1. (a) 4 cos (4x). (b) 2 sin (π -2x) = 2 sin (2x).

(c) cos²x - sin²x = cos 2x. (d) cos x e^{sin x}.

(e) - tan x. (f) - cosec² x.

2. If the displacement is: x = x₀ sin (2t).

Then the velocity is given by:

,

which will be a maximum at the beginning of each cycle including at t = 0. The maximum kinetic energy is given in terms of the maximum velocity by ½ mv².

Insertion of the molecular parameters then leads to:

,

and hence to a molar energy:

.

Note that the mass of a mole of H atoms must be given in kg for this calculation.

3. (a) x³ + k (b) (c) x - ¼x⁸ + k

4. (a) +k (b) 2m⁴ - 2m + k (c)

(d) +k (e) ¼φ⁴ + 2φ² -2φ + k

5. (a) sq. units

(b) sq. units

(c) (= 13 / 6) sq. units

6. (a) + k        (b) 5ln x - + k        (c) + k         (d) + k

(e) + k         (f) (-1/3)exp(-3x) + k        (g) 2e^2x + k         (h) 6ln x+ k

(i) e^x +3ln x - 6x+ k         (j) 3sinx + k         (k) -2cos x -3tan x + k

(l) -cos5x + k        (m) 2sin2x + k