The Differential Calculus

Excerpts

• The Differential Calculus

  The really inspiring reflection suggested by the history of mathematics is the unity of thought and interest among men of so many epochs, so many nations, and so many races.

• The Differential Calculus

  This idea is immediately presented to us by the study of nature; velocity is the rate of increase of the distance travelled, and acceleration is the rate of increase of velocity.

• The Differential Calculus

  When $x$ increases to $x + h$, the function $x^{2}$ increases to $(x + h)^{2}$; so that the total increase has been $(x + h)^{2} - x^{2}$, due to an increase $h$ in the argument. Hence throughout the interval $x$ to $(x + h)$ the average increase of the function per unit increase of the argument is $\dfrac{(x + h)^{2} - x^{2}}{h}$.

• The Differential Calculus

  In reading over the Newtonian method of statement, it is tempting to seek simplicity by saying that $2x + h$ is $2x$, when $h$ is zero. But this will not do; for it thereby abolishes the interval from $x$ to $x + h$, over which the average increase was calculated.

• The Differential Calculus

  A function $f(x)$ has the limit $l$ at a value $a$ of its argument $x$, when in the neighbourhood of $a$ its values approximate to $l$ within *every* standard of approximation.

• The Differential Calculus

  Thus the limit of $\dfrac{2x}{x}$ at $x = 0$ is $2$, and it has no value at $x = 0$.

• The Differential Calculus

  It is a well-founded historical generalization, that the last thing to be discovered in any science is what the science is really about.

Equations

Problems

No exercises in this chapter.