- Find a parametrization for the intersection of the planes 2x+y+3z = −2 and 3x−2y+z = −4.
- Find a parametrization for the intersection of the plane 3x + y + z = −3 and the circular
cylinder x
^{2}+ y^{2}= 1. - Find a parametrization for the intersection of the plane 4x + 2y + 3z = 2 and the elliptic
cylinder x
^{2}+ 4z^{2}= 9. - Find a parametrization for the straight line joining and.
- Find a parametrization for the intersection of the surfaces 3y + 3z = 3x
^{2}+ 2 and 3y + 2z = 3. - Find a formula for the curvature of the curve y = sinx in the xy plane.
- An object moves over the curve where t ∈ ℝ and a is a positive constant. Find the value of t at which the normal component of acceleration is largest if there is such a point.
- Find a formula for the curvature of the space curve in ℝ
^{2},. - An object moves over the helix, . Find the normal and tangential components of the acceleration of this object as a function of t and write the acceleration in the form a
_{T}T + a_{N}N. - An object moves over the helix, . Find the normal and tangential components of the acceleration of this object as a function of t and write the acceleration in the form a
_{T}T + a_{N}N. - An object moves in ℝ
^{3}according to the formula. Find the normal and tangential components of the acceleration of this object as a function of t and write the acceleration in the form a_{T}T + a_{N}N. - An object moves over the helix, . Find the osculating plane at the point of the curve corresponding to t = π∕4.
- An object moves over a circle of radius r according to the formula
where v = rω. Show that the speed of the object is constant and equals to v. Tell why a

_{T}= 0 and find a_{N}, N. - Suppose = c where c is a constantR. Show the velocity, R
^{′}is always perpendicular to R. - An object moves in three dimensions and the only force on the object is a central force. This means
that if ris the position of the object, a= krwhere k is some function. Show that if this happens, then the motion of the object must be in a plane. Hint: First argue that a × r = 0. Next show that=
^{′}. Therefore,^{′}= 0. Explain why this requires v × r = c for some vector c which does not depend on t. Then explain why c ⋅ r = 0. This implies the motion is in a plane. Why? What are some examples of central forces? - Let R=i +j +k. Find the arc length, s as a function of the parameter t, if t = 0 is taken to correspond to s = 0.
- Let R= 2 i +j + 4tk. Find the arc length, s as a function of the parameter t, if t = 0 is taken to correspond to s = 0.
- Let R= e
^{5t}i + e^{−5t}j + 5tk. Find the arc length, s as a function of the parameter t, if t = 0 is taken to correspond to s = 0. - Consider the curve obtained from the graph of y = f. Find a formula for the curvature.
- Consider the curve in the plane y = e
^{x}. Find the point on this curve at which the curvature is a maximum. - An object moves along the x axis toward and then along the curve y = x
^{2}in the direction of increasing x at constant speed. Is the force acting on the object a continuous function? Explain. Is there any physically reasonable way to make this force continuous by relaxing the requirement that the object move at constant speed? If the curve were part of a railroad track, what would happen at the point where x = 0? - An object of mass m moving over a space curve is acted on by a force F. Show the work done by this
force equals ma
_{T}. In other words, it is only the tangential component of the force which does work. - The edge of an elliptical skating rink represented in the following picture has a light at its left end
and satisfies the equation += 1. (Distances measured in yards.)
A hockey puck slides from the point T towards the center of the rink at the rate of 2 yards per second. What is the speed of its shadow along the wall when z = 8? Hint: You need to find

at the instant described.

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