Overview
This lecture reviews calculating the difference quotient for various functions, including linear, quadratic, rational, and radical types, emphasizing careful algebraic manipulation and simplification.
Linear Functions
- For f(x) = 5x - 3, the difference quotient simplifies to 5.
- For f(x) = 3x - 7, the difference quotient simplifies to 3.
Polynomial Functions
- For f(x) = 3x² - 5x - 1, the difference quotient is 6x + 3h - 5.
- For f(x) = x² - 25, the difference quotient is 2x + h.
- For f(x) = x³ + 1, the difference quotient is 3x² + 3xh + h².
- For f(x) = -2x³ - x - 7, the difference quotient is -6x² - 6xh - 2h² - 1.
Rational Functions
- For f(x) = 7/(x-1), the difference quotient is -7/[(x+h-1)(x-1)].
- For f(x) = 4x/(x-5), the difference quotient is -20/[(x+h-5)(x-5)].
Radical Functions
- For f(x) = √(x-4), the difference quotient is 1/[√(x+h-4) + √(x-4)].
- For f(x) = √(7-x²), the difference quotient is (-2x-h)/[√(7-(x+h)²) + √(7-x²)].
Key Terms & Definitions
- Difference Quotient — The expression [f(x+h) - f(x)] / h, used to compute the average rate of change or the derivative of a function.
- Distributive Property — Algebraic rule for expanding expressions: a(b + c) = ab + ac.
- Rationalizing the Numerator — Multiplying by a conjugate to remove radicals from the numerator in difference quotients.
- Factoring — Taking out common factors in an expression to simplify or prepare for cancellation.
Action Items / Next Steps
- Practice the difference quotient for additional functions.
- Review algebraic techniques: expanding polynomials, rationalizing numerators, and factoring.
- Complete any assigned homework on difference quotients.