Understanding Domains of Vector-Valued Functions

The video tutorial explains how to find the domain of a vector valued function by determining the domains of its components X(t), Y(t), and Z(t). It uses both graphical and algebraic methods to find these domains and then identifies the intersection of these domains to determine the overall domain of the vector valued function. The tutorial concludes with a visualization of the function's graph.

What is the domain of a vector-valued function?

• The domain of the last component only

• The intersection of the domains of its components

• The union of the domains of its components

• The domain of the first component only

What is the domain of X(t) = t^3?

• t > 0

• t ≤ 0

• All real numbers

• t ≥ 0

How do you determine the domain of Y(t) = √(t + 4)?

• t ≤ -4

• t > -4

• t < -4

• t ≥ -4

What is the domain of Z(t) = √(5 - t)?

• t ≥ 5

• t ≤ 5

• t < 5

• t > 5

What is the intersection of the domains of X(t), Y(t), and Z(t)?

• (-∞, ∞)

• [-4, 5]

• [0, 5]

• (-4, 5)

In interval notation, how is the domain of Y(t) = √(t + 4) expressed?

• (-∞, -4]

• [-4, ∞)

• (-4, ∞)

• [0, ∞)

What does a closed interval mean in the context of domain?

• The interval includes its endpoints

• The interval excludes its endpoints

• The interval is infinite

• The interval is undefined

How is the domain of the vector-valued function expressed in set-builder notation?

• {t | t > -4 and t < 5}

• {t | t ≥ -4 and t ≤ 5}

• {t | t < -4 and t > 5}

• {t | t ≤ -4 and t ≥ 5}

What is the significance of the intersection of domains in vector-valued functions?

• It determines the integral of the function

• It determines the range of the function

• It determines the domain of the function

• It determines the derivative of the function

What does the graph of the vector-valued function represent?

• A line in space

• A point

• A plane

• A space curve