.

• Approximations by the Differential Calculus

  \phi x + \phi' x\, k

  With the value x replaced by x + k, where k is a small error, phi(x + k) is very nearly phi x plus phi' x times k.

• Approximations by the Differential Calculus

  \phi' x\, h

  The error committed in taking phi x, when the error h has been made in x, is very nearly phi' x times h, so errors of the result are in the proportion of the errors of x.

• Approximations by the Differential Calculus

  \sqrt{3^{2} + 4^{2}}

  The hypotenuse of a right-angled triangle with base 3 and other side 4 is the square root of 3 squared plus 4 squared, which is 5.

Problems

No exercises in this chapter.