1.2 Slope of a Curve at a Point

We want to extend the concept of slope from straight lines to more general curves. To do this, we must introduce the notion of a tangent line to a curve at a point.

The tangent line to a circle at a point P is the straight line that touches the circle just at P.

If we enlarge the region near P, we will notice that the magnified section of the circle looks almost straight. Further magnifications would make the section of the circle near P more closely resemble the tangent line at P.

We notice that the tangent line at P reflects the steepness of the circle at P. It is reasonable to define the slope of the circle at P to be the slope of the tangent line at P. Similar reasoning leads to a suitable definition of slope for an arbitrary curve at a point P.

If we magnify the region close to a point P for an arbitrary curve, we will notice that like the circle, as we increase magnification, the region of the curve near P more closely resembles a certain straight line.

This straight line is called the tangent line to the curve at P. This line best approximates the curve near P. We can then define the slope of a curve at a point P to be the slope of the tangent line to the curve at P.

We can usually use calculus to compute slopes by using formulas. For now, suppose we have the graph of y = x 2. The slope at any point on this graph is given by [slope of the graph y = x 2 at point (x,y)] = 2x So, the slope at (3,9) is 2(3) = 6, and the slope at (0,0) is 2(0) = 0.

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