Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9378ef80-7767-4c68-bae6-da61c50171e5%2F71615b66-cf7f-404b-8725-f497f2db6327%2Foiv7498_processed.jpeg&w=3840&q=75)
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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