3-8 (a) Sketch the plane curve with the given...

\begin{equation} \begin{array}{l}{3-8} \\ {\text { (a) Sketch the plane curve with the given vector equation. }} \\ {\text { (b) Find } \mathbf{r}^{\prime}(t) .} \\ {\text { (c) Sketch the position vector } \mathbf{r}(t) \text { and the tangent vector } \mathbf{r}^{\prime}(t) \text { for }} \\ {\text { the given value of } t \text { . }}\end{array} \end{equation} $$\mathbf{r}(t)=\left\langle t-2, t^{2}+1\right\rangle, \quad t=-1$$

\begin{equation}
\begin{array}{l}{3-8} \\ {\text { (a) Sketch the plane curve with the given vector equation. }} \\ {\text { (b) Find } \mathbf{r}^{\prime}(t) .} \\ {\text { (c) Sketch the position vector } \mathbf{r}(t) \text { and the tangent vector } \mathbf{r}^{\prime}(t) \text { for }} \\ {\text { the given value of } t \text { . }}\end{array}
\end{equation}
$$\mathbf{r}(t)=\left\langle t-2, t^{2}+1\right\rangle, \quad t=-1$$

Key Concepts

Vector-Valued Functions

A vector-valued function assigns a vector to every value in its domain. In the context provided, it maps the parameter t to a vector in the plane, representing points on a curve. This concept is fundamental in describing motion and curves in space, where each vector represents a position in the coordinate system.

Parametric Equations

Parametric equations describe a curve by specifying each coordinate as a function of a parameter, in this case t. This representation allows for a clear definition of curves, especially when the relationship between the coordinates is not given by a single function, and is useful in both analytical and graphical interpretations of curves.

Derivatives of Vector Functions

The derivative of a vector function gives the rate of change of the vector with respect to the parameter. It is computed by differentiating each component of the vector-valued function, resulting in a new vector function that describes the velocity or the direction in which the original function is changing.

Tangent Vectors

A tangent vector to a curve at a given point is the derivative of the vector function at that point. It indicates the instantaneous direction of the curve and is crucial for understanding the local behavior of a curve. Tangent vectors play an important role in differential geometry and dynamics.

Sketching Curves

Sketching curves from parametric equations involves plotting points using the given functions for each coordinate and understanding the overall shape and behavior of the curve as the parameter varies. This process is a practical application of vector-valued functions and helps in visualizing abstract mathematical concepts geometrically.

Position Vectors

A position vector specifies the location of a point in space relative to an origin. In this context, the vector-valued function itself represents the position vector of points on the curve. Visualizing position vectors helps in understanding the spatial distribution and directionality of curves in the coordinate plane.

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