Notes on Course Logistics and Rates of Change

Course logistics and class structure

  • Next week and ongoing: most classes include group work or a quiz (revamp of the course; material remains the same).
  • Office hours:
    • In person in the instructor's office after class. If you’re unsure how to get to ECA 205, you can follow the instructor.
    • Schedule: arrives around 6:00 PM (class ends at 5:45 PM). Waits 10 minutes. If no one shows, leaves.
    • If you expect to arrive after 6:00 PM, tell the instructor during class that you’ll arrive at 6:15 PM or 6:20 PM so they can wait.
    • If there are questions, the instructor will stay for a full hour.
    • Alternatives: Zoom office hours or office-by-appointment; notify if you need hours outside the posted times.
  • Course materials:
    • All materials live in Canvas under Modules.
    • There is no traditional textbook for this class.
  • Edfinity:
    • Edfinity is available alongside course materials; layout may look different depending on user.
    • You can set up Edfinity from the Canvas link; the instructor has been away all summer and will record sessions.
  • Recording and access:
    • Classes are recorded via Locked Zoom sessions; recordings cover everything that happens in class.
    • If you miss something, you can rewatch the Zoom recording and scan for the relevant portion.
    • Recordings are usually processed within about an hour after class and linked to Canvas; announcements are sent via email.
    • You can also access the recording through Canvas announcements/emails.
  • Calculators and testing:
    • A calculator is allowed for this class; graphing calculators are recommended.
    • Allowed: TI-84 (TI-89 is not allowed as it performs algebraic steps that are not permitted).
  • Extra credit:
    • A 5% extra credit is available if an assignment is submitted at least 24 hours before it’s due.
    • The extra credit scheme is described in the course materials; it can significantly affect borderline grades.
    • Edfinity scoring is automatic and you’ll see real-time feedback as you work.
    • In Canvas, all problems are treated as 10-point items once they are synced from Edfinity; the numbers shown in Edfinity may differ from Canvas, but the final grade reflects the Canvas point values.
  • Attendance policy:
    • Attendance is an advantage under departmental policy.
    • The policy states that, for classes meeting twice a week, you cannot have more than two unexcused absences (absences without notice or documentation).
    • Gap notes exist to address missed material; the notes themselves are not graded, but group work and quizzes are.
  • Grading components (as described):
    • Pre-exams: 15% each (dates in the syllabus).
    • Group work and quizzes: graded components (details in the syllabus).
    • Attendance policy contributes to overall grade as described in policy.
  • Quick student support note:
    • There is a reference to a support person or resource; the exact wording in the transcript is unclear ("psychic support person" appears to be a mishearing). If you need help, ask the instructor or the department for the appropriate support resources.
  • Final notes on structure:
    • The instructor indicated that today is mostly lecture-focused and there will be no formal report today; you will be filling in prior sections as usual.
    • Several sentences in the transcript were hurried or fragmented, but the core logistics are: use Canvas and Edfinity, attend posted office hours, take advantage of recorded lectures, and follow the attendance and grading policies.

Rates of change: definitions and concepts

  • What we’re studying: rates of change, focusing on the average rate of change and the instantaneous rate of change.
  • Average rate of change (ARC): the change in the dependent variable per unit change in the independent variable.
  • Notation and basics:
    • Define the change in y as riangle y = y2 - y1 and the change in x as riangle x = x2 - x1.
    • The average rate of change is ext{ARC} = rac{ riangle y}{ riangle x}.
    • If we view y as a function of x (i.e., y = f(x)), then we can also write ext{ARC} = rac{f(x2) - f(x1)}{x2 - x1}.
  • Dependent vs independent variables:
    • The dependent variable is the output (often denoted y or f(x)).
    • The independent variable is the input (often denoted x or another quantity like time or quantity produced).
    • Delta represents a total change over an interval, while the endpoint values represent the start and end of that interval.
  • Geometric interpretation:
    • The ARC corresponds to the slope of the secant line through two points on the graph of the function.
    • Secant line: the line that passes through the two points
      (x1, f(x1)) and (x2, f(x2)).
    • Slope of the secant line is the ARC: ext{slope}{ ext{secant}} = rac{f(x2) - f(x1)}{x2 - x_1}.
  • Two-point notation:
    • Using points \(x1, y1) and \(x2, y2):
      ext{slope}{ ext{secant}} = rac{y2 - y1}{x2 - x_1}.
  • A note on sign:
    • The ARC can be negative if the graph is decreasing over the interval.
    • A negative ARC corresponds to a downward-sloping secant line.

Example framework: quadratic context and a quantity- or profit-based function

  • A common setup uses a quadratic function where the quantity q represents the input (e.g., units produced/sold) and the price or profit is the output (e.g., P(q) or profit).
  • General method to compute ARC for a function P(q):
    • Given two quantities q1 and q2, compute
      ext{ARC} = rac{P(q2) - P(q1)}{q2 - q1}.
  • Worked illustration (as described in transcript):
    • To compute P(4) you substitute q = 4 into the function P(q).
    • If you had a calculator setup, you would enter the function values and compute:
    • Example form (as described): replace with concrete function values, e.g., if P(q) is given by a simple expression, compute $P(10)$ and $P(4)$, then form the ratio
      rac{P(10) - P(4)}{10 - 4}.
    • The transcript suggested a calculation pattern like evaluating "P(10) + 5" and "P(4) + 2" as part of the process and then dividing by 6; this illustrates the method rather than a fixed numeric result. The general takeaway is:
      ext{ARC} = rac{P(10) - P(4)}{10 - 4}.
  • Interpretation of the result:
    • The ARC gives the average rate of change of profit with respect to quantity over the interval from q = 4 to q = 10.
    • A positive ARC implies increasing profit on that interval; a negative ARC would imply decreasing profit.
    • If P is measured in thousands of dollars, then the ARC is in thousands of dollars per unit of q (or the appropriate units given by the function).

