Day 34 - Extrema/Critical Numbers - 02.22.16

Update

  1. What are extrema?

    1. extreme values

    2. minima

    3. maxima

    4. all of the above

    5. none of the above

  2. What is the difference between relative and absolute extrema?

    1. Absolute extrema and relative extrema are always the same.

    2. Absolute extrema are based on the points that are either side of the extrema.

    3. Relative extrema are always the biggest or smallest values overall.

    4. Relative extrema can never be absolute extrema.

    5. none of the above

  3. Which of the following represent the interval notation for the following: 0 < x < 5

    1. (0, 5)

    2. [0, 5)

    3. (0, 5]

    4. [0, 5]

    5. none of the above

  4. Which of the following represent the interval notation for the following:

    1. (0, 5)

    2. [0, 5)

    3. (0, 5]

    4. [0, 5]

    5. none of the above

  5. Which of the following represent the interval notation for the following:

    1. (0, 5)

    2. [0, 5)

    3. (0, 5]

    4. [0, 5]

    5. none of the above


    Review
    • Pre-calculus
      • Extrema
        • Minima
        • Maxima
      • Absolute vs. Relative Extrema
      • Interval Notation

          Lesson
          • Objective
            • How can extrema be defined for a function?
            • How can critical numbers be calculated using derivatives?
            • How are critical numbers related to extrema?
          • Challenge 3.1
            • Extreme Value Theorem
              • Sketch a graph that has no maxima or minima.
              • Sketch a graph on an open interval that has no maxima or minima.
              • Sketch a graph on a closed interval that has no maxima or minima.
              • Sketch a graph on a closed interval that has a maxima, but no minima.
            • Critical Numbers
              • Sketch a graph that has an extremum that is differentiable.
              • Sketch a graph that has an extremum that is not differentiable.
              • Sketch a graph that has a point in which the derivative is zero, but is not an extrema.
          • Videos
          • Practice

            Exit Ticket
            • Posted on the board at the end of the block.

            Homework
            • Study!


            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.