COSC/MATH 201 · Exam 1 Study Guide

Friday, October 9, 2026

What to prepare

Exam 1 is Friday, October 9, 2026. It starts with 20 multiple-choice questions (80 points) and ends with two short calculations (20 points). It is designed to take about 30–35 minutes, not the full 50-minute period. Any permitted aids will be announced in class before the exam. The practice below uses different numbers and contexts from the exam.

Review the course modules for R Fundamentals; Modules 2.2, 2.3, 4.1, 4.2, 4.3, and 5.2; and Growth Models in Practice. Project 1 and Project 2 are useful practice, but you will not have to write a whole Quarto report or run R on the paper exam.

From a story to an update

A stock is an amount at a time; a flow changes it. Know whether a flow adds to or subtracts from a stock. For a time step \(\Delta t\),

\[ \text{new stock}=\text{old stock}+\Delta t(\text{inflow rate}-\text{outflow rate}). \]

If the rate is per day, \(\Delta t\) must be in days. A rate of 6 fish/day over half a day adds 3 fish, not 6. When a model has several stocks, compute all flows from the old values before updating any stock. Otherwise, the order of your R statements changes the model.

Recognize and reason about the models

Model Typical rate of change What to explain
Unconstrained growth or decay \(rP\) The rate is proportional to the current amount; a negative \(r\) gives decay. This model has no resource limit.
Constrained growth \(rP(1-P/K)\) \(K\) is carrying capacity. Below \(K\) the stock grows; above \(K\) it declines (for positive \(r\)).
Competition Each population’s growth is reduced by interaction with the other An encounter term such as \(cAB\) increases when either population increases. Both competitors are harmed by competition, though not necessarily equally.
Predator–prey Prey lose and predators gain from encounters The same encounter can have opposite signs in the two equations. Predator peaks often lag prey peaks.
SIR Susceptible \(\to\) infectious \(\to\) recovered Infection transfers people from \(S\) to \(I\); recovery transfers people from \(I\) to \(R\). In the simple closed model, \(S+I+R\) stays constant.

Recognize these models from a short story, use an equation when the question supplies it, and predict the direction of a change. You will not be asked to reproduce the textbook’s diagrams or memorize its differential equations. Distinguish a model’s output from evidence that the model accurately describes the world.

For a simple SIR model, this guide uses \(\text{infection}=\beta SI/N\) and \(\text{recovery}=\gamma I\). The book and our module discuss another transmission convention; use the convention explicitly stated in a question. Know which group loses and gains from each transfer.

Read small R simulations

Review vectors made with c(), indexing with [ ], assigning with <-, and tracing one short update. Know why old <- stock[i - 1] preserves the previous value. You will not write a complete R program or need to memorize plot commands, Quarto syntax, or loop syntax for this exam.

Numerical error and evidence

  • Absolute error is \(|\text{estimate}-\text{reference}|\) in the original units. Relative error divides that difference by \(|\text{reference}|\) (when the reference is nonzero); multiply by 100 for a percentage.
  • A computer stores only finitely many numbers. R’s 0.1 + 0.2 may not compare exactly equal to 0.3; arithmetic order can matter. Roundoff from one step can carry forward into later steps.
  • A finite-step Euler simulation approximates continuous change. Compare with a known solution when one exists, and repeat with a smaller \(\Delta t\) to see whether the result stabilizes. A smaller step is evidence about numerical approximation, not proof that the model’s assumptions match reality.
  • Overflow means a result is too large for the numeric representation; underflow means it is too small to represent normally. They are different from using a time step that is too coarse.

