Vector-Valued Functions and Integration

The video tutorial explains how to determine a vector-valued function R of T given its derivative and initial conditions. It covers the integration of each component separately, using integration as the inverse of differentiation. The tutorial demonstrates finding constants for each component by substituting initial conditions, resulting in the complete vector-valued function R of T.

What are the components of the derivative of the vector-valued function R(T)?

• 2t, sin T, e^2T

• 6t, 2 cos T, 3 e^T

• 2t + 6, sin T, 3 e^2T

• 2 + 6t, 2 sin T, 3 e^2T

What is the initial condition given for the vector-valued function R(T)?

• R(0) = (2, 5, 1)

• R(0) = (1, 5, 2)

• R(0) = (2, 1, 5)

• R(0) = (5, 1, 2)

What is the purpose of integrating the derivative R'(T)?

• To find the original vector-valued function R(T)

• To find the derivative of R(T)

• To solve for T

• To determine the constant of integration

What is the integrated form of the X component of R(T)?

• 2T^2 + 3T + 5

• 3T^2 + 2T + 1

• 2T + 3T^2 + 5

• 2T + 6T^2 + 1

How is the constant C1 determined for the X component?

• By using the initial condition R(0) = (5, 1, 2)

• By solving the equation 2T + 3T^2 = 5

• By differentiating the X component

• By setting T = 1

What is the integrated form of the Y component of R(T)?

• -2 cos T + 3

• 2 sin T + 3

• -2 sin T + 3

• 2 cos T + 3

How is the constant C2 determined for the Y component?

• By using the initial condition R(0) = (5, 1, 2)

• By differentiating the Y component

• By setting T = 1

• By solving the equation -2 cos T = 1

What substitution is used to integrate the Z component?

• U = 3T

• U = 2T

• U = e^T

• U = T^2

What is the integrated form of the Z component of R(T)?

• 3 e^(T) + 2

• 3 e^(T) + 1/2

• 3 e^(2T) + 1/2

• 3 e^(2T) + 2

How is the constant C3 determined for the Z component?

• By solving the equation 3 e^(2T) = 2

• By setting T = 1

• By differentiating the Z component

• By using the initial condition R(0) = (5, 1, 2)