Vector Valued Functions and Limits

The video tutorial covers vector-valued functions, including their definition, domain, evaluation, and graphing. It explains how to convert 3D equations into vector form and discusses limits and continuity in vector-valued functions. The tutorial provides examples and exercises to reinforce understanding.

What is a vector valued function composed of?

• Functions inside vector components

• Single variable equations

• Ordered pairs

• Only numbers

Which part of the vector valued function r(t) = sqrt(4 - t^2)i + t^2j - 6tk has a domain issue?

• 6tk

• None of the above

• sqrt(4 - t^2)i

• t^2j

What is the domain of the function r(t) = sqrt(4 - t^2)i + t^2j - 6tk?

• [-2, 2]

• t < -2

• t > 2

• All real numbers

How do you evaluate the vector function r(t) = cos(t)i + sin(t)j at t = 0?

• 1i + 1j

• 0i + 1j

• 0i + 0j

• 1i + 0j

What is the orientation of the graph of r(t) = 2cos(t)i + sin(t)j + 3tk?

• Random

• Counterclockwise

• No orientation

• Clockwise

What is the name of the shape formed by the graph of r(t) = 2cos(t)i + sin(t)j + 3tk?

• Circle

• Ellipse

• Parabola

• Elliptical Helix

How do you represent the curve x^2 + y^2 + z^2 = 4, x + z = 2 in vector form with parameter x = 1 + sin(t)?

• r(t) = (1 + sin(t))i + sqrt(2 - 2sin^2(t))j + (1 - sin(t))k

• r(t) = (1 + sin(t))i + (2 - sin(t))j + (1 - sin(t))k

• r(t) = (1 + sin(t))i + (1 - sin(t))j + sqrt(2 - 2sin^2(t))k

• r(t) = (1 + sin(t))i + (1 - sin(t))j + (2 - sin(t))k

What is the definition of the limit of a vector valued function r(t) = f(t)i + g(t)j + h(t)k as t approaches a?

• The product of the component functions

• The sum of the component functions

• None of the above

• The limit of each component function as t approaches a

What are the conditions for a vector valued function r(t) to be continuous at t = a?

• Both r(a) and the limit of r(t) as t approaches a exist and are equal

• The limit of r(t) as t approaches a exists

• r(a) exists

• None of the above

What is the limit of the vector function r(t) = sin(t)/t i + cos(t) j + (t+1)/(t-1) k as t approaches 0?

• (1, 1, -1)

• (0, 0, 0)

• (0, 1, 1)

• (1, 0, 1)