Lesson 4.2: Introduction to Instantaneous Rates of Change

Key Objectives

• Understand and practice calculating difference quotients.
• Recognize the difference quotient as an average rate of change on an interval ( x ) to ( x + h ).
• Introduction to instantaneous rates of change.

Difference Quotient

• Definition: Difference quotient is the average rate of change of a function on the interval ( x ) to ( x + h ).
• Formula: [ \frac{f(x+h) - f(x)}{h} ]
  • This represents the slope of the secant line between the points ( (x, f(x)) ) and ( (x+h, f(x+h)) ).

Examples

Example 1: Linear Function

• Function: ( f(x) = -10x + 7 )
• Find the average rate of change from ( x ) to ( x + h ).
• Steps:
  1. Calculate ( f(x+h) - f(x) ):
     • ( = -10(x+h) + 7 - (-10x + 7) )
     • Simplifies to ( -10h )
  2. Divide by ( h ):
     • ( \frac{-10h}{h} = -10 )
  3. Result: Consistent with the slope of the line.

Example 2: Quadratic Function

• Function: ( f(x) = 3x^2 - 2 )
• Calculate from ( x = 2 ) to ( x = 2 + h ).
• Steps:
  1. Evaluate ( f(2+h) - f(2) ):
     • ( = 3(2+h)^2 - 2 - (3 \times 2^2 - 2) )
     • Simplifies to ( 11h + 3h^2 )
  2. Divide by ( h ):
     • ( \frac{11h + 3h^2}{h} = 11 + 3h )

Example 3: Rational Function

• Function: ( f(x) = \frac{2}{x} )
• Calculate from ( x = 3 ) to ( x = 3 + h ).
• Steps:
  1. Evaluate ( f(3+h) - f(3) ):
     • ( = \frac{2}{3+h} - \frac{2}{3} )
     • Simplifies to ( \frac{-2h}{3(3+h)} )
  2. Divide by ( h ):
     • ( \frac{-2}{3(3+h)} )

Instantaneous Rate of Change

• Conceptualized as the rate of change at a precise moment.
• Calculation: Let ( h ) approach 0 in the difference quotient.
• Caution: Direct substitution of ( h = 0 ) leads to an indeterminate form (( \frac{0}{0} )).
  • Use limits to resolve.

Example: Velocity

• Function: ( p(t) = -16t^2 + 50t + 5 )
• Average velocity from ( t=0 ) to ( t=1 ):
  • ( \frac{p(1) - p(0)}{1} = 34 ) feet per second.
• Instantaneous velocity at ( t=1 ):
  • Evaluate ( \frac{p(1+h) - p(1)}{h} ) and let ( h \to 0 ):
  • Result: ( 18 ) feet per second.

Derivative

• Definition: Derivative ( f'(x) ) of a function is the instantaneous rate of change.
• Calculated using the difference quotient approach.

Final Example: Rational Function

• Function: ( g(x) = \frac{1}{2x + 3} )
• Calculate instantaneous rate of change at ( x=1 ).
• Steps:
  1. Evaluate ( g(1+h) - g(1) ) over ( h ).
  2. Simplify and evaluate as ( h \to 0 ):
  • Result: (-\frac{2}{25} ).

Conclusion

• Mastery of the difference quotient is essential for understanding rates of change in calculus.
• Practice with various functions to gain familiarity.
• Understand that the derivative represents the instantaneous rate of change.