Question

Sketch the plane curve represented by the vector-valued function, and sketch the vectors \(\mathbf{r}\left(t_{0}\right)\) and \(\mathbf{r}^{\prime}\left(t_{0}\right)\) for the given value of \(t_{0} .\) Position the vectors such that the initial point of \(\mathbf{r}\left(t_{0}\right)\) is at the origin and the initial point of \(\mathbf{r}^{\prime}\left(t_{0}\right)\) is at the terminal point of \(\mathbf{r}\left(t_{0}\right) .\) What is the relationship between \(\mathbf{r}^{\prime}\left(t_{0}\right)\) and the curve? $$ \mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}, \quad t_{0}=\frac{\pi}{2} $$

Short answer

First, determine \(\mathbf{r}(t_{0}) = \mathbf{i}\) and \(\mathbf{r}^{\prime}(t_{0}) = -\mathbf{j}\). After sketching the circle and vectors, it can be seen that \(\mathbf{r}^{\prime}(t_{0})\) is a tangent vector to the curve at the point corresponding to \(t_{0}\).

Step-by-step solution

Find \(\mathbf{r}(t_0)\) and \(\mathbf{r}^\prime(t_0)\)\

For \(t_0 = \frac{\pi}{2}\), the vector \(\mathbf{r}(t_0)\) can be computed by substituting \(t_0\) into the expression given for \(\mathbf{r}(t)\). The derivative \(\mathbf{r}^\prime(t)\) can be computed by taking the derivative of \(\mathbf{r}(t)\) with respect to \(t\), and then substituting \(t_0\) into \(\mathbf{r}^\prime(t)\).

Sketch the curve

The vector-valued function \(\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}\) represents a circle in the plane with radius 1, where \(t\) is the parameter corresponding to the angle. Thus the circle should be sketched on a plane.

Sketch vectors

\(\mathbf{r}(t_{0})\) should be sketched from the origin to the point on the circle corresponding to \(t_0 = \frac{\pi}{2}\). Then, \(\mathbf{r}^{\prime}(t_{0})\) should be sketched from the terminal point of \(\mathbf{r}(t_{0})\) tangent to the curve.

Describe the relationship of \(\mathbf{r}^{\prime}\) to the curve

Looking at the sketched vectors and the curve, it is evident that vector \(\mathbf{r}^{\prime}\) is tangent to the curve at the point determined by \(\mathbf{r}(t_{0})\). This shows the instantaneous rate of change at that point.