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