Day 20 - Derivative - 09.15.14

Bell Ringer
  • Go over Quiz 3
  • All tests need to be remediated by Friday!
      1. Find the slope of the tangent line at x = 2. Prove using two different methods. f(x) = (x - 2)2

        1. 0

        2. 1

        3. -1

        4. does not exist

        5. none of the above

      2. Find the slope of the tangent line at x = 2. Prove using two different methods. f(x) = |x - 2|

        1. 0

        2. 1

        3. -1

        4. does not exist

        5. none of the above

      3. Find the derivative at x = 0. Prove using two different methods. f(x) = x1/3

        1. 0

        2. -1

        3. 1

        4. none of the above

      4. Find the derivative of the following function: f(x) = x3 / 4 - 3x

        1. dy/dx = 3x2 / 4

        2. dy/dx = (3x2 / 4) - 3

        3. dy/dx = (3x / 4) - 3

        4. dy/dx = 3x2 - 3

        5. none of the above

      5. At what points does the following function have a horizontal tangent line: y = sin x - x, 0 ≤ x ≤ 2π

        1. 0

        2. π

        3. all of the above

        4. none of the above

      Review
      • Secant vs. Tangent Line
        • Slope
      • Definition of Derivative
        • Importance of Derivative
          • What does it allow us to do?
      • Drawing a Tangent Line on a Graph
      • Basic Differentiation Rules
        • Constant Rule
        • Power Rule
        • Sum and Difference Rule
        • Sine and Cosine Derivatives

      Lesson
      • Derivative Notation
      • Differentiation
      • Rates of Change
        • Position Function
        • Average Velocity
        • Instantaneous Velocity
        • Free Fall Problems
      • Practice (Section 2.2)
        • Checkpoint #1
          • page 115 #20, #22, #24
        • Checkpoint #2
          • page 115 #36 and #38
        • Checkpoint #3
          • page 116 #54 and #64
        • Checkpoint #4
          • page 116 #78 and #80
        • Checkpoint #5
          • page 116 #100

      Exit Ticket
      • Exit TIcket will be posted on the board in class.
      Lesson Objective(s)
      • How can derivatives be used to find rates of change?

      Standard(s)
      • APC.5
        • Investigate derivatives presented in graphic, numerical, and analytic contexts and the relationship between continuity and differentiability.

          • The derivative will be defined as the limit of the difference quotient and interpreted as an instantaneous rate of change.

      • APC.6

        • ​The student will investigate the derivative at a point on a curve.

          • Includes:

            • finding the slope of a curve at a point, including points at which the tangent is vertical and points at which there are no tangents

            • using local linear approximation to find the slope of a tangent line to a curve at the point

            • ​defining instantaneous rate of change as the limit of average rate of change

            • approximating rate of change from graphs and tables of values.

      • 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

      • APC.8

        • Apply the derivative to solve problems.

          • Includes:

            • 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

      • APC.9

        • Apply formulas to find derivatives.

          • Includes:

            • derivatives of algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions

            • derivations of sums, products, quotients, inverses, and composites (chain rule) of elementary functions

            • derivatives of implicitly defined functions


      Mathematical Practice(s)
      • #1 - Make sense of problems and persevere in solving them
      • #2 - Reason abstractly and quantitatively
      • #3 - Construct viable arguments and critique the reasoning of others
      • #4 - Model with mathematics