Class Notes on Upcoming Test and Rates of Change in Business Calculus

Test Date Changes

  • The test has been rescheduled to next Wednesday.
  • Assignments' due dates remain unchanged for now.
  • Students must bring their computers to class for the test.
  • The test will be conducted on Pearson and will involve scratch work.

Test Information

  • A formula sheet is allowed, although it might not be strictly necessary.
  • Students may bring notes written on their formula sheet.
  • Formulae will be provided during the section starting today, which should be incorporated into the personal formula sheets for future reference (important for the final exam preparation).

Assessments Overview

  • Tests, quizzes, and homework are part of the assessment structure.
  • A quiz will be available online and can be completed at home.
  • The quiz will be opened for a few days; expected to open by the end of the week.

New Section: Business Calculus

  • The class will begin with a focus on "Rates of Change" in Section 11.3 as per the syllabus.
  • Concepts from previous sections (11.1 and 11.2) will reappear on the midterm and final exams.

Concept of Average Rate of Change

  • The formula for average rate of change is defined as:
    • ext{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
  • This concept is related to speed in the context of distance and time.

Example Problem 1: Average Speed Calculation

  • A driver traveled 168 miles from Cleveland to Columbus in 3 hours.
    • Setup: Calculating average speed from time t = 1.5 to t = 2.
  • Inputted values into the formula:
    • f(2) = 118 (miles at 2 hours)
    • f(1.5) = 86 (miles at 1.5 hours)
  • Calculation:
    • = \frac{118 - 86}{2 - 1.5} = \frac{32}{0.5} = 64 ext{ miles per hour}

Example Problem 2: Average Rate of Change from a Function

  • Given a quadratic function: f(x) = x^2 + 4x + 5, find average rate of change from a = -2 to b = 3.
  • Setup:
    • Using the formula:
    • = \frac{f(3) - f(-2)}{3 - (-2)} = \frac{(3^2 + 43 + 5) - ((-2)^2 + 4(-2) + 5)}{3 + 2}
  • Calculation steps:
    • f(3) = 9 + 12 + 5 = 26
    • f(-2) = 4 - 8 + 5 = 1
  • Therefore:
    • = \frac{26 - 1}{5} = \frac{25}{5} = 5

Average Rate of Change in Real Life

  • Practical example: Calculate the average rate of change of data from 2013 to 2015 based on economic data:
    • f(2015) = 17833 and f(2013) = 13000
    • Setup:
    • Average Rate of Change:
    • = \frac{17833 - 13000}{2015 - 2013} = \frac{4833}{2} = 2416.5

Connection to Calculus and Slope

  • The average rate of change formula is analogous to calculating the slope of a line:
    • Standard slope formula:
    • ext{slope} = \frac{y2 - y1}{x2 - x1}
  • In calculus, concepts of change will involve curves rather than straight lines.
  • The foundational idea remains the same, focusing on incremental values to find the average.

Upcoming Concepts

  • The next focus is on velocity and its computation, which will use the limit definition of average velocity:
  • v(t) = \lim_{h \to 0} \frac{s(t + h) - s(t)}{h}
  • Note: 's' represents the position function related to time.

Homework and Study Reminder

  • Students should utilize available time for homework to prepare for the test effectively.
  • Bring questions for clarification in subsequent classes.