Day 47 - Unit 3 Synthesis - 10.23.14

Updates
  • Unit 3 Test tomorrow, 10/24!
  • Recording
Bell Ringer
  • Review

  1. Locate the absolute extrema of the function f(x) = 2x2 + 12x - 4 on the closed interval [-6, 6].

    1. no absolute max; absolute min: f(6) = 140

    2. absolute max: f(-3) = -22; absolute min: f(6) = 140

    3. absolute max: f(6) = 140; no absolute min

    4. absolute max: f(6) = 140; absolute min: f(-3) = -22

    5. no absolute max or min

  2. Find all the relative extrema for the following function: f(x) = sin(x) + cos(x) on the interval [0, 2π].

    1. Relative Max: (5π / 4, sqrt(2)); Relative Min: (π / 4, -sqrt(2))

    2. Relative Max: (π / 4, sqrt(2)); Relative Min: (-π / 4, -sqrt(2))

    3. Relative Max: (π / 4, sqrt(2)); Relative Min: (5π / 4, -sqrt(2))

    4. Relative Max: (π / 4, sqrt(2)); Relative Min: (5π / 4, sqrt(2))

    5. none of the above

  3. Determine whether Mean Value Theorem can be applied to the function f(x) = x3 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 f’(c) = [f(b) - f(a)] / (b - a).

    1. MVT applies; c = 4

    2. MVT applies; c = -16sqrt(3) / 3

    3. MVT applies; c = 8

    4. MVT applies; c = 16sqrt(3) / 3

    5. MVT does not apply

  4. 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: (-∞, π/4) and (5π/4, ∞) | Decreasing: (π/4, 5π/4)

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

    3. Increasing: (-∞, π/4) | Decreasing: (π/4, 5π/4) and (5π/4, ∞)

    4. Decreasing: (-∞, π/4) | Increasing: (π/4, 5π/4) and (5π/4, ∞)

    5. none of the above

  5. Find all intervals on which f(x) = (x + 1) / (x - 3) is concave downward.

    1. (1. ∞)

    2. (-∞, ∞)

    3. (-∞, 3)

    4. (3, ∞)

    5. none of the above

Review
Lesson
  • Review
    Exit Ticket
    • N/A
    Lesson Objective(s)
    • How can derivatives be used to find optimum conditions?
    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
    • Concavity and Points of Inflection (page 195)
      • P - #6
      • Q - #16
      • R - #18
      • S - #20
      • T - #24
      • U - #32
      • V - #38
      • W - #52
    • Limits at Infinity
    • Optimization