91Ó°ÊÓ

In the context of multivariable calculus, the velocity function of a particle describes the rate of change of its position vector over time. When given a position vector \( \vec{r}(t) \) such as \( t^{2} \vec{i} + (1-t) \vec{j} \), the velocity function \( \vec{v}(t) \) is derived by differentiating the position vector with respect to time, \( t \). This is crucial as it tells us how fast and in what direction an object is moving at any point in time.

To find the velocity function from \( \vec{r}(t) \), differentiate each component of the position vector independently. In this exercise, the velocity function becomes \( \vec{v}(t) = 2t \vec{i} - \vec{j} \). This indicates that at any given time \( t \):

• The object's velocity in the \( \vec{i} \) (or x) direction is \( 2t \), linearly dependent on time.
• The velocity in the \( \vec{j} \) (or y) direction is constant at \( -1 \), meaning it has a consistent downward motion.

Understanding and computing the velocity function is fundamental in predicting and analyzing motion.

Differentiation is a core operation in calculus used to determine how a function changes at any given point. In the realm of multivariable calculus, differentiation extends to vector functions, such as the position vector \( \vec{r}(t) \), to find derivatives like velocity.

For scalar functions, you differentiate to find the slope of the function at any point. For vector functions, like our position vector \( \vec{r}(t) \), you differentiate each component separately. Thus:

• For \( t^{2} \vec{i} \), differentiation with respect to \( t \) gives \( 2t \vec{i} \). This part of the velocity function indicates acceleration or deceleration along the \( \vec{i} \) axis with a magnitude dependent on time.
• For \( (1-t) \vec{j} \), differentiation yields \( -1 \vec{j} \), a constant, showing a steady motion upward or downward in the \( \vec{j} \) direction.

Differentiation allows us to translate geometric information about position into analytic insights about motion, such as speed and direction.

The position vector \( \vec{r}(t) \) describes a particle's location in the plane at any given time \( t \). It's a fundamental concept in multivariable calculus as it provides a trajectory path for moving particles. In this exercise, the position vector is given by \( t^{2} \vec{i} + (1-t) \vec{j} \).

A position vector:

• Contains components that specify the particle's coordinates, here along the \( \vec{i} \) and \( \vec{j} \) directions.
• Changes over time, representing how the particle's location varies.
• Serves as the basis from which we derive the velocity function, as the velocity is the time derivative of this vector.

The role of the position vector is akin to drawing a map of a particle's journey. At any time \( t \), you can plug in the value to \( \vec{r}(t) \) and find the exact point in the plane where the particle resides. Knowing the start (when \( t = 0 \)) and changes with time enables exploration of dynamic behaviors through calculus operations like differentiation.