Differential Equations Final Review
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Questions and Answers

How many students initially brought the disease back from Spring Break?

  • 0 students
  • 50 students
  • 1 student (correct)
  • 99 students

    • According to the model, how many students are infected 4 days after Spring Break?

  • 10 students
  • 99 students
  • 50 students
  • 90 students (correct)

    • What does the limit of I(t) represent in the context of the model?

  • The initial number of infected people
  • The number of people who recover from the virus
  • Total capacity of the dormitory
  • The number of people who will ever get infected (correct)

    • When is the disease spreading most rapidly based on the graph's behavior?

    <p>When half the dormitory is infected</p> Signup and view all the answers

    How long until 90% of the small town's population has heard the rumor?

    <p>At least 6 hours</p> Signup and view all the answers

    What is the solution form for the limited growth model with initial condition y(0) = y0?

    <p>y(t) = M - C e^{-kt}</p> Signup and view all the answers

    What does lim y(t) indicate in the limited growth model?

    <p>It describes long-term stable population size</p> Signup and view all the answers

    What is the limiting behavior of the Gompertz model as t approaches infinity?

    <p>A constant value for y(t)</p> Signup and view all the answers

    What type of calculator is required for the final exam?

    <p>Scientific calculator</p> Signup and view all the answers

    What is the weight of the final exam towards the overall course grade?

    <p>35%</p> Signup and view all the answers

    When will homework from Chapters 8 & 9 be accepted?

    <p>Only until the last class</p> Signup and view all the answers

    What materials are not allowed during the final exam?

    <p>Notes, books, laptops, etc.</p> Signup and view all the answers

    If a student missed class, what should they do to catch up?

    <p>Get notes from someone in the class</p> Signup and view all the answers

    How will partial credit be awarded on the exam?

    <p>If calculations are shown</p> Signup and view all the answers

    Which chapters are the focus of the Final Review Sheet?

    <p>Chapters 8 &amp; 9</p> Signup and view all the answers

    What is a suggestion for students who struggled on a previous test?

    <p>Use that test as a study guide</p> Signup and view all the answers

    What is the term for the value $x.95$ in the context of financial loss in a portfolio?

    <p>Value at Risk</p> Signup and view all the answers

    To compute $E(X|X \geq c)$, which function is used to renormalize $f(x)$?

    <p>$\hat{f}(x)$</p> Signup and view all the answers

    How would you compute the median using the formula for $\alpha$-quantiles?

    <p>Set $\alpha = 0.5$</p> Signup and view all the answers

    What represents the expected tail loss $E(X|X \geq x.95)$ specifically?

    <p>Value at Risk</p> Signup and view all the answers

    Which condition must be met for a function $g(x)$ to be a valid probability density function (pdf)?

    <p>Must equal one when integrated over its domain</p> Signup and view all the answers

    What is the consequence of a negative variance for a random variable $X$?

    <p>Indicates a data error</p> Signup and view all the answers

    In which scenario does the mean and median fall in the same range of $[a, b]$?

    <p>When the distribution is uniform</p> Signup and view all the answers

    What must be true for a function $f(x)$ defined on $[0, 6]$ to be successfully normalized into a pdf?

    <p>It must be continuous</p> Signup and view all the answers

    What is the mean of an exponential random variable with parameter 𝑎?

    <p>1/𝑎</p> Signup and view all the answers

    For an exponential random variable 𝑋 with the probability density function 𝑓(𝑥) = 3e^{-3𝑥}, what is the variance?

    <p>1/9</p> Signup and view all the answers

    What is the probability that a livery service is more than 15 minutes late, if the service follows an exponential distribution with mean 5 minutes?

    <p>0.2231</p> Signup and view all the answers

    What is the value of 𝑧 for 𝑃(𝑋 ≤ 4) if 𝑋 is a normal random variable with mean 7 and standard deviation 2?

    <p>-1.5</p> Signup and view all the answers

    What is the probability of a normal random variable 𝑋 falling between its mean and one standard deviation above it, given 𝜇 = 7 and 𝜎 = 2?

    <p>0.6826</p> Signup and view all the answers

    Which of the following values corresponds to the 90th percentile of a normal distribution with mean 7 and standard deviation 2?

    <p>9.28</p> Signup and view all the answers

    What is the median of a continuous probability density function 𝑓(𝑥) = 3𝑒^{-3𝑥}$ on [0, ∞)?

    <p>0.693</p> Signup and view all the answers

    In a normal distribution, which statement accurately describes the relationship between the mean, median, and mode?

    <p>Mean, median, and mode are all equal</p> Signup and view all the answers

    What are the two properties a function must satisfy to be considered a probability density function (pdf) on the interval [a, b]?

