USING AND UNDERSTANDING MATHEMATICS W/
7th Edition
ISBN: 9780135239919
Author: BENNETT AND BR
Publisher: PEARSON
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Chapter 8.B, Problem 35E
To determine
The doubling time of the population.
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Refer to page 12 for a problem on solving a homogeneous differential equation.
Instructions:
• Simplify the equation into a homogeneous form.
Use appropriate substitutions to reduce complexity.
Solve systematically and verify the final result with clear back-substitutions.
Link:
[https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]
Refer to page 36 for solving a bang-bang control problem.
Instructions:
•
Formulate the problem, identifying the control constraints.
•
•
Apply Pontryagin's Maximum Principle to derive the switching conditions.
Clearly illustrate the switching points in the control trajectory. Verify the solution satisfies the
optimality criteria.
Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]
Total marks 16
5.
Let (N,F,P) be a probability space and let X : N → R be a
random variable such that the probability density function is given by
f(x)=ex for x € R.
(i)
Find the characteristic function of the random variable X.
[8 Marks]
(ii) Using the result of (i), calculate the first two moments of
the random variable X, i.e., E(X") for n = 1,2.
(iii)
What is the variance of X.
[6 Marks]
[2 Marks]
Chapter 8 Solutions
USING AND UNDERSTANDING MATHEMATICS W/
Ch. 8.A - A town’s population increases in one year from...Ch. 8.A - A town’s population increases in one year from...Ch. 8.A - The balance owed your credit card doubles from...Ch. 8.A - The number Of songs in your iPod has increased...Ch. 8.A - Which of the following is in example of...Ch. 8.A - On a chessboard with 64 squares, you place 1 penny...Ch. 8.A - At 11:00 you place a single bacterium in a bottle,...Ch. 8.A - Consider the bacterial population described in...Ch. 8.A - Consider the bacterial population described in...Ch. 8.A - Which of the following is not true of any...
Ch. 8.A - Describe basic differences between linear growth...Ch. 8.A - 2. Briefly explain how repeated doublings...Ch. 8.A - Briefly summarize the Story Of the bacteria in the...Ch. 8.A - Explain the meaning Of the two key facts about...Ch. 8.A - Decide whether each of the following statements...Ch. 8.A - Suppose you had a magic hank account in which your...Ch. 8.A - A small town that grows exponentially can become a...Ch. 8.A - H. Human population has been growing exponentially...Ch. 8.A - Linear or Exponential? State whether the growth...Ch. 8.A - Prob. 10ECh. 8.A - Linear or Exponential? State whether the growth...Ch. 8.A - Prob. 12ECh. 8.A - Linear or Exponential? State whether the growth...Ch. 8.A - Linear or Exponential? State whether the growth...Ch. 8.A - Linear or Exponential? State whether the growth...Ch. 8.A - 9-16: Linear or Exponential? State whether the...Ch. 8.A - Chessboard Parable. Use the chessboard parable...Ch. 8.A - Chessboard Parable. Use the chessboard parable...Ch. 8.A - Chessboard Parable. Use the chessboard parable...Ch. 8.A - Chessboard Parable. Use the chessboard parable...Ch. 8.A - Magic Penny Parable. Use the magic penny parable...Ch. 8.A - Magic Penny Parable. Use the magic penny parable...Ch. 