Parametrization and Properties of Ellipses

The video tutorial explains how to determine a parametrization for an ellipse traced counterclockwise. It starts with the ellipse equation and uses the trigonometric identity cosine squared plus sine squared equals one to derive the parametric equations x(t) = 2cos(t) and y(t) = 4sin(t). The tutorial checks the orientation by evaluating the parametric equations at specific points and confirms the correct orientation. Finally, it expresses the parametrization as a vector valued function r(t) with components 2cos(t) and 4sin(t).

What is the equation of the ellipse discussed in the video?

• x^2 + y^2 = 1

• x^2/2 + y^2/8 = 1

• x^2/16 + y^2/4 = 1

• x^2/4 + y^2/16 = 1

Which trigonometric identity is used to help find the parametrization of the ellipse?

• cos^2(θ) + sin^2(θ) = 1

• cos(2θ) = cos^2(θ) - sin^2(θ)

• tan^2(θ) + 1 = sec^2(θ)

• sin(2θ) = 2sin(θ)cos(θ)

How is the x-component of the parametrization expressed?

• x(t) = 2sin(t)

• x(t) = 2cos(t)

• x(t) = 4cos(t)

• x(t) = 4sin(t)

What is the expression for y(t) in the parametrization?

• y(t) = 2cos(t)

• y(t) = 4sin(t)

• y(t) = 2sin(t)

• y(t) = 4cos(t)

What is the range of t to trace the entire ellipse?

• 0 to π

• 0 to 4π

• 0 to 2π

• 0 to π/2

At t = 0, what are the coordinates of the point on the ellipse?

• (2, 0)

• (0, 4)

• (4, 0)

• (0, 2)

What is the y-coordinate when t = π/2?

• 4

• 1

• 0

• 2

What is the x-coordinate when t = π/2?

• 0

• 2

• 4

• 1

What is the vector-valued function for the parametrization?

• r(t) = (4cos(t), 2sin(t))

• r(t) = (2cos(t), 4sin(t))

• r(t) = (2sin(t), 4cos(t))

• r(t) = (4sin(t), 2cos(t))

What is the orientation of the ellipse as traced by the parametrization?

• Counterclockwise

• Clockwise

• Vertical

• Horizontal