\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
For the function \(f(x)=\cos(x)\),
  1. Find \(f^{(1)}(x)\), \(f^{(2)}(x)\), \(f^{(3)}(x)\) and \(f^{(4)}(x)\).
  2. Find \(f(0)\), \(f'(0)\), \(f^{(2)}(0)\), \(f^{(3)}(0)\) and \(f^{(4)}(0)\).
  3. Show that the Maclaurin series for \(\cos x\) is$$ \cos x =1 -\frac{x^2}{2!}+\frac{x^4}{4!}+\dots=\sum_{k=0}^\infty \frac{(-1)^{k}}{(2k)!}x^{2k} $$

Capture an image of your work. AI teacher feedback takes approximately 10 seconds.