8.A - Magic Penny Parable. Use the magic penny parable...Ch. 8.A - Magic Penny Parable. Use the magic penny parable...Ch. 8.A - Bacteria in a Bottle Parable. Use the bacteria...Ch. 8.A - Bacteria in a Bottle Parable. Use the bacteria...Ch. 8.A - Bacteria in a Bottle Parable. Use the bacteria...Ch. 8.A - Bacteria in a Bottle Parable. Use the bacteria...Ch. 8.A - 29. Human Doubling. Human population in the year...Ch. 8.A - Doubling Time versus Initial Amount. a. Would you...Ch. 8.A - Facebook Users. The table shows the number of...Ch. 8.A - Prob. 32ECh. 8.A - Exponential Growth. Identify at least two news...Ch. 8.A - Prob. 34ECh. 8.A - Prob. 35ECh. 8.B - Prob. 1QQCh. 8.B - Suppose your salary increases at a rate of 4.5%...Ch. 8.B - Which of the following is not a good approximation...Ch. 8.B - A town’s population doubles in 23 years. Its...Ch. 8.B - Radioactive tritium (hvdrogen-3) has a halt-life...Ch. 8.B - Radioactive uramum-235 has a hall-life of about...Ch. 8.B - The population of an endangered species decreases...Ch. 8.B - log10108= a.100,000,000 b. 108 c.8Ch. 8.B - A rural popular ion decreases at a rate of 20% per...Ch. 8.B - Prob. 10QQCh. 8.B - What is a doubling tune? Suppose a population has...Ch. 8.B - Prob. 2ECh. 8.B - State the approximate doubting time formula and...Ch. 8.B - Prob. 4ECh. 8.B - Prob. 5ECh. 8.B - 6. State the approximate hall-life formula and the...Ch. 8.B - 7. Briefly describe exact doubling time and...Ch. 8.B - 8. Give an example in which it is important to use...Ch. 8.B - Our town is growing with a doubling time of 25...Ch. 8.B - Our town is growing at a rate of 7% per year, so...Ch. 8.B - A toxic chemical decays with a hall-life of 10...Ch. 8.B - The hall-life of plutomum-239 is about 24,000...Ch. 8.B - Logarithms. Refer to the Brief Review on p. 488....Ch. 8.B - Logarithms. Refer to the Brief Review on p. 488....Ch. 8.B - Prob. 15ECh. 8.B - Prob. 16ECh. 8.B - 13-24: Logarithms. Refer to the Brief Review on p....Ch. 8.B - Prob. 18ECh. 8.B - Logarithms. Refer to the Brief Review on p. 488....Ch. 8.B - Logarithms. Refer to the Brief Review on p. 488....Ch. 8.B - 13-24: Logarithms. Refer to the Brief Review on p....Ch. 8.B - Prob. 22ECh. 8.B - Prob. 23ECh. 8.B - 23-24. Logarithm Calculations
24. Using the...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - 25-32: Doubling Time. Each exercise gives a...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - 25-32: Doubling Time. Each exercise gives a...Ch. 8.B - World Population. In 2017, estimated world...Ch. 8.B - World Population. In 2017, estimated world...Ch. 8.B - 31. Rabbits. A community of rabbits begins with an...Ch. 8.B - Prob. 36ECh. 8.B - Approximate Doubling Time Formula. Use the...Ch. 8.B - 37-40: Approximate Doubling Time Formula. Use the...Ch. 8.B - Approximate Doubling Time Formula. Use the...Ch. 8.B - 37-40: Approximate Doubling Time Formula. Use the...Ch. 8.B - Half-Life. Each exercise gives a half-life for an...Ch. 8.B - Half-Life. Each exercise gives a half-life for an...Ch. 8.B - Half-Life. Each exercise gives a half-life for an...Ch. 8.B - 41 -48: Half-Life. Each exercise gives a half-life...Ch. 8.B - 41-48: Half-Life. Each exercise gives a half-life...Ch. 8.B - 41 -48: Half-Life. Each exercise gives a half-life...Ch. 8.B - 41-48: Half-Life. Each exercise