Day 35 - Increasing/Decreasing Functions - 02.23.16

Update

  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 and Relative Extrema (video)
      • Interval Notation
    • Extrema (video)/Critical Numbers (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?
      • How does Extreme Value Theorem work?
          Lesson
          • Challenge 3.2
            • Using only calculus, how can we find extrema?
            • Sketch a function with extrema.
            • See me for hints.
          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.