The place of the elementary calculus in the senior high-school mathematics : and suggestions for a modern presentation of the subject . slope of the tangent line. Infact, under these conditions theposition of the tangent line isthe limit of the position of thesecant turning about the point P, and the slope of the tangent lineis the limit of the slope of the secant. Next, turn this into the notation of the calculus. Represent thecoordinates of P by (x,y). Then by giving x an increment Ax, thecoordinates of P will be (x + Ax, y + Ay). This idea was developedunder the preceding topics. Then, sinc
The place of the elementary calculus in the senior high-school mathematics : and suggestions for a modern presentation of the subject . slope of the tangent line. Infact, under these conditions theposition of the tangent line isthe limit of the position of thesecant turning about the point P, and the slope of the tangent lineis the limit of the slope of the secant. Next, turn this into the notation of the calculus. Represent thecoordinates of P by (x,y). Then by giving x an increment Ax, thecoordinates of P will be (x + Ax, y + Ay). This idea was developedunder the preceding topics. Then, since PL = Ax and PL = Ay,the slope of the secant is Ay /Ax. Also, as the point P moves alongthe curve toward P, Ax approaches zero as its limit. The symbolfor expressing the limit of the slope of the secant as Ax approaches zero as its limit is nm —• This symbol is the geometric ratioAx->o Ax form of the derivative. This is read the limit of the ratio Ay to Ax as Ax approaches zero as its limit. The Symbol f (x). At this early stage in the learners knowledge of the calculus, it would be well to use only / (x) to represent the. Sca/e: lunit* 2spaces Fig. 14 Suggestions for Presentation of Calculus 61 derivative of the function of This form is used to a large extentin the more advanced calculus and hence it is desirable from thepoint of view of increased knowledge of mathematical language. Then the derivative may be expressed as / (x) = lim —• x ->- o Ax This symbol is a quick way of denning the derivative, gives adesirable point of view, and is not an entirely new way of present-ing a definition since the definitions of the trigonometric functionswere also explained by stating that they mean such and such thingsmerely because it is agreed that they shall mean those Second Step. This second step in the meaning of the deriv-ative is the development of Cauchys fractional form of the de-rivative. The same curve is usedas in the preceding step. The or-dinate
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