The book’s notation, precision, and magnitude

Module 5.2 uses a particular normalized exponential notation: put the decimal point immediately before the first nonzero digit. For example,

\[ 0.0004500=0.4500\times10^{-3}. \]

Ordinary scientific notation would write the same value as \(4.500\times10^{-4}\); that is mathematically correct but not the book’s normalized form. In the book’s terminology, the digits 4500 are the significand, the precision is 4 significant digits, and the magnitude is \(10^{-3}\) (not just \(-3\)). Leading zeros do not count; trailing zeros after a decimal point do. For an integer written without a decimal point, the book does not count trailing zeros as significant: \(3{,}704{,}000=0.3704\times10^7\) has precision 4 and magnitude \(10^7\).

Multiple-choice practice

Try these before opening the answers. Each has one best answer.

  1. A savings balance earns a fixed percentage of its current value each year, with no stated cap. Which model best fits? A. unconstrained growth; B. constrained growth; C. competition; D. SIR.
  2. A logistic population is at its carrying capacity. The model’s immediate growth rate is A. positive; B. negative; C. zero; D. unknown.
  3. A stock gains 12 units/day. With a quarter-day step, its gain is A. 3; B. 12; C. 24; D. 48 units.
  4. Two animal species eat the same limited food. Their interaction is most naturally A. infection; B. competition; C. predator–prey; D. unconstrained decay.
  5. In a predator–prey model, more prey with predators fixed initially makes encounter terms A. smaller; B. larger; C. negative; D. disappear.
  6. In a closed SIR model, recovery transfers a person from A. \(S\) to \(I\); B. \(I\) to \(R\); C. \(R\) to \(S\); D. the population to outside it.
  7. If x <- c(3, 8, 11), then x[3] is A. 3; B. 8; C. 11; D. 22.
  8. A reference value is 200 and an approximation is 190. Percent relative error is A. 1%; B. 5%; C. 10%; D. 50%.
  9. A value is too large for the computer’s numeric representation. This is A. overflow; B. validation; C. a model assumption; D. time-step error.
  10. Reducing \(\Delta t\) changes a simulation result substantially. That tells you most directly that A. the model is correct in nature; B. the numerical answer is sensitive to step size; C. the parameter values are measured correctly; D. the graph is mislabeled.
  11. In the book’s normalized notation, \(0.00620\) is A. \(6.20\times10^{-3}\); B. \(0.620\times10^{-2}\); C. \(0.0620\times10^{-1}\); D. \(0.620\times10^2\).
  12. The precision of the written number \(0.00620\) is A. 2; B. 3; C. 4; D. 5 significant digits.
  13. The magnitude of \(0.620\times10^{-2}\) is A. \(-2\); B. \(0.620\); C. \(10^{-2}\); D. \(620\).

Two short calculation practices

  1. A population starts at 80. Its rate is \(0.25P(1-P/160)\) individuals/year. With a one-year step, find the initial rate and next population.
  2. A closed SIR population has \(S=800\), \(I=200\), \(R=0\), and \(N=1000\). For this problem, infection is \(0.1SI/N\) people/day and recovery is \(0.05I\) people/day. With a one-day step, find both flows and the next \(S\), \(I\), and \(R\).

Multiple choice: 1 A (percentage of current balance); 2 C (the logistic factor is zero at capacity); 3 A (\(12\times0.25=3\)); 4 B (shared resource); 5 B (encounters increase); 6 B (\(I\to R\)); 7 C (R’s third element); 8 B (\(10/200=5\%\)); 9 A (overflow); 10 B (numerical sensitivity does not validate the real-world model); 11 B (decimal before the first nonzero digit); 12 B (6, 2, and the final 0 count); 13 C (magnitude is a power of 10).

Short calculations: 1. Rate \(=0.25(80)(1-80/160)=10\) individuals/year; after one year, \(P=90\). 2. Infection \(=0.1(800)(200)/1000=16\) people/day; recovery \(=0.05(200)=10\) people/day. Next stocks: \(S=784\), \(I=206\), \(R=10\); total remains 1000.

Outside this exam

No textbook-diagram labeling, memorized differential equations, complete R program, Quarto syntax, exact analytic solution of a differential equation, calculus derivation of logistic growth, or the unheld strep practice module.

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