9.2.2 a) and g) 9.2 NATURAL LOGARITHMS173This supply function is defined only for p 4: the assumption is that price mustexceed 4 for firms to be tempted into the industry. Elasticity of supply is very large foronly just greater than 4 and falls as p increases, being close to 3 if p is very large.Exercises02.1 We showed in Section 9.1 that an instantaneous rate of interest of r yields an APR(annual percentage rate) of 100f% per annum, where ↑ = e" – 1. If compound-ing is continuous and you know that the APR is 100s% per annum, what is theinstantaneous rate of interest?9.2.2 Differentiate(a) In(ax), (b) In(xª +1), (c) xlna, (d) x²,(e) In(e" +1), (f) exp(e" +1), (g) Inx4 + 1'9.2.3 (a) In Example 1 of this section we found the derivative of c. Now do thisanother way by taking natural logarithms of both sides of the equation y =Ca and then differentiating with respect to x.(b) In Example 4 of this section we found the derivative of exp(-x²). Now dothis another way, using the same method as in part (a) of this exercise.9.2.4 Find the critical points of the functiony = x? – 2 In(1+ x²)and determine their nature. Sketch the graph.9.2.5 Find the critical point ofy = xe-*and determine its nature. Find also the point of inflexion and thefor which the function is (a) convex, (b) concave.ranges of values[Note that we are not quite in a position to sketch the graph of this function:although its behaviour for x large and negative is obvious, that for x large andpositive is not. This gap will be filled in Exercise 10.4.3.]2.6 Suppose that the supply function of a competitive industry isq = Apª + Bp°,wheroiuo onctants Jwith a hDind th

Question

9.2.2 a) and g) 9.2 NATURAL LOGARITHMS173This supply function is defined only for p > 4: the assumption is that price mustexceed 4 for firms to be tempted into the industry. Elasticity of supply is very large foronly just greater than 4 and falls as p increases, being close to 3 if p is very large.Exercises02.1 We showed in Section 9.1 that an instantaneous rate of interest of r yields an APR(annual percentage rate) of 100f% per annum, where ↑ = e&#34; – 1. If compound-ing is continuous and you know that the APR is 100s% per annum, what is theinstantaneous rate of interest?9.2.2 Differentiate(a) In(ax), (b) In(xª +1), (c) xlna, (d) x²,(e) In(e&#34; +1), (f) exp(e&#34; +1), (g) Inx4 + 1'9.2.3 (a) In Example 1 of this section we found the derivative of c. Now do thisanother way by taking natural logarithms of both sides of the equation y =Ca and then differentiating with respect to x.(b) In Example 4 of this section we found the derivative of exp(-x²). Now dothis another way, using the same method as in part (a) of this exercise.9.2.4 Find the critical points of the functiony = x? – 2 In(1+ x²)and determine their nature. Sketch the graph.9.2.5 Find the critical point ofy = xe-*and determine its nature. Find also the point of inflexion and thefor which the function is (a) convex, (b) concave.ranges of values[Note that we are not quite in a position to sketch the graph of this function:although its behaviour for x large and negative is obvious, that for x large andpositive is not. This gap will be filled in Exercise 10.4.3.]2.6 Suppose that the supply function of a competitive industry isq = Apª + Bp°,wheroiuo onctants Jwith a hDind th

Accepted Answer

We have to differentiate the given functions.

(a) In(ax), (b) In(x* + 1), (c) æ In a, (d) æ", (e) In(e" + 1), (f) exp(e" +1), (8) In +1