Day 33 - Related Rates - 10.02.14

Bell Ringer
  • Reminders
    • Differentiation Test 2 on this Friday, 10.03.14!
    • Summative Exam 1 on next Friday, 10.10.14!

Implicit Differentiation

  1. Find the derivative of the following implicitly: x3 + y2 = 5

    1. dy/dx = 3x2 / (2y)

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

    3. dy/dx = -3x2 / y

    4. dy/dx = -x2 / y

    5. none of the above

  2. Find the second derivative of the following implicitly: x3 + y2 = 5

    1. dy/dx = (-12xy - 9x4y-1) / (4y2)

    2. dy/dx = (12xy - 9x4y-1) / (4y2)

    3. dy/dx = (-12xy - 9x4y-1) / y2

    4. dy/dx = (-12xy + 9x4y-1) / (4y2)

    5. none of the above

  3. Find the equation of the tangent line at (1, -2) for the following: x3 + y2 = 5

    1. y = ¾x + (5/4)

    2. y = ¾x - (5/4)

    3. y = -¾x + (5/4)

    4. y = -¾x - (5/4)

    5. none of the above

Related Rates

  1. Grain is poured into a conical pile at a rate of 20 ft3/min. The diameter of the base of the cone is approximately 4 times the height. At what rate is the height of the pile changing when the pile is 10 ft high?

    1. 12.5/π

    2. 0.8/π

    3. 1/(20π)

    4. 1/(2π)

    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
  • Implicit Differentiation
Lesson
  • Related Rates
    • Brainstorming Activity
    • Discussion
    • Checkpoints
      • A - page 154 #2
      • B - page 154 #16
      • C - page 154 #18
      • D - page 154 #20
      • E - page 154 #14
Exit Ticket
  • Exit Ticket will be posted on the board in class.
Lesson Objective(s)
  • How can differentiation be used to find the related rates?
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