Exam 14: Calculus of Vector-Valued Functions

Determine the radius, center, and plane containing the circle parametrized by .

Consider the linear paths and , described by and . Do the lines traced by and intersect? If so, where?

Find the points on the curve where the tangent line is parallel to the plane

A satellite has an orbit about Earth with a radius of m above Earth's surface. The radius of Earth is m, and its mass is kg. Compute the period of motion (in hours). Recall .

An object orbiting the Sun has an orbital period of 12 years. Determine the length of the semimajor axis of the orbit (The mass of the Sun is kg). Recall .

The curvature of at is:

Parameterize the curve of intersection of the surfaces and .

Find the center and radius of the osculating circle at if .

Consider the curve traced by . This curve lies on both a sphere and a plane. A) Find the equation of the sphere. B) Find the equation of the plane. C) Explain why the curve traced by lies on a circle.

Suppose an asteroid traverses a circular orbit of radius 12,000 km about a planet. If the period of the orbit is 67 hours, what is the mass of the planet? Recall .

Find the unit tangent of the curve at .

The curve intersects the plane at which of the following points?

Find a vector parametrization for the tangent line to the curve at the point .

Find , where and : A) by first computing the cross product and then differentiating. B) using the cross product rule.

Parameterize the curve of intersection of the hemisphere and the parabolic cylinder .

Find the arc length parametrization of the curve .

Find the length of the curve described by the vector function .

Describe the curve traced by the following vector valued function, and sketch the graph of this curve.

A particle's position at time is given by . Find at

Find an arc length parametrization of the curve .