Vector Valued Functions and Graphs

This video tutorial introduces vector-valued functions, which map real numbers to vectors. It covers their representation in 2D and 3D spaces, using both IJ notation and component form. The video discusses the uniqueness of graphs generated by different functions and explains how to determine the domain of vector-valued functions. An example is provided to illustrate finding the domain, and a graphing exercise demonstrates how to plot a vector-valued function and determine its orientation. The tutorial aims to enhance understanding of vector-valued functions and their applications.

What is a vector valued function?

• A function that maps a vector to another vector

• A function that maps a real number to a vector

• A function that maps a vector to a real number

• A function that maps a real number to a real number

In the XY plane, how is a vector valued function typically expressed?

• Using polar coordinates

• Using I and J unit vectors

• Using spherical coordinates

• Using cylindrical coordinates

What additional component is included in vector valued functions in 3D space?

• M unit vector

• L unit vector

• N unit vector

• K unit vector

What do the endpoints of vectors in a vector valued function trace out?

• A straight line

• A curve

• A point

• A plane

Can different vector valued functions produce the same graph?

• Yes, always

• Only if they are linear

• No, never

• Yes, sometimes

What happens when you graph the function R(t) = (cos(t), sin(t)) for t in [0, π]?

• It traces a parabola

• It traces a semicircle

• It traces a full circle

• It traces a line

What is the domain of a vector valued function?

• The domain of the first component only

• The intersection of the domains of its components

• The domain of the last component only

• The union of the domains of its components

For the function R(t) = (1/t, 1/(t-1), cos(t)), what values of t are excluded from the domain?

• t = 1 and t = 3

• t = 0 and t = 2

• t = 1 and t = 2

• t = 0 and t = 1

What is the first step in graphing a vector valued function?

• Convert to polar form

• Convert to cylindrical form

• Convert to rectangular form

• Convert to spherical form

What type of graph is represented by the equation x^2/4 + y^2/9 = 1?

• A circle

• An ellipse

• A parabola

• A hyperbola