Chord Mask Video - Stage 3 Points

Overview

• Two main objectives:
  1. Use differentiation to find the coordinates of stationary points.
  2. Use the second derivative to determine whether a stationary point is a maximum or minimum.
• Key Concepts:
  • Stationary points, also known as turning points, are where the graph turns.
  • Maximum points (maxima) have a positive gradient that turns negative.
  • Minimum points (minima) have a negative gradient that turns positive.
  • The gradient (dy/dx) is zero at stationary points.
  • The second derivative helps determine the nature of turning points.
    • Less than zero for maxima.
    • Greater than zero for minima.

Finding Coordinates of Stationary Points

Example 1: Quadratic Function

• Function: y = x² - 6x + 4
• Process:
  1. Differentiate to find gradient function: dy/dx = 2x - 6
  2. Set dy/dx = 0 to find stationary point:
     • 2x - 6 = 0
     • Solve for x: x = 3
  3. Substitute x = 3 back into the original equation to find y:
     • y = 3² - 18 + 4 = -5
  4. Result: Minimum point at (3, -5)

Example 2: Cubic Function

• Function: y = x³ - 3x² - 24x
• Process:
  1. Differentiate to find gradient function: dy/dx = 3x² - 6x - 24
  2. Set dy/dx = 0:
     • 0 = 3x² - 6x - 24
     • Divide by 3: 0 = x² - 2x - 8
     • Factorize: (x-4)(x+2) = 0
     • Solve: x = 4, x = -2
  3. Substitute x-values back into original equation for y:
     • x = 4: y = -80
     • x = -2: y = 28
  4. Results: Maximum point at (-2, 28), Minimum point at (4, -80)

Determining Nature of Stationary Points

Second Derivative Test

• For maxima: d²y/dx² < 0
• For minima: d²y/dx² > 0

Example: Using Second Derivative

• Initial Example: Maximum (-2, 28), Minimum (4, -80)
• Function: d²y/dx² = 6x - 6
  • Substitute x = -2:
    • d²y/dx² = -18 (negative, confirms maximum)
  • Substitute x = 4:
    • d²y/dx² = 18 (positive, confirms minimum)

Typical Question: Find Coordinates and Nature

• Function: y = x - x² - x³
• Process:
  1. Differentiate: dy/dx = 1 - 2x - 3x²
  2. Set dy/dx = 0:
     • Solve: 3x² + 2x - 1 = 0
     • Factorize and solve: x = 1/3, x = -1
  3. Substitute x-values for y-values:
     • x = -1: y = -1
     • x = 1/3: y = 5/27
  4. Determine nature using second derivative:
     • d²y/dx² = -2 - 6x
     • x = -1: d²y/dx² = 4 (minimum)
     • x = 1/3: d²y/dx² = -4 (maximum)
  5. Results: Minimum at (-1, -1), Maximum at (1/3, 5/27)

Summary

• Use dy/dx = 0 to find stationary points.
• Use d²y/dx² to find the nature (maxima or minima).
• Maxima: d²y/dx² < 0
• Minima: d²y/dx² > 0