Average rate of change vs. instantaneous rate of change (conceptual)

  • Average rate of change (ARC): slope of the secant line through two points on the curve.
  • Instantaneous rate of change (IROC): slope of the tangent line at a single point; measures how fast the quantity is changing at that exact point.
    • The IROC is the limit of the ARC as the interval between the two points shrinks to zero:
      ext{IROC at } x = a ext{ is } rac{dy}{dx}igg|_{x=a} =
      abla y ext{ (in derivative form) }.
    • The tangent line touches the curve at exactly one point and has slope equal to the IROC at that point.
  • Practical distinction:
    • ARC answers: "On average, how fast is y changing with x between x1 and x2?"
    • IROC answers: "At the exact point x = a, how fast is y changing at that instant?"

Revenue example: interpreting rates of change in a quadratic model

  • Setup context: weekly revenue (in millions of dollars) as a function of time (weeks), with t measured in weeks.
  • Question: Which rate of change helps answer: "How fast is the company’s revenue changing between the 20th and 30th week?"
    • Answer: average rate of change over the interval [20, 30], because it uses two distinct time points.
    • If we want the instantaneous rate at a specific week, we would look at the derivative (or estimate it via the tangent line) at that week.
  • Shrinking intervals to approach instantaneous rate:
    • Consider shrinking the time step Δt and looking at values like Δt = 1, 0.5, 0.1 weeks, etc.
    • The corresponding ARC values approach the instantaneous rate of change as Δt → 0.
  • Numerical illustration notes from the transcript (paraphrased):
    • The instructor attempted to compare values like 20.1 and 21.0 to illustrate how the ARC changes as you tighten the interval.
    • There was a reference to a computed slope around 1.2 (or 1.2 units per week), illustrating the idea that the average rate of change can approach a tangent slope as the interval becomes very small.
    • There was a brief, unclear discussion about degree settings (degrees vs radians) on a calculator, which is relevant when computing derivatives analytically or with certain numerical methods.
  • Key takeaway for the revenue example:
    • Use ARC for the interval [20, 30] to measure average rate of change in revenue over that span.
    • To estimate IROC (instantaneous rate) near a week, narrow the interval around that week and observe the trend of ARC values; conceptually, this approximates the derivative at that week.

Tangent line and instantaneous rate of change (concept and example)

  • Tangent line interpretation:
    • The slope of the tangent line at a point on the graph is the instantaneous rate of change at that point.
    • The tangent line touches the curve at exactly one point and provides a local linear approximation of the function near that point.
  • In the transcript: the phrase about a tangent line is used to distinguish from the secant line (ARC).
  • Practical statement: as the interval used for the ARC becomes very small, the secant line more closely approximates the tangent line, and the ARC approaches the IROC (derivative).

Practical notes and calculator usage

  • Calculator rules:
    • Allowed: graphing calculators (e.g., TI-84). Not allowed: TI-89 (because of algebraic capabilities not permitted in coursework).
    • When used with Edfinity and real-time grading, students can see immediate feedback on their work.
  • Early submission and extra credit:
    • A 5% extra credit exists for submissions at least 24 hours before due date.
    • Extra credit policies may impact borderline grades and overall course performance.
  • Recording and accessibility:
    • Zoom recordings are stored and linked to Canvas; they are available to review missed material.
    • Recordings are processed typically within an hour after class and announced via email.
  • Attendance and gaps:
    • Attendance is valuable and governed by departmental policy; two unexcused absences are allowed for classes meeting twice per week.
    • Gap notes exist to address missed content; they are not graded; group work and quizzes remain graded components.
  • About the workflow and materials:
    • All course materials live in Canvas under Modules.
    • There is no traditional textbook for this class.
    • Edfinity is used for some of the assignments and is integrated with Canvas.
  • Clarifications on phrasing in the transcript:
    • There is a mention of a "support person" or a similar resource; a phrase in the transcript appears garbled ("psychic support person"). If you need help, check with the instructor or department for the correct support resources.
  • Summary of final notes:
    • The class will continue with a mix of group work and quizzes.
    • Office hours are available in-person or via Zoom/appointment.
    • All materials live in Canvas; use Edfinity for practice and automatic grading.
    • Review the attendance policy and grading breakdown as described in the syllabus.
    • For rates of change, remember the definitions of ARC and IROC, the secant line slope, and the tangent line interpretation as you move from average to instantaneous rates.

Quick reference formulas (LaTeX)

  • Average rate of change (two points):
    ext{ARC} = rac{f(x2) - f(x1)}{x2 - x1} = rac{y2 - y1}{x2 - x1}.
  • Change notation:
    riangle y = y2 - y1, ag*{} \ riangle x = x2 - x1.
  • Tangent / instantaneous rate of change at x = a:
    ext{IROC at } x=a ext{ (slope of tangent)} = rac{dy}{dx}igg|_{x=a}.
  • Example ARC for a profit function P(q) over q1 to q2:
    ext{ARC} = rac{P(q2) - P(q1)}{q2 - q1}.
  • Slope of secant line through points \(x1, f(x1)) and \(x2, f(x2)):
    rac{f(x2) - f(x1)}{x2 - x1}.
  • Conceptual relation between ARC and IROC:
    • ARC is an average slope over an interval.
    • IROC is the limit of ARC as the interval length goes to zero.

End of notes