Problem 1

Consider the curve with the parameterization .

(a). If is the Position of the Particle at time Compute the Velocity and Speed of the Particle at time . Simplify Your Answer

(b). Compute the Curvature of at

(c). Compute the Magnitude of the Normal and Tangential Components of the Acceleration at

(d). Compute , the Arc Length of the Curve from to

(e). Re-parameterize with respect to Arc Length Starting at point

Problem 2

Let be the line segment from the origin to the point in the plane. Compute the integral .

Problem 3

(a). Which of the following is a Parametrization of the Quarter-circle with Radius Two in the Upper Right Quadrant

Parametrization	Yes	No
	x
		x
	x
	x
	x
		x

We can start by checking to ensure that both their components stay positive throughout their range, this lets us rule out the second parameterization

A circle of radius must satisfy the equation for all values of so we can test the parameterizations on these rules.

All of the parameterizations besides the last one conforms to this rule so we can rule the last out.

Next we need to check that the parameterizations trace from to , they all do this besides the ones we already ruled out.

(b). The Curve is the Intersection of the Cylinder and the Plane . Give a Parameterization such that is Oriented Counter-clockwise when Viewed from above

Problem 4

Suppose is a parameterized curve not passing through the origin, such that

Show that there exists a function such that

For to be equal to they must be parallel this means there is some scalar valued function where for every point.