Mathematics Field of Specialization Exam
16 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the dependent variable in the differential equation 𝑑𝑃 = 𝑘𝑃 + 𝜔?

  • 𝑘
  • 𝜔
  • 𝑃 (correct)
  • 𝑡
  • Which of the following is NOT an autonomous differential equation?

  • 𝑑𝑥 = 1 + 𝑥² (correct)
  • 𝑑𝑤 = 𝑤
  • 𝑑𝑥 = 𝑒𝑥
  • 𝑑𝑦 = 2 − 𝑦
  • What technique of solving differential equations involves using geometry to visualize the behavior of the model?

  • Euler's method
  • Numerical approach
  • Analytic approach
  • Qualitative approach (correct)
  • What is the step size of the problem given by 𝑑𝑦 = 𝑡 − 𝑦², 𝑦(0) = 1, 0 ≤ 𝑡 ≤ 1, ∆𝑡 = 0.25?

    <p>∆𝑡 = 0.25</p> Signup and view all the answers

    Which of the following qualities refers to a differential equation having a unique solution?

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

    What value of 𝑦 will NOT result in an equilibrium point of the differential equation 𝑑𝑦 = sin(𝑦)?

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

    What is the purpose of finding the equilibrium points in a differential equation?

    <p>To determine the long-term behavior of the solution</p> Signup and view all the answers

    What is the term for a differential equation that does not explicitly contain the independent variable?

    <p>Autonomous differential equation</p> Signup and view all the answers

    What is the value of P for which the population would be constant in the logistic model?

    <p>P = N</p> Signup and view all the answers

    What is the solution to the differential equation 𝑑𝑦 = 𝑦2−1?

    <p>𝑦(𝑡) = 1 − 𝑡</p> Signup and view all the answers

    How many solutions are there in the differential equation 𝑑𝑣𝑐 = − 𝑣𝑐?

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

    What should be done to have an accurate approximate in Euler’s method?

    <p>Decrease the value of the step size</p> Signup and view all the answers

    What does it mean if all the solutions for the differential equation are the same and are within the same interval of time?

    <p>The solution to the differential equation is unique</p> Signup and view all the answers

    Which of the following values of y will result to the equilibrium points of the differential equation 𝑑𝑦 = (𝑦 − 2)(𝑦 + 3)?

    <p>I. -3 and II. 0</p> Signup and view all the answers

    What type of problem is the population model 𝑑𝑃 = 0.5𝑃(1 − 𝑃/200)?

    <p>Logistic growth</p> Signup and view all the answers

    What is the carrying capacity of the population in the model 𝑑𝑃 = 0.5𝑃(1 − 𝑃/200)?

    <p>200</p> Signup and view all the answers

    Study Notes

    Differential Equations

    Field of Specialization

    • Major in Mathematics
    • 25 items

    Instructions

    • 1.5 hours to complete the test
    • Shade only one answer per question
    • Avoid erasures

    Multiple Choice Questions

    Dependent Variables

    • In the differential equation dP = kP + ω, the dependent variable is P

    Autonomous Differential Equations

    • An autonomous differential equation does not involve time explicitly
    • Example: dy/dt = 2 - y
    • Example: dx/dt = e^x
    • NOT autonomous: dx/dt = 1 + x^2

    Techniques of Solving Differential Equations

    • Qualitative approach: uses geometry to overview the behavior of the model
    • Analytic approach
    • Numerical approach: Euler's method
    • Euler's method: decreases the step size for accurate approximation

    Existence and Uniqueness

    • Existence: the differential equation has a valid solution
    • Uniqueness: the differential equation has only one solution

    Phase Lines

    • Equilibrium points: f(y) = 0
    • Finding equilibrium points in a differential equation
    • Example: dy/dt = sin(y)

    Logistic Model

    • Population model: dP/dt = k(1 - P/N)P
    • Population is constant when P = N

    Solution to a Differential Equation

    • Example: dy/dt = y^2 - 1
    • Solution: y(t) = 1 + t

    Electrical Circuit

    • Differential equation: dv_c/dt = -v_c/RC
    • Initial condition: v_c(0) = 6
    • Slope field: number of solutions is 1

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Description

    This 25-item exam is designed to test one's knowledge and skills in mathematics, with a time limit of 1.5 hours. It covers various topics in mathematics and is a major assessment for mathematics students.

    More Like This

    Numeracy/Mathematics Exam Instructions
    12 questions
    Mathematics Exam Questions Summary
    24 questions

    Mathematics Exam Questions Summary

    AppreciatedRetinalite9006 avatar
    AppreciatedRetinalite9006
    Mathematics Exam Instructions - Parts 1 & 2
    48 questions
    Use Quizgecko on...
    Browser
    Browser