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## Calculus

The function  is defined by
for all real values of  x except 0.

1. Find the coordinates of the points where the graph of the function meets the axes.
2. Show that the function  has exactly one stationary point whose   lies in the interval.
3. Use Newton Raphson Method to find the  of the stationary point giving your answer correct to 6 decimal places. Determine the nature of the stationary point.
4. Find the equations of the linear vertical and horizontal asymptotes for.
1. Use your answers from 1-5 to produce a hand drawn sketch of the graph of .

[50]

For a given firm let Q stand for the amount produced to meet a market demand. Let P stand for the price per unit of quantity produced. Assume that Q is a differentiable function of P.  Let  be the total revenue function and  the marginal revenue.

• Give the definition of the price elasticity of demand.

Use the definition of to show that if the demand is elastic (i.e. ) then      will fall if the price increases;  and that if the demand is inelastic

(i.e. ) then  will rise as the price increases.

• Given that .  Solve the resulting differential equation to find the demand equation if when ,

• Use an expression of  as a function of   and  to find the value of that maximises the total revenue

The supply equation is given by    .  Use the decimal Search method to find the equilibrium price giving your answer correct to five decimal places.

[50]

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