Problems

• Find the differentials of the following functions:
  • $f = t - \sin(t)\cos(t))$,
  • $g = \sin(e^t)$,
  • $h = \frac{t-1}{t+1}$,
  • $k = \frac{e^t - 1}{e^t +1}$
  • $j = \int_x^{x^2} e^{t^2} dt$
• A particle moves in a force field $\underline{F} = [y^2, -x^2]$.
  Its motion is given by $\underline{X} = [\cos(t), \sin(t)]$.
  The associated work differential is $dW = \underline{F}.d\underline{X} = \underline{F}.(\frac{d}{dt}\underline{X}) dt = \underline{F}.\underline{V} dt$.
  Find $dW$ for this motion.