Problems

In Problems 1-9 find the derivative.

10            Find the line tangent to the curve X = sin²ti + cos²tj + sin t cos tk at t = π/3.

11             Find the line tangent to the curve X = ti + t²j + t³k at the point (1, 1, 1).

12            Find the line tangent to the cycloid X = (t - sin t)i + (1 - cos t)j at t = π/4.

In Problems 13-25 find the velocity, speed, and acceleration.

20            A point on the rim of a wheel of radius one in the (x, y) plane which is spinning counterclockwise at one radian per second and whose center at time t is at (t, 0). (At t = 0, S = i.)

21             A bug which is crawling outward along a spoke of a wheel at one unit per second while the wheel is spinning at one radian per second. The center of the wheel is at the origin, and at t = 0, the bug is at the origin and the spoke is along the x-axis. (A spiral of Archimedes.)

22            The point at distance one from the origin in the direction of the vector

t > 0.

23            A car going counterclockwise around a circular track x² + y² = 1 with speed |2t| at time t. At t = 0 the car is at (1,0).

24            A point moving at speed one along the parabola y = x², going from left to right. (S = 0 at t = 0.)

25            A point moving at speed y along the curve y = e^x going from left to right. (S = j at t = 0.) In Problems 26-33 find the length of the given curve.

In Problems 34-37 find the position vector of a particle with the given velocity vector and initial position.

38            Find the position vector of a particle whose acceleration vector at time r is A = i + tj + e^tk, if at t = 0 the velocity and position vectors are both zero.

39            Find the position vector S if A = sin ti + cos tj + k, and at t = 0, V = 0 and S = 0.

40            Show that if U is the unit vector of X, then

41             Show using the Chain Rule that if X is the position vector of a curve and s is the length from 0 to t, then dX/ds is a unit vector tangent to the curve.

42            Suppose a particle moves so that its speed is constant and its distance from the origin at time t is e^t. Show that the angle between the position and velocity vectors is constant.

43            Prove that if F(t) is perpendicular to a constant vector C for all t, then F'(t) is also perpendicular to C.

44             Prove that if F(t) is parallel to a constant vector C for all t, then F'(t) is also parallel to C.

45            Prove the following differentiation rule for scalar multiples:

46            Prove the vector product rule