Exam 13: Vector-Valued Functions

Which of the following is an arc length parameterization of the given two-dimensional curve? The circle of radius 2 centered at the origin

Explain your answer or give a counterexample. [Hint - it may be useful to think about circular motion about the origin.]

Evaluate the given integral.

Find the binormal vector at .

Find the landing point for a projectile launched under the given conditions. Round distances to the nearest foot. Assign your coordinate system as indicated in the figure below. initial speed: initial height: h = 7 ft Angle: launch direction: due east Magnus force: northerly force of 19 pounds Projectile mass: 3 slug

Ted tells his incredulous nephews that he can turn a bucket of water upside down without the water pouring out! To illustrate, he puts a small amount of water in a bucket and swings it around in a circle so that the bucket travels from his knees, up over his head, and swings back down to his knees (see figure). If the water is 2.3 feet from the center of the circle of motion and is weightless at the top of the loop, what is the linear speed of the bucket? [Ignore the fact that the water in the bucket has depth; that is, pretend all the water is 2.3 feet from the loop's center.]

Find the acceleration function for the given position function.

Which of the following is the graph of the given parametric surface?

Find the unit tangent vector to the curve at the indicated point.

For , show that each center of curvature lies on the curve traced out by . Show your work.

Use the graph of below to predict whether the curvature would be larger at or .

For the circle with radius c centered at , show that the curvature at any point is equal to . Show all of your work.

Find the unit normal vector at the given point.

Find the position function from the given acceleration function.

Find t such that and are perpendicular.

Find t such that and are perpendicular.

Find the osculating circle at the given point.

Find the tangential and normal components of acceleration for the given position function at the indicated point.

Find parametric equations for the indicated curve. The intersection of and z = 4

A stationary merry-go-round of radius 8 feet is started in motion by a push consisting of a force of 7 pounds on the outside edge, tangent to the circular edge of the merry-go-round, for 4 seconds. The moment of inertia of the merry-go-round is . Find the resulting angular velocity of the merry-go-around.