27. Determine the domain of definition of the given function. (a) f(1) = In(In 1) (b) f(t)= In(In(In 1)) 28. Find the equations of the tangent lines to the graph of y= In |x| at x = 1 and x = -1. 29. Find the coordinates of the relative extreme point of y = x² In x, x>0. Then, use the second derivative test to decide if the point is a relative maximum point or a relative minimum point. 30. Repeat the previous exercise with y=√x In x. 31. The graphs of y=x+ ln x and y = ln 2x are shown in Fig. 6. (a) Show that both functions are increasing for x > 0. (b) Find the point of intersection of the graphs. 2 1 I -2 2 y=x+lnx -2 1.0 y-In 2r Figure 6 32. Repeat Exercise 31 with the functions y = x + ln x and y = In 5x. (See Fig. 7). 1.5 1.0 y=x+Inx 2.0 y-In 5x + 1.5 2.0 Figure 7 33. The graph of the function y=x²- In x is shown in Fig. 8. Find the coordinates of its minimum point. 3 2 f(x)=x²-Inx 0.5 1.0 1.5 Figure 8 34. The function y = 2x² - In 4x (x>0) has one minimum point. Find its first coordinate. 35. A Demand Equation If the demand equation for a certain commodity is p = 45/(In x), determine the marginal revenue function for this commodity, and compute the marginal rev- enue when x = 20. 36. Total Revenue Suppose that the total revenue function for a manufacturer is R(x) = 300 ln(x + 1), so the sale of x units of a product brings in about R(x) dollars. Suppose also that the total cost of producing x units is C(x) dollars, where C(x) = 2x. Find the value of x at which the profit function R(x) - C(x) will be maximized. Show that the profit function has a relative maximum and not a relative minimum point at this value of x. 37. An Area Problem Find the maximum area of a rectangle in the first quadrant with one corner at the origin, an opposite corner on the graph of y=-In x, and two sides on the coordinate axes. TECHNOLOGY EXERCISES 38. Analysis of the Effectiveness of an Insect Repellent Human hands covered with cotton fabrics impregnated with the insect repellent DEPA were inserted for 5 minutes into a test chamber containing 200 female mosquitoes. The function. f(x) = 26.48 - 14.09 In x gives the number of mosquito bites received when the concentration was x percent. [Note: The answers to parts (b)-(e) can be obtained either algebraically or from the graphs. You might consider trying both methods.] (Source: Journal of Medical Entomology.) (a) Graph f(x) and f'(x) for 0 < x≤ 6. (b) How many bites were received when the concentration. was 3.25%? (c) What concentration resulted in 15 bites? (d) At what rate is the number of bites changing with respect to concentration of DEPA whenx=2.75? (e) For what concentration does the rate of change of bites with respect to concentration equal -10 bites per per- centage increase in concentration?
27. Determine the domain of definition of the given function. (a) f(1) = In(In 1) (b) f(t)= In(In(In 1)) 28. Find the equations of the tangent lines to the graph of y= In |x| at x = 1 and x = -1. 29. Find the coordinates of the relative extreme point of y = x² In x, x>0. Then, use the second derivative test to decide if the point is a relative maximum point or a relative minimum point. 30. Repeat the previous exercise with y=√x In x. 31. The graphs of y=x+ ln x and y = ln 2x are shown in Fig. 6. (a) Show that both functions are increasing for x > 0. (b) Find the point of intersection of the graphs. 2 1 I -2 2 y=x+lnx -2 1.0 y-In 2r Figure 6 32. Repeat Exercise 31 with the functions y = x + ln x and y = In 5x. (See Fig. 7). 1.5 1.0 y=x+Inx 2.0 y-In 5x + 1.5 2.0 Figure 7 33. The graph of the function y=x²- In x is shown in Fig. 8. Find the coordinates of its minimum point. 3 2 f(x)=x²-Inx 0.5 1.0 1.5 Figure 8 34. The function y = 2x² - In 4x (x>0) has one minimum point. Find its first coordinate. 35. A Demand Equation If the demand equation for a certain commodity is p = 45/(In x), determine the marginal revenue function for this commodity, and compute the marginal rev- enue when x = 20. 