Learning Outcome 3: Vector-Valued Functions and Parametric Motion

Understand vector-valued functions of one variable and their derivatives, perform associated computations, and apply understanding and computations to solve problems.

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Performance Criteria:
  1. Plot positions at various times for parametric motion given by a multiple-valued function of one variable. Find the rectangular equation of the path for parametric motion in two dimensions and identify its ``shape" (line, circle, ellipse, etc.), or vice-versa. Find parametric equations for a given path.

  2. Find the average velocity, as a vector, over a time interval for a particle with given equations of parametric motion in 2- or 3-space.

  3. Find the average rate of change, with respect to time, of function values experienced by a particle traveling in 2- or 3-space.

  4. Find vectors representing displacement and average velocity for a vector-valued function.

  5. Find velocity, speed and acceleration for parametric motion in two or three dimensions.

  6. Find displacement and distance traveled for parametric motion in two or three dimensions.

  7. Integrate a vector valued function, solve an initial value problem for parametric motion in two or three dimensions.

  8. Solve projectile motion problems.

  9. Find the curvature of a path at a given point. Given a graph of a path in the $xy$-plane, determine whether the curvature at one point is less, or more, than the curvature at another point. Determine points where the curvature is zero.

  10. Given the path and direction of motion of a particle and information about whether it is speeding up, slowing down, or moving at a constant speed at a point, sketch possible velocity and acceleration vectors at that point. Sketch possible tangential and normal components of the acceleration at that point.

  11. Given the velocity and normal and tangential components of acceleration for a particle, determine whether the particle is (a) speeding up, slowing down or moving at constant speed and (b) whether the path of the particle is straight or curved.

  12. Find the unit tangential and normal vectors   T   and   N.   Find the tangential and normal vector components of the acceleration vector; find the tangential and normal scalar components of the acceleration vector. Write the acceleration vector at some time in the form   a = aT T + aN N

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