These are to be discussed in your tutorial in week 7.

1. Write down the derivatives of:

(a) 2e^x (b) x² - e^x (c) e^-5x (d) 5ln x (e) ln x² (f) ln (1/x)

2. Find the gradient of each of the following curves at the specified value of x :

(a) y = e^x - 2x where x = 0

(b) y = x² + 2e^x where x = 2

(c) y = e^x(x³ - 2) where x = 1

3. Differentiate each of the following functions with respect to x :

(a) (3x + 1)² (b) (2x⁴ - 5)^½ (c) x(2x + 3)³

(d) (e) (f)

(g) ln (x + 1)² (h) 5exp(x² + 1) (i)

4. The Maxwell-Boltzmann distribution law for the speeds of molecules in a gas at temperature T states that:

where A is a proportionality constant and B is given by:

Calculate the most probable velocity for an N₂ molecule at a temperature of 300 K given the parameters: m = 4.65x10^-26 kg, k = 1.38x10^-23 J / deg.

[Hint: You will need to use the rule for a product of functions and the chain rule to find the value of for which N(ν) is a maximum.]

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