Day 30 - Implicit Differentiation - 09.29.14

Bell Ringer
  • New Website Additions
    • Availability
    • Post-Assessment Reflection
  • Differentiation Test 2 on this Friday, 10.03.14!
  • Summative Exam 1 on next Friday, 10.10.14!

Calculate dy/dx for the following:

  1. y = x

    1. dy/dx = x

    2. dy/dx = y

    3. dy/dx = 1

    4. dy/dx = 0

    5. none of the above

  2. 2y = x

    1. dy/dx = x/2

    2. dy/dx = 2y

    3. dy/dx = 2

    4. dy/dx = 0

    5. none of the above

  3. xy = 1

    1. dy/dx = -x2

    2. dy/dx = -1/x2

    3. dy/dx = 1/x2

    4. dy/dx = -2/x2

    5. none of the above

  4. x2y = x + 1

    1. dy/dx = (-x - 2)/x3

    2. dy/dx = (x - 2)/x3

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

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

    5. none of the above

  5. y3 + y2 - 5y - x2 = -4

    1. dy/dx = (2x) / (3y2 + 2y - 5)

    2. dy/dx = (2x) / (3y2 + 2y + 5)

    3. dy/dx = (-2x) / (3y2 + 2y - 5)

    4. dy/dx = 2 / (3y2 + 2y - 5)

    5. 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
  • Derivative Notation
  • Differentiation
  • Rates of Change
    • Position Function
    • Average Velocity
    • Instantaneous Velocity
    • Free Fall Problems
  • Product Rule
  • Quotient Rule
  • Chain Rule
Lesson
  • Implicit Differentiation
    • Checkpoints
      • A - page 146 #6
      • B - page 146 #10
      • C - page 146 #26
      • D - page 146 #30
      • E - page 147 #48
      • F - page 147 #52
      • G - page 147 #66
      • H - page 148 #74
      • I - page 148 #78
Exit Ticket
  • Exit Ticket will be posted on the board in class.
Lesson Objective(s)
  • How can implicit differentiation be used to find the derivative?
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