5. Show that F =(y cos(x),sin(x)+ y) is a conservative vector field and use this to calculate the work done by F in moving a particle from (7/4,1) to (27/3,3).

2. Let f : R" → R be a continuously differentiable function, while a E R" is its non-stationary point (i.e., Vf(x) # 0). Moreover, let d* be the vector defined by Vf(x) ||Vf(x)||' d* = - while d is any unit vector in R" different from d* (i.e., d e R", ||d|| = 1, d + d*). Show that there exist real numbers d E (0, 00), ɛ E (0, 0) such that %3D f(x + td*) – f(x + td) 0. (ii) For t z 0, approximate f(x+td*) and f(x+td) using the first-order Taylor polynomial of f(x) at x.

6. Compute fF dr where f(x, y) = (y² cos x, 2y sinx) and C is the straight line path starting at (1,1) and ending at (1,3). Hint: is the vector field conservative? (Answer: 8 sin 1)

1) A particle moves in a velocity field (2xy + 10, ײ) starting from point (0, 1) at t = 1. Where will it be when t = 1.1?