Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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9.2.2 a) and g)
9.2 NATURAL LOGARITHMS 173 This supply function is defined only for p > 4: the assumption is that price must exceed 4 for firms to be tempted into the industry. Elasticity of supply is very large for only just greater than 4 and falls as p increases, being close to 3 if p is very large. Exercises 02.1 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 the instantaneous 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) In x4 + 1' 9.2.3 (a) In Example 1 of this section we found the derivative of c. Now do this another 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 do this another way, using the same method as in part (a) of this exercise. 9.2.4 Find the critical points of the function y = x? – 2 In(1+ x²) and determine their nature. Sketch the graph. 9.2.5 Find the critical point of y = xe-* and determine its nature. Find also the point of inflexion and the for 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 and positive is not. This gap will be filled in Exercise 10.4.3.] 2.6 Suppose that the supply function of a competitive industry is q = Apª + Bp°, whero iuo onctants Jwith a h Dind th
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Transcribed Image Text:9.2 NATURAL LOGARITHMS 173 This supply function is defined only for p > 4: the assumption is that price must exceed 4 for firms to be tempted into the industry. Elasticity of supply is very large for only just greater than 4 and falls as p increases, being close to 3 if p is very large. Exercises 02.1 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 the instantaneous 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) In x4 + 1' 9.2.3 (a) In Example 1 of this section we found the derivative of c. Now do this another 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 do this another way, using the same method as in part (a) of this exercise. 9.2.4 Find the critical points of the function y = x? – 2 In(1+ x²) and determine their nature. Sketch the graph. 9.2.5 Find the critical point of y = xe-* and determine its nature. Find also the point of inflexion and the for 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 and positive is not. This gap will be filled in Exercise 10.4.3.] 2.6 Suppose that the supply function of a competitive industry is q = Apª + Bp°, whero iuo onctants Jwith a h Dind th
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