Lesson on Average Rate of Change of a Function

Formula for Average Rate of Change

• Expression: ( \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} )
• Interval: ( [a, b] )
  • a and b: x-values
  • f(a) and f(b): y-values

Example Calculations

Example 1

• Given:
  • ( a = 1 )
  • ( b = 3 )
• Calculate f(a) and f(b):
  • ( f(1) = 1^2 + 4 \times 1 - 5 = 0 )
  • ( f(3) = 3^2 + 4 \times 3 - 5 = 16 )
• Average Rate of Change:
  • ( \frac{f(3) - f(1)}{3 - 1} = \frac{16 - 0}{2} = 8 )
• Interpretation: Represents the slope of the secant line.

Example 2

• Function: ( f(x) = x^3 - 4 )
• Interval: ( x = 2 ) to ( x = 5 )
• Calculate f(a) and f(b):
  • ( f(5) = 5^3 - 4 = 121 )
  • ( f(2) = 2^3 - 4 = 4 )
• Average Rate of Change:
  • ( \frac{f(5) - f(2)}{5 - 2} = \frac{121 - 4}{3} = 39 )

Concepts

• Secant Line:
  • Touches two points on the graph
  • The average rate of change represents its slope.
• Tangent Line:
  • Touches only one point on the graph.

Additional Resources

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