Page:Calculus Made Easy.pdf/97

increase by a small increment $dx$, to the right, it will be observed that y also (in this particular curve) increases by a small increment $dy$ (because this particular curve happens to be an ascending curve). Then the ratio of $dy$ to $dx$ is a measure of the degree to which the curve is sloping up between the two points $Q$ and $T$. As a matter of fact, it can be seen on the figure that the curve between $Q$ and $T$ has many different slopes, so that we cannot very well speak of the slope of the curve between $Q$ and $T$. If, however, $Q$ and $T$ are so near each other that the small portion $QT$ of the curve is practically straight, then it is true to say that the ratio ${\dfrac {dy}{dx}}$ is the slope of the curve along $QT$. The straight line $QT$ produced on either side touches the curve along the portion $QT$ only, and if this portion is indefinitely small, the straight line will touch the curve at practically one point only, and be therefore a tangent to the curve.

This tangent to the curve has evidently the same slope as $QT$, so that ${\dfrac {dy}{dx}}$ is the slope of the tangent to the curve at the point $Q$ for which the value of ${\dfrac {dy}{dx}}$ is found.

We have seen that the short expression “the slope of a curve” has no precise meaning, because a curve has so many slopes–in fact, every small portion of a curve has a different slope. “The slope of a curve at a point” is, however, a perfectly defined thing; it is