Theorem 1: Direction of a Curve at the Origin

Derivatives can be used to measure direction in polar as well as in rectangular coordinates. We begin with two theorems, one about the direction of a curve at the origin (an unusual point in polar coordinates) and the other about the direction of a curve elsewhere. Then we shall use these theorems for sketching curves.

The theorem tells us that if r = 0 at θ₀, the curve must approach the origin from the θ₀ direction. Figure 7.8.1 shows two cases.

(a)    If r has a local maximum or minimum at θ₀, then r has the same sign on both sides of θ₀. In this case the curve has a cusp at θ₀.

Figure 7.8.1 (a)

(b)    If r has no local maximum or minimum at θ₀, then r is positive on one side of θ₀ and negative on the other side. In this case the curve crosses the origin at θ₀.

Figure 7.8.1 (b)

We now consider points other than the origin. In rectangular coordinates, the slope of a curve y = f(x) at a point P is dyjdx = tan φ where φ is the angle between the x-axis and the line tangent to the curve at P as shown in Figure 7.8.2.

Figure 7.8.2

When r ≠ 0 in polar coordinates, a useful measure of the direction of the curve at a point P is tan ψ, where ψ is the angle between the radius OP and the tangent line at P (see Figure 7.8.3).

Figure 7.8.3