gives a half-life...Ch. 8.B - 41 -48: Half-Life. Each exercise gives a half-life...Ch. 8.B - Approximate Half-Life Formula. Use the approximate...Ch. 8.B - 49-52: Half-Life Formula. Use the approximate...Ch. 8.B - Approximate Half-Life Formula. Use the approximate...Ch. 8.B - 49-52: Half-Life Formula. Use the approximate...Ch. 8.B - Prob. 53ECh. 8.B - Exact Formulas. Compare the doubling times found...Ch. 8.B - Exact Formulas. Compare the doubling times found...Ch. 8.B - Exact Formulas. Compare the doubling times found...Ch. 8.B - Prob. 57ECh. 8.B - 58. Nuclear Weapons. Thermonuclear weapons use...Ch. 8.B - Prob. 59ECh. 8.B - Yucca Mountain. The U.S. government spent nearly...Ch. 8.B - Crime Rate. The homicide rate decreases at a rate...Ch. 8.B - Drug Metabolism. A particular antibiotic is...Ch. 8.B - Atmospheric Pressure. The pressure of Earth's...Ch. 8.B - Prob. 64ECh. 8.B - 65. Radioactive Half-Life. Find a news story that...Ch. 8.B - Prob. 66ECh. 8.B - Prob. 67ECh. 8.B - Prob. 68ECh. 8.B - Prob. 69ECh. 8.C - World population is currently rising by about 80...Ch. 8.C - The most recent doubling of world population took...Ch. 8.C - The primary reason for the rapid growth of human...Ch. 8.C - The carrying capacity of the Earth is defined as...Ch. 8.C - Which of the billowing would cause estimates of...Ch. 8.C - 6. Recall the bacteria in a bottle example from...Ch. 8.C - When researchers project that human population...Ch. 8.C - Prob. 8QQCh. 8.C - Prob. 9QQCh. 8.C - Prob. 10QQCh. 8.C - Based on Figure 8.3, contrast the changes in human...Ch. 8.C - Prob. 2ECh. 8.C - Haw do today’s birth and death rates compare to...Ch. 8.C - Prob. 4ECh. 8.C - Prob. 5ECh. 8.C - What is overshot and collapse? Under what...Ch. 8.C - Prob. 7ECh. 8.C - 8. If birth rates fall more than death rates, the...Ch. 8.C - The carrying capacity of our planet depends only...Ch. 8.C - to rapid increases in computing technology, we...Ch. 8.C - In the wild, we always expect the population of...Ch. 8.C - 12. Past civilizations that no longer exist must...Ch. 8.C - Varying Growth Rates. Starting from a 2017 world...Ch. 8.C - Varying Growth Rates. Starting from a 2017 world...Ch. 8.C - Varying Growth Rates. Starting from a 2017 world...Ch. 8.C - Varying Growth Rates. Starting from a 2017 world...Ch. 8.C - Birth and Death Rates. The following table gives...Ch. 8.C - 17-20: Birth and Death Rates. The following table...Ch. 8.C - 17-20: Birth and Death Rates. The following table...Ch. 8.C - Birth and Death Rates. The following table gives...Ch. 8.C - 21. Logistic Growth. Consider a population that...Ch. 8.C - Logistic Growth. Consider a population that begins...Ch. 8.C - U.S Population. Starting from an estimated U.S....Ch. 8.C - Prob. 24ECh. 8.C - Prob. 25ECh. 8.C - Prob. 26ECh. 8.C - Prob. 27ECh. 8.C - 28. Slower Growth. Repeat Exercise 27, but use a...Ch. 8.C - Prob. 29ECh. 8.C - World Carrying Capacity. For the given carrying...Ch. 8.C - 29-32: World Carrying Capacity. For the given...Ch. 8.C - 29-32: World Carrying Capacity. For the given...Ch. 8.C - Prob. 33ECh. 8.C - Prob. 34Ch. 8.C - Population Predictions. Find population...Ch. 8.C - Prob. 36ECh. 8.C - Prob. 37ECh. 8.C - Prob. 38ECh. 8.C - Prob. 39ECh. 8.C - Prob. 40ECh. 8.D - Prob. 1QQCh. 8.D - Prob. 2QQCh. 8.D - 3. What is a 0-decibel sound?