    <p>It must be non-negative and integrate to 1 over [a, b].</p> Signup and view all the answers

    Which normalization factor is required to convert the function g(x) = 5x - x² defined on [0, 5] into a valid pdf?

    <p>1/15</p> Signup and view all the answers

    What occurs when trying to calculate the mean and variance for the pdf f(x) = x² on the interval [1, ∞)?

    <p>Both mean and variance are infinite.</p> Signup and view all the answers

    For a uniform random variable X on the interval [a, b], which of the following is true about its mean?

    <p>Mean is equal to the midpoint of the interval.</p> Signup and view all the answers

    What is the cumulative distribution function (CDF) associated with the pdf f(x) = 3e^(-3x) on the interval [0, ∞)?

    <p>1 - e^(-3x)</p> Signup and view all the answers

    Given the pdf f(x) = 4x on the interval [1, 3], what is the CDF F(x) for this interval?

    <p>2x - 1</p> Signup and view all the answers

    For the uniform distribution of X on [0, 10], what is the probability P(4 ≤ X ≤ 8)?

    <p>0.4</p> Signup and view all the answers

    What is the key distinction between a probability density function (pdf) and its cumulative distribution function (CDF)?

    <p>A pdf represents probabilities for specific outcomes while CDF represents cumulative probabilities.</p> Signup and view all the answers

    What is the primary limitation of the exponential growth model?

    <p>It fails to account for limited resources affecting population growth.</p> Signup and view all the answers

    In the Doomsday Equation, what happens when the exponent is greater than 1?

    <p>Population growth becomes infinite in a finite time.</p> Signup and view all the answers

    What initial condition is used to solve for 'C' in the Doomsday Equation?

    <p>$y(0) = 2$</p> Signup and view all the answers

    Which of the following equations is separable?

    <p>$ rac{dy}{dx} = 2x - y$</p> Signup and view all the answers

    What is true about the solution to the differential equation $ rac{dy}{dx} = 2y(1-5)$ with the initial condition $y(0) = 1$?

    <p>It approaches 5 as $t o ext{infinity}$.</p> Signup and view all the answers

    Which of the following statements about the general solutions of the equations provided is correct?

    <p>The function $y = rac{ ext{ln} x}{x^2}$ is not a solution of the differential equation $x^2 rac{dy}{dx} + xy = 1$.</p> Signup and view all the answers

    What is the expected behavior of the population model described by the Doomsday Equation?

    <p>Population will become infinite almost immediately.</p> Signup and view all the answers

    In the context of the differential equations presented, which equation represents a first-order linear differential equation?

    <p>$ rac{dy}{dx} + 3y = 5x + 2$</p> Signup and view all the answers

    Study Notes

    Final Review Sheet

    • The final exam is cumulative, covering all class material and homework assignments from the entire semester.
    • Missed homework problems should be completed.
    • Notes from classmates are useful if you missed any classes.
    • The final exam will be in the classroom.
    • No make-up or rescheduling of the final exam is allowed.
    • A scientific or graphing calculator is required for the final exam.
    • Partial credit will be awarded only if work is shown.
    • Chapter 8 & 9 homework is due by the last class.

    Practice Problems for Final Exam

    • Extra practice problems are provided for review, not for grading
    • Problems are similar to quiz and final format

    Specific Questions

    • Key terms and concepts, including differential equations, separable differential equations, autonomous differential equations, and first-order linear differential equations are defined, including examples and methods to solve them.
    • Finding the specific solutions of different order linear differential equations is discussed.
    • Methods of calculating quantities for exponential decay models (e. g. half-lives, remaining amounts ) are detailed and illustrated with examples.
    • Exponential growth model problems, including finding doubling time and specific amounts over time, are illustrated.
    • Properties of a probability density function (pdf) are provided (e.g. how to determine a function is a pdf based on the criteria of the function)
    • Questions regarding the calculation of the mean, variance, and standard deviation for probability density functions are included.
    • How to find a Cumulative distribution function (CDF) given a pdf
    • Questions on Uniform random variables and related calculations such as finding the CDF, PDF, mean, variance, and standard deviation for uniformly distributed variables are explained
    • Questions relating to exponential random variables.
    • Finding probabilities and expected value are included.
    • Questions regarding Normal random variables and their distributions are included.
    • Calculating the percentages of values in specified intervals.
    • Problems with normal random variables are illustrated.
    • Finding Medians for specified functions.
    • Calculation of conditional expectations are detailed with examples.

    Economic Concepts

    • Consumer and producer surplus are defined and calculated given particular functions

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    Related Documents

    MA139 Final Review Sheet NEW 2024 PDF

    Description

    Prepare for your cumulative final exam with this comprehensive review sheet on differential equations. Focus on key terms and concepts, such as separable differential equations and first-order linear differential equations. Practice problems similar to the exam format are also included to enhance your understanding.

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