Day 42 - Concavity and Second Derivative Test - 10.16.14

Updates
  • Quiz 7 on Friday!
    • Covers the entire new unit so far
      • Extrema
      • Rolle's Theorem
      • Mean Value Theorem
      • Increasing and Decreasing Functions
      • First Derivative Test
      • Concavity
      • Second Derivative Test
  • Today's Schedule
    • Block 1 - 8:25-9:55
    • Block 2 - 10:02-11:06
    • Block 3 - 11:13-1:19
      • Lunch
        • A - 11:06-11:38
        • B - 12:04-12:35
        • C - 12:50-1:19
    • Block 4 - 1:26-2:30

Bell Ringer


  • Increasing/Decreasing Functions
  1. Find the open intervals where the function f(x) = sin(x) + cos(x) is increasing or decreasing on the closed interval [0, 2π]

    1. Increasing: (0, π/4) and (5π/4, 2π) | Decreasing: (π/4, 5π/4)

    2. Decreasing: (0, π/4) and (5π/4, ) | Increasing: (π/4, 5π/4)

    3. Increasing: (0, π/4) | Decreasing: (π/4, 5π/4) and (5π/4, 2π)

    4. Decreasing: (0, π/4) | Increasing: (π/4, 5π/4) and (5π/4, 2π)

    5. none of the above

  2. Identify the open intervals where the function f(x) = -4x2 - 6x - 4 is increasing or decreasing.

    1. Increasing: (-∞, -¾) | Decreasing: (-¾, ∞)

    2. Decreasing: (-∞, -¾) | Increasing: (-¾, ∞)

    3. Increasing on (-∞, ∞)

    4. Decreasing on (-∞, ∞)

    5. none of the above


Review

Lesson
    • Concavity and Inflection Points
    • Second Derivative Test
      • How is the concavity of a function related to its first derivative?
      • How is the concavity of a function related to its second derivative?
      • Checkpoints (page 195)
        • P - #6
        • Q - #16
        • R - #18
        • S - #20
        • T - #24
        • U - #32
        • V - #38
        • W - #52

    Exit Ticket
    • Posted on the board at the end of the class.
    Lesson Objective(s)
    • How can derivatives be used to find concavity?
    • How can derivatives be used to find points of inflection? 

    Standard(s)
    • 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.*


    Mathematical Practice(s)
    • #1 - Make sense of problems and persevere in solving them
    • #2 - Reason abstractly and quantitatively
    • #5 - Use appropriate tools strategically
    • #6 - Attend to precision
    • #8 - Look for and express regularity in repeated reasoning


    Past Checkpoints
    • Extrema (page 169)
      • A - #4
      • B - #6
      • C - #18
      • D - #22
      • E - #34
    • Rolle's Theorem (page 176)
      • F - #4
      • G - #10
      • H - #26
    • Mean Value Theorem (page 176-177)
      • I - #34
      • J - #42
      • K - #44
    • Increasing/Decreasing Functions (page 186)
      • L - #6
      • M - #20
      • N - #44
      • O - #48