36. Total Revenue Suppose that the total revenue function for a manufacturer is R(x) = 300 ln(x + 1), so the sale of x units of a product brings in about R(x) dollars. Suppose also that the total cost of producing x units is C(x) dollars, where C(x) = 2x. Find the value of x at which the profit function R(x) - C(x) will be maximized. Show that the profit function has a relative maximum and not a relative minimum point at this value of x. 37. An Area Problem Find the maximum area of a rectangle in the first quadrant with one corner at the origin, an opposite corner on the graph of y=-In x, and two sides on the coordinate axes. TECHNOLOGY EXERCISES 38. Analysis of the Effectiveness of an Insect Repellent Human hands covered with cotton fabrics impregnated with the insect repellent DEPA were inserted for 5 minutes into a test chamber containing 200 female mosquitoes. The function. f(x) = 26.48 - 14.09 In x gives the number of mosquito bites received when the concentration was x percent. [Note: The answers to parts (b)-(e) can be obtained either algebraically or from the graphs. You might consider trying both methods.] (Source: Journal of Medical Entomology.) (a) Graph f(x) and f'(x) for 0 < x≤ 6. (b) How many bites were received when the concentration. was 3.25%? (c) What concentration resulted in 15 bites? (d) At what rate is the number of bites changing with respect to concentration of DEPA whenx=2.75? (e) For what concentration does the rate of change of bites with respect to concentration equal -10 bites per per- centage increase in concentration?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Solve Q33&Q35 ASAP
Please solve both in the order to get positive feedback
![27. Determine the domain of definition of the given function.
(a) f(t)= In(In 1) (b) f(t) In(ln(In 1))
28. Find the equations of the tangent lines to the graph of
y = ln |x| at x = 1 and x = -1.
29. Find the coordinates of the relative extreme point of
y = x² lnx, x > 0. Then, use the second derivative test to
decide if the point is a relative maximum point or a relative
minimum point.
30. Repeat the previous exercise with y = √x ln x.
31. The graphs of y = x + ln x and y = In 2x are shown in Fig. 6.
(a) Show that both functions are increasing for x > 0.
(b) Find the point of intersection of the graphs.
3+
2
1
-1+
-2
-3
3.
2
1
Figure 6
32. Repeat Exercise 31 with the functions y = x + ln x and
y = In 5x. (See Fig. 7).
-1
هم
-2-
0.5
-3
Figure 7
y = x + ln x
1.0
0.5
y = In 2r
+
1.5
y = x + ln x
1.0
y = In 5
2.0
1.5
F
2.0
33. The graph of the function y = x² - In x is shown in Fig. 8.
Find the coordinates of its minimum point.
4x-3
=-[In(x + 5)]².4x³
x4 +5
Solutions to Check Your Understanding 4.5
1. Here, f(x) = [In(x4 + 5)]¹. By the chain rule,
f'(x) = (-1) [In(x4 + 5)]².In(x4 + 5)
dx
5-
4
3
31
2+
1+
0
2. f'(x) = n(In x) =
4.5 The Derivative of In x 245
f(x)=x²-In x
0.5 1.0
Figure 8
34. The function y = 2x² - In 4x (x > 0) has one minimum
point. Find its first coordinate.
1.5
35. Demand Equation If the demand equation for a certain
commodity is p = 45/(In x), determine the marginal revenue
function for this commodity, and compute the marginal rev-
enue when x = 20.
36. Total Revenue Suppose that the total revenue function for a
manufacturer is R(x) = 300 ln(x + 1), so the sale of x units of a
product brings in about R(x) dollars. Suppose also that the total
cost of producing x units is C(x) dollars, where C(x) = 2x.
Find the value of x at which the profit function R(x) - C(x)
will be maximized. Show that the profit function has a relative
maximum and not a relative minimum point at this value of x.
37. An Area Problem Find the maximum area of a rectangle in the
first quadrant with one corner at the origin, an opposite corner
on the graph of y=-In x, and two sides on the coordinate axes.
TECHNOLOGY EXERCISES
38. Analysis of the Effectiveness of an Insect Repellent Human
hands covered with cotton fabrics impregnated with the
insect repellent DEPA were inserted for 5 minutes into a test
chamber containing 200 female mosquitoes. The function
f(x) = 26.48 14.09 In x gives the number of mosquito bites
received when the concentration was x percent. [Note: The
answers to parts (b)-(e) can be obtained either algebraically
or from the graphs. You might consider trying both methods.]
