Parametrized curves and their derivatives

If c(t) is a vector-valued function in two dimensions (i.e., c : R →R2), then c(t) parametrizes a curve in the plane. (This is similar to how we parametrized a line.) For example, the function c(t) = (3 cos t)i + (2 sin t)j parametrizes an ellipse, as illustrated in the following demo. I’ve drawn c(t) as a blue arrow. As you change t from 0 to 2π (by dragging the blue point on the slider), the head of the arrow traces out the ellipse.

Can you show mathematically that c(t) traces out an ellipse? To do so, let (x,y) be the point defined by (x,y) = c(t) for some value of t. Since c(t) = (3 cos t, 2 sin t), we conclude that x = 3 cos t and y = 2 sin t. Can you see that x and y satisfy

x2 32 + y2 22 = 1,

which is the equation for the above ellipse? (You remember that cos 2t + sin 2t = 1 for any t. You can calculate expresions x2∕32 and y2∕22, and add them together.)

The derivative of a parametrized curve

The definition of the derivative of a parametrized curve is analogous to what you learned in one-variable calculus:

c'(t) = lim h→0c(t + h) −c(t) h .  (1)

In one-variable calculus, the derivative was the slope of the graph. Is this true for parametrized curves? In this case, the derivative is a vector, so it can’t just be the slope (which is a scalar). Instead, the derivative c'(t) is the tangent vector of the curve traced by c(t). In this way, the direction of the derivative c'(t) specifies the slope of the curve traced by c(t). The length (or magnitude) of the derivative ||c'(t)|| specifies how fast c(t) traces out the curve as you change t.

Let’s see why the derivative defined by equation (1) is tangent to the curve. Below, I’ve plotted c(t) in blue and c(t + h) in green. In red, I’ve plotted the estimate of the derivative for a given h, which we’ll denote by

gh(t) = c(t + h) −c(t) h .

Note that gh(t) is defined so that

lim h→0gh(t) = c'(t).

For any value of t, you can see that when h = 1, then gh(t) = c(t + h) −c(t) (since the red vector joins the heads of the blue and green vectors). As you move h toward zero (by moving the second slider), the red vector approaches the tangent of the ellipse. When h is zero, the blue and green vectors are identical. In this limit, the red vector is the derivative c'(t) and is tangent to the ellipse.

We can use the fact that c'(t) is tangent to the curve c(t) to calculate the equation for the tangent line to a curve.