91Ó°ÊÓ

Parametric curve sketching involves illustrating a set of points in a plane or space that are defined by one or more variables, known as parameters. In this context, we examine a vector-valued function given by \( \mathbf{r}(t) = t \mathbf{i} + \frac{1}{t} \mathbf{j} \), which describes the position of a point in the plane as a function of the parameter \( t \).

To sketch the curve, you would begin by evaluating \( \mathbf{r}(t) \) for various values of \( t \) and plotting the resulting points on a coordinate grid. This visual representation helps to understand the behavior of the curve as the parameter changes. It's particularly beneficial to plot points where \( t \) equals whole numbers, as well as where \( t \) is a fraction or a negative number, to get a comprehensive view of the curve's shape.

For \( t_0 = 2 \) specifically, we categorize the point determined by \( \mathbf{r}(t_0) \) and use it as a reference for the vector sketches in subsequent steps.

The tangent vector of a curve at a given point indicates the direction in which the curve is heading at that instant. Mathematically, the tangent vector is the derivative of the vector-valued function. For our function \( \mathbf{r}(t) \), the derivative \( \mathbf{r}'(t) \) provides the components of the tangent vector for any value of \( t \).

Finding the Tangent Vector at \( t_0 \)

By evaluating \( \mathbf{r}'(2) \)—that is, the derivative at \( t_0 = 2 \)—we can find the tangent vector at the specified point. This vector, often normalized to be a unit vector \( \mathbf{T} \) for simplification and comparison purposes, is an essential tool for understanding the immediate motion along the curve. Sketching this vector on your graph at \( \mathbf{r}(t_0) \) visually depicts the curve's direction and rate of change at that point.

Plotting \( \mathbf{T} \) reinforces the concept of instant velocity, where the vector's direction corresponds to a tangent line on the curve, and its magnitude relates to the speed with which a particle would move along that line if it were following the curve's path.

The normal vector at a particular point on a curve provides a perpendicular direction to the tangent vector at the same point. In a two-dimensional setting, this vector typically points towards the center of curvature, essentially dictating the 'inside' or concave side of the curve. To identify the normal vector \( \mathbf{N} \) for our function, one strategy is to find the derivative of the tangent vector \( \mathbf{T} \) and then evaluate it at \( t_0 \).

Determining the Normal Vector at \( t_0 \)

Once \( \mathbf{T} \) has been found, we'll calculate its derivative and then normalize the resultant vector to derive \( \mathbf{N} \) at \( t_0 = 2 \). Sketching \( \mathbf{N} \) alongside \( \mathbf{T} \) at the point \( \mathbf{r}(t_0) \) aids in visualizing how the curve bends at that location. The crossed arrangement of \( \mathbf{T} \) and \( \mathbf{N} \) encapsulates the motion of traversing the curve with \( \mathbf{T} \) aligned in the direction of travel and \( \mathbf{N} \) indicating the direction in which the path is curving.