(Source: Journal of Medical Entomology.)
(a) Graph f(x) and f'(x) for 0 < x≤ 6.
(b) How many bites were received when the concentration
was 3.25%?
(c) What concentration resulted in 15 bites?
(d) At what rate is the number of bites changing with respect
to concentration of DEPA when .x = 2.75?
(e) For what concentration does the rate of change of bites
with respect to concentration equal -10 bites per per-
centage increase in concentration?
1 d,
In x =
In x dx
11 1
In x x
x ln x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda8ca31e-ab8e-4c2e-96b3-361b44dfca23%2F87d3e3af-788f-4dd6-9f6e-2141ff2e6b9b%2Fcz0dkow_processed.jpeg&w=3840&q=75)
Transcribed Image Text:27. Determine the domain of definition of the given function.
(a) f(t)= In(In 1) (b) f(t) In(ln(In 1))
28. Find the equations of the tangent lines to the graph of
y = ln |x| at x = 1 and x = -1.
29. Find the coordinates of the relative extreme point of
y = x² lnx, x > 0. Then, use the second derivative test to
decide if the point is a relative maximum point or a relative
minimum point.
30. Repeat the previous exercise with y = √x ln x.
31. The graphs of y = x + ln x and y = In 2x are shown in Fig. 6.
(a) Show that both functions are increasing for x > 0.
(b) Find the point of intersection of the graphs.
3+
2
1
-1+
-2
-3
3.
2
1
Figure 6
32. Repeat Exercise 31 with the functions y = x + ln x and
y = In 5x. (See Fig. 7).
-1
هم
-2-
0.5
-3
Figure 7
y = x + ln x
1.0
0.5
y = In 2r
+
1.5
y = x + ln x
1.0
y = In 5
2.0
1.5
F
2.0
33. The graph of the function y = x² - In x is shown in Fig. 8.
Find the coordinates of its minimum point.
4x-3
=-[In(x + 5)]².4x³
x4 +5
Solutions to Check Your Understanding 4.5
1. Here, f(x) = [In(x4 + 5)]¹. By the chain rule,
f'(x) = (-1) [In(x4 + 5)]².In(x4 + 5)
dx
5-
4
3
31
2+
1+
0
2. f'(x) = n(In x) =
4.5 The Derivative of In x 245
f(x)=x²-In x
0.5 1.0
Figure 8
34. The function y = 2x² - In 4x (x > 0) has one minimum
point. Find its first coordinate.
1.5
35. Demand Equation If the demand equation for a certain
commodity is p = 45/(In x), determine the marginal revenue
function for this commodity, and compute the marginal rev-
enue when x = 20.
36. Total Revenue Suppose that the total revenue function for a
manufacturer is R(x) = 300 ln(x + 1), so the sale of x units of a
product brings in about R(x) dollars. Suppose also that the total
cost of producing x units is C(x) dollars, where C(x) = 2x.
Find the value of x at which the profit function R(x) - C(x)
will be maximized. Show that the profit function has a relative
maximum and not a relative minimum point at this value of x.
37. An Area Problem Find the maximum area of a rectangle in the
first quadrant with one corner at the origin, an opposite corner
on the graph of y=-In x, and two sides on the coordinate axes.
TECHNOLOGY EXERCISES
38. Analysis of the Effectiveness of an Insect Repellent Human
hands covered with cotton fabrics impregnated with the
insect repellent DEPA were inserted for 5 minutes into a test
chamber containing 200 female mosquitoes. The function
f(x) = 26.48 14.09 In x gives the number of mosquito bites
received when the concentration was x percent. [Note: The
answers to parts (b)-(e) can be obtained either algebraically
or from the graphs. You might consider trying both methods.]
(Source: Journal of Medical Entomology.)
(a) Graph f(x) and f'(x) for 0 < x≤ 6.
(b) How many bites were received when the concentration
was 3.25%?
(c) What concentration resulted in 15 bites?
(d) At what rate is the number of bites changing with respect
to concentration of DEPA when .x = 2.75?
(e) For what concentration does the rate of change of bites
with respect to concentration equal -10 bites per per-
centage increase in concentration?
1 d,
In x =
In x dx
11 1
In x x
x ln x
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