Day 36 - Increasing/Decreasing Functions - 10.06.15

Update
  • Summative Exam 1 will be graded sometime today!

Bell Ringer

  1. What are critical numbers?

    1. numbers needed to solve problems

    2. values of x that cause a derivative to be equal to zero

    3. values of y that cause a derivative to be equal to zero

    4. values of x that cause a derivative to be undefined

    5. both b and d

  2. How are relative extrema related to critical numbers?

    1. There is no relationship between critical numbers and relative extrema.

    2. Every relative extrema occurs at a critical number.

    3. Every critical number has a relative extrema.

    4. Critical numbers are only found at relative extrema.

    5. none of the above

  3. Find the critical number(s) of the following:

    1. x = 1

    2. x = -1

    3. x = 0

    4. undefined

    5. none of the above

  4. Find the relative extrema for the following function:

    1. (1, 6)

    2. (-1, 4)

    3. (0, 5)

    4. no relative extrema

    5. none of the above

  5. Find and describe all extrema for the following function: on the interval

    1. Absolute max: (-1, 4) | Absolute min: (2, 13) | No relative extrema

    2. Absolute min: (-1, 4) | Absolute max: (2, 13) | Relative min: (0, 5)

    3. Absolute min: (-1, 4) | Absolute max: (2, 13) | No relative extrema

    4. No extrema

    5. none of the above

Review
  • Pre-calculus
    • Extrema
      • Minima
      • Maxima
      • Absolute
      • Relative
    • Interval Notation
  • Extrema (video) (checkpoints)
    • How can extrema be defined for a function?
    • How can critical numbers be calculated using derivatives?
    • How are critical numbers related to extrema?
    • Absolute and Relative Extrema (video)
    • Critical Numbers (video)

Lesson
Exit Ticket
  • Posted on the board at the end of the block (link).

Homework
  • N/A

In-Class Help Requests





Standard(s)
  • APC.7
    • Analyze the derivative of a function as a function in itself.
      • Includes:
        • comparing corresponding characteristics of the graphs of f, f', and f''
        • ​defining the relationship between the increasing and decreasing behavior of f and the sign of f'
        • ​translating verbal descriptions into equations involving derivatives and vice versa
        • analyzing the geometric consequences of the Mean Value Theorem;
        • defining the relationship between the concavity of f and the sign of f"; and ​identifying points of inflection as places where concavity changes and finding points of inflection.
  • APC.8
    • Apply the derivative to solve problems.
      • Includes:
        • ​analysis of curves and the ideas of concavity and monotonicity
        • optimization involving global and local extrema;
        • modeling of rates of change and related rates;
        • use of implicit differentiation to find the derivative of an inverse function;
        • interpretation of the derivative as a rate of change in applied contexts, including velocity, speed, and acceleration; and
        • differentiation of nonlogarithmic functions, using the technique of logarithmic differentiation.*
          • *AP Calculus BC will also apply the derivative to solve problems.
            • Includes:
              • ​analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors;
              • ​numerical solution of differential equations, using Euler’s method;
              • ​l’Hopital’s Rule to test the convergence of improper integrals and series; and
              • ​geometric interpretation of differential equations via slope fields and the relationship between slope fields and the solution curves for the differential equations.