Exam 12: Vector-Valued Functions

Describe the graph of

Find the velocity, speed, and acceleration of a particle whose position is given by r(t) = at seconds.

r(t) = 31t + t³ i - 2t² j is the position vector of a particle moving in a plane. Find the acceleration at an arbitrary time t.

If x = t³, y = 12 + t², then k(t) at t = 1 is

Let r(t) = 2t i + (10 + 3t)j. Find T(t).

Find the arc length of the graph of

Find the parametric equation of the line which is perpendicular to r(t) = $\langle$ 5t, 6 sin 2t, 6 cos 2t $\rangle$ at .

Find parametric equations for using arc length, s, as a parameter. Use the point on the curve where t = 0 as the reference point.

Express as a parametric equation.

If v = 2i and a = 7i - 9j, find a_N .

Find parametric equations for using arc length as a parameter. Use the point on the curve where as a reference point.

Find the unit tangent and unit normal vectors to the curve r(t) = (-8 + 4 cos t) i + sin t j at . Sketch a portion of the curve showing the point of tangency.

Let r(t) = (t² + 9)i + e ^t j + e ^t k. Find N(t) for t = 0. Approximate your calculations to three decimal places.

Let . Find the equation of the line tangent to the graph of at the point .Write your answer using parametric equations.

Find the velocity of a particle moving along the curve r(t) = (-8 + t³)i + 4t j - t² k at t = 1.

r(t) = (-1 + t³)i - 2t j; 1 $\le$ t $\le$ 2. Find the distance.

Find the radius of curvature for r(t) = $\langle$ 4 sin t, 2t - sin 2t - 3, cos 2t $\rangle$ at .

Find the unit tangent and unit normal vectors to the curve r(t) = t² i + 2t j at t = 1.

Find the curvature for r(t) = e ^t i + e ^t j + 30k.

If , find .