Differential Equations Final Review

The final exam contributes 35% towards your course grade, making it a significant portion of your overall score.

Your previous test scores, if you did not perform well on them, will only contribute a small percentage (10%) to your final course grade.

The Final Exam covers all materials from the entire semester, including all chapters, homework problems, and topics discussed in class.

You are only allowed to use a scientific or graphing calculator during the Final Exam. No notes, books, laptops, or other electronic devices are permitted.

Partial credit might be awarded on the Final Exam if you show your work clearly and demonstrate your understanding of the problem-solving process.

The Final Exam must be submitted by the specified deadline as outlined on the syllabus. There are no makeups or reschedulings allowed.

Chapters 8 & 9 homework will be accepted only until the last class. Make sure to catch up and complete all assignments.

Previous quizzes, tests, and review sheets with solutions are available to help you prepare for the Final Exam. Use them to your advantage!

A mathematical model used to describe the growth of a population that is limited by carrying capacity. It assumes that the growth rate is proportional to both the current population size and the difference between the carrying capacity and the current population size.

The maximum population size that a particular environment can sustain indefinitely, given available resources.

The starting population size at the beginning of the observation period (time = 0).

The point on a curve where the rate of change (slope) is at its maximum. In the context of the logistic model, it represents the point where the population growth rate is the highest.

A mathematical model used to describe growth that starts exponentially but slows down over time, approaching a limiting value. It assumes that the growth rate is proportional to both the current population size and the natural logarithm of the ratio of the carrying capacity to the current population size.

A mathematical model that describes growth with an upper limit. Assuming a growth rate proportional to the difference between the carrying capacity and the current population size.

A solution to a differential equation that satisfies a particular initial condition.

The value of the dependent variable (population size in this case) at the initial time (t=0).

A function 𝑓(𝑥) that describes the probability distribution of a continuous random variable 𝑋. It satisfies two properties: 1) 𝑓(𝑥) ≥ 0 for all 𝑥, and 2) The area under the curve of 𝑓(𝑥) from 𝑎 to 𝑏 is equal to 1.

The process of scaling a function 𝑔(𝑥) on an interval [𝑎, 𝑏] to create a PDF 𝑓(𝑥) by dividing 𝑔(𝑥) by the integral of 𝑔(𝑥) over the interval.

The expected value of a continuous random variable 𝑋 with PDF 𝑓(𝑥) is calculated by integrating 𝑥 * 𝑓(𝑥) over the entire range of 𝑋.

The variance of a continuous random variable 𝑋 with PDF 𝑓(𝑥) measures the spread of the distribution around the mean. It is calculated by integrating (𝑥 - 𝜇)² * 𝑓(𝑥) over the entire range of 𝑋.

The standard deviation of a continuous random variable 𝑋 is the square root of the variance. It gives a measure of the typical deviation of values from the mean.

A function 𝐹(𝑥) that gives the probability that a continuous random variable 𝑋 is less than or equal to a given value 𝑥. It is defined as the integral of the PDF from 𝑎 to 𝑥.

The PDF 𝑓(𝑥) can be obtained by taking the derivative of the CDF 𝐹(𝑥).

A continuous random variable where each value in a given interval has an equal probability of occurring.

A differential equation where population growth becomes infinite in a finite amount of time. This occurs when the exponent in the growth term is slightly greater than 1.

A model of population growth where the rate of growth is proportional to the size of the population, resulting in a continuous increase at an increasingly rapid rate.

A growth model that assumes unlimited resources, resulting in an unrealistic scenario of exponential growth without any limits.

A differential equation where the variables can be separated on opposite sides of the equation, allowing for integration to solve for the solution.

A differential equation that can be expressed in the form dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x.

The rate of change of a curve at a particular point, determined by the derivative of the function at that point, which represents the instantaneous rate of change.

The general solution of a differential equation represents all possible solutions, while the specific solution is a particular solution obtained by applying an initial condition to the general solution.

For an exponential random variable with parameter 'a', the mean is calculated as 1/a. This represents the average value of the variable.

The variance of an exponential random variable with parameter 'a' is 1/a². This measures the spread of the distribution.

The standard deviation of an exponential random variable with parameter 'a' is √(1/a²). It provides another measure of the spread.

A normal random variable is defined by two parameters: the mean (µ) and standard deviation (σ). These parameters determine the shape and position of the distribution.

In a normal distribution, 68% of the data falls within one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

The z-score is a standardized value that represents how many standard deviations a data point is from the mean. It is calculated as (x - µ) / σ.

The median of a random variable 𝑋 is the value 'm' that divides the total probability into two equal halves. It is the point where 𝑃(𝑋 ≤ m) = 0.5.

The 'α'-quantile of a random variable 𝑋 is the value 'xα' where 𝑃(𝑋 ≤ xα) = α. It represents the point below which a specified proportion 'α' of the data falls.

The 95th percentile (𝑥.95 ) of a probability distribution, representing the maximum potential financial loss within a specified confidence level (usually 95%). It's the worst-case scenario for a portfolio, taking into account the probability of loss.

Specific points on a probability distribution that divide the data into equal intervals. Each 𝛼-Quantile represents the value of the random variable that corresponds to a specific probability 𝛼. For example, the 0.25-quantile, or the 25th percentile, is the value below which 25% of the data lies.

The 0.5-quantile, or the 50th percentile, representing the midpoint of a probability distribution. It separates the data into equal halves, with 50% of the data falling below and 50% above it.

The difference between the 75th percentile (Q3) and the 25th percentile (Q1) of a probability distribution. It measures the spread of the middle 50% of the data, indicating variability.

The expected value of a random variable 𝑋 given that another event has already occurred. In context of finance, this event often involves exceeding a certain loss threshold.

The expected value of a random variable 𝑋 given that 𝑋 is above the Value at Risk (𝑥.95). In finance, it represents the average loss that is incurred when the financial loss exceeds the VaR threshold.

Adjusting a PDF to ensure it remains a valid probability distribution over a subset of its original domain. This involves scaling the PDF by a constant factor to ensure that the area under the curve on the new domain equals 1.

The average value of a random variable 𝑋, calculated by summing all possible values of 𝑋 and dividing by the total number of values. It represents the center of the distribution.