Day 41 - Optimization - 10.14.15

Update

Bell Ringer
  1. Find all of the critical numbers for .

    1. –3, 0, 3

    2. 3

    3. 3, –3

    4. 0

    5. none of the above

  2. Locate the absolute extrema of the function on the closed interval [–6, 6]

    1. No absolute max, Absolute min:  

    2. Absolute max: , Absolute min:  

    3. Absolute max: , No absolute min

    4. Absolute max: , Absolute min:

    5. No absolute max or min

  3. Determine whether Rolle's Theorem can be applied to the function on the closed interval [–1, 5].  If Rolle´s Theorem can be applied, find all values of c in the open interval (–1, 5) such that .

    1. Rolle’s Theorem applies;  c = –2

    2. Rolle’s Theorem applies;  c = 0.5

    3. Rolle’s Theorem does not apply

    4. Rolle’s Theorem applies;  c = 2

    5. both a and d

  4. Determine whether the Mean Value Theorem can be applied to the function on the closed interval [0, 16]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (0, 16) such that .

    1. MVT applies;  c = 4

    2. MVT applies;  c =

    3. MVT applies;  c = 8

    4. MVT applies;  c =

    5. MVT does not apply

  5. Find all intervals on which    is concave upward.

    1. none of these
  • 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)
  • Increasing/Decreasing Functions (video) (checkpoints)
    • How are derivatives related to functions increasing and decreasing?
  • Rolle's Theorem & Mean Value Theorem
    • How are Rolle's Theorem and Mean Value Theorem related to differentiation?
  • Concavity and Inflection Points
    • How are derivative related to intervals of concave upwards and downwards?

Lesson

  • Posted on the board at the end of the block

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.