These are to be discussed in your tutorial in week 5.

1. Differentiate the following expressions:

(a) - 2 (b) 3 - (c) + 6 (d) -

(e) - (the Lennard-Jones potential) (f) 3

(g) - (h) 3 (i) + (j) - + -

2. What is the value of the gradient of these functions when:

(a) y = 5x² + , when x = 2 (b) p = + 6 , when q = 3

(c) φ = x - , when x = 1.2 (d) Z = 3ζ² + ζ^½ - ζ^7/3 , when ζ = 0.1

3. Find the value(s) of x at which the following functions have stationary values, and the corresponding value of y, and determine the character of the stationary point (max, min or point of inflection).

(a) y = 3 - x + x² (b) y = x + 1/x

(c) y = 8 - x³ (d) y = 4x³ - x⁴

4. Differentiate the following (without multiplying out the terms!):

(a) x(x² + 1)      (b) 3x(5 - x )       (c) (x² + 1)(3x³ -2x + 2)

(d) x(5x - 1)(7x² + 4)      (e) (x² + ) x      (f)

(g)       (h)

5. When isobutyl bromide and sodium ethoxide at equal concentration react in ethanol solution each concentration (C) is found to vary with time (t) according to the equation:

          (1)

where C₀ is its initial concentration and A is a constant.

(a) Derive an expression for the rate of consumption of isobutyl bromide or sodium ethoxide, given by .

(b) If the reactant concentrations C₀ are chosen to be 0.06 mol dm^-3, and the temperature is 95ºC, C(t) falls to 0.05 mol dm^-3 after 10 minutes. Use eqn.(1) to derive a value of A. Hence calculate the initial (t = 0) rate of consumption (in mol dm^-3 min^-1). What will be the concentrations after 25 minutes?