the softest sound...Ch. 8.D - 4. A sound of 85 decibels is defined to be
a. 85...Ch. 8.D - Prob. 5QQCh. 8.D - Prob. 6QQCh. 8.D - Prob. 7QQCh. 8.D - Prob. 8QQCh. 8.D - Prob. 9QQCh. 8.D - Prob. 10QQCh. 8.D - What is the magnitude scale for earthquakes? What...Ch. 8.D - What is the decibel scale? Describe how it is...Ch. 8.D - What is pH? What pH values define an acid, a base,...Ch. 8.D - What is acid rain? Why is it a serious...Ch. 8.D - An earthquake of magnitude 6 will do twice as much...Ch. 8.D - A 120-dB wand is 20% louder than a 100-dB sound.Ch. 8.D - If I double the amount of water in the cup, I'll...Ch. 8.D - The lake water was crystal clear, so It could not...Ch. 8.D - Earthquake Magnitudes. Use the earthquake...Ch. 8.D - Earthquake Magnitudes. Use the earthquake...Ch. 8.D - Earthquake Magnitudes. Use the earthquake...Ch. 8.D - Earthquake Magnitudes. Use the earthquake...Ch. 8.D - Earthquake Magnitudes. Use the earthquake...Ch. 8.D - 9-14: Earthquake Magnitudes. Use the earthquake...Ch. 8.D - The Decibel Scale. Use the decibel scale to answer...Ch. 8.D - 15-20: The Decibel Scale. Use the decibel scale to...Ch. 8.D - 15-20: The Decibel Scale. Use the decibel scale to...Ch. 8.D - The Decibel Scale. Use the decibel scale to answer...Ch. 8.D - The Decibel Scale. Use the decibel scale to answer...Ch. 8.D - Prob. 20ECh. 8.D - Prob. 21ECh. 8.D - Prob. 22ECh. 8.D - Prob. 23ECh. 8.D - Inverse Square Law. Use the inverse square law for...Ch. 8.D - 25-32: The pH Scale. Use the pH scale to answer...Ch. 8.D - 25-32: The pH Scale. Use the pH scale to answer...Ch. 8.D - 25-32: The pH Scale. Use the pH scale to answer...Ch. 8.D - Prob. 28ECh. 8.D - Prob. 29ECh. 8.D - Prob. 30ECh. 8.D - Prob. 31ECh. 8.D - Prob. 32ECh. 8.D - Logarithmic Thinking. Briefly describe, in words,...Ch. 8.D - 33-38: Logarithmic Thinking. Briefly describe, in...Ch. 8.D - Logarithmic Thinking. Briefly describe, in words,...Ch. 8.D - Logarithmic Thinking. Briefly describe, in words,...Ch. 8.D - Prob. 37ECh. 8.D - Prob. 38ECh. 8.D - 39. Sound and Distance.
The decibel level for...Ch. 8.D - 40. Variation in Sound with Distance. Suppose that...Ch. 8.D - Toxic Dumping in Acidified Lakes. Consider a...Ch. 8.D - Prob. 42ECh. 8.D - Earthquakes in the News. Find a recent news story...Ch. 8.D - Disasters. Find the death lolls for some of the...Ch. 8.D - Prob. 45ECh. 8.D - Prob. 46ECh. 8.D - Ocean Acidification. Research ocean acidification...
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- 6. Let P be the standard normal distribution, i.e., P is the proba- bility measure on (R, B(R)) given by 1 dP(x) = 를 = e dx. √2πT Consider the random variables 21 fn(x) = (1 + x²) en+2, x Є R, n Є N. Using the dominated convergence theorem, prove that the limit Total marks 9 exists and find it. lim E(fn) n∞ [9 Marks]arrow_forwardRefer to page 38 for solving an optimal control problem using dynamic programming. Instructions: • Define the value function and derive the Hamilton-Jacobi-Bellman (HJB) equation. • Solve the HJB equation explicitly, showing all intermediate steps and justifications. Verify the solution satisfies the boundary conditions and optimality. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 18 for solving a second-order linear non-homogeneous differential equation. Instructions: Solve the associated homogeneous equation first. Use either the method of undetermined coefficients or variation of parameters for the particular solution. • Provide detailed steps for combining solutions into the general solution. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forward
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