1. (a) Differentiate \(\cos x\) from first principles.
   (b) Differentiate \(\mathrm { e } ^ { 3 x } \sin 4 x\) with respect to \(x\).
   (c) Find \(\int x ^ { 2 } \sin 2 x \mathrm {~d} x\).
   
2. Showing all your working, evaluate
   (a) \(\quad \sum _ { r = 3 } ^ { 50 } ( 4 r + 5 )\),
   (b) \(\quad \sum _ { r = 2 } ^ { \infty } \left( 540 \times \left( \frac { 1 } { 3 } \right) ^ { r } \right)\).
   
3. The function \(f\) is defined by

$$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x - 1$$ (a) Show that the equation \(f ( x ) = 0\) has a root in the interval \([ 0,1 ]\).
(b) Using the Newton-Raphson method with \(x _ { 0 } = 0 \cdot 8\),
(i) write down in full the decimal value of \(x _ { 1 }\) as given in your calculator,
(ii) determine the value of this root correct to six decimal places.
Explain why the Newton-Raphson method does not work if \(x _ { 0 } = \frac { 1 } { 3 }\).