AS

lecture recording on 10 September 2025 at 13.26.21 PM

Difference quotient and derivative for f(x) = x^2 - 8x + 9

  • Define the function: f(x) = x^2 - 8x + 9.
  • Compute a specific value: f(2) = 4 - 16 + 9 = -3.
  • Evaluate at 2 plus a small increment:
    • f(2+ h) = (2+h)^2 - 8(2+h) + 9 = 4 + 4h + h^2 - 16 - 8h + 9 = -3 - 4h + h^2.
  • Form the difference quotient for the point a = 2 (the form using h):
    • rac{f(2+h) - f(2)}{h} = rac{(-3 - 4h + h^2) - (-3)}{h} = rac{-4h + h^2}{h} = -4 + h.
  • Take the limit as $h o 0$ to get the derivative at 2:
    • f'(2) =
      abla{h o 0} rac{f(2+h) - f(2)}{h} = abla{h o 0}(-4 + h) = -4.
  • Insight: the constant term $f(2) = -3$ contributes the subtraction of a negative number in the numerator, hence the appearance of "minus negative three" in some writeups of the difference quotient.

Derivative at a general point (h-form) and an equivalent x-form

  • The derivative at a point $a$ is defined by
    • f'(a) =
      abla_{h o 0} rac{f(a+h) - f(a)}{h}.
  • For the given $f(x)$, expanding around a generic $x$:
    • f(x+h) = (x+h)^2 - 8(x+h) + 9 = x^2 + 2xh + h^2 - 8x - 8h + 9.
    • f(x+h) - f(x) = (x^2 + 2xh + h^2 - 8x - 8h + 9) - (x^2 - 8x + 9) = 2xh + h^2 - 8h.
      • rac{f(x+h) - f(x)}{h} = rac{2xh + h^2 - 8h}{h} = 2x + h - 8.
    • Taking the limit gives the derivative function: f'(x) =
      abla_{h o 0} (2x + h - 8) = 2x - 8.
  • An alternate, point-specific view uses the fixed value $f(a)$ in the numerator but varies the input: for $a=2$,
    • f'(2) =
      abla_{x o 2} rac{f(x) - f(2)}{x-2}.
    • Here, f(x) - f(2) = (x^2 - 8x + 9) - (-3) = x^2 - 8x + 12 = (x-2)(x-6).
    • Thus,
      abla{x o 2} rac{(x-2)(x-6)}{x-2} = abla{x o 2} (x-6) = -4.
  • Both approaches agree: f'(x) = 2x - 8 ext{ and } f'(2) = -4.$$

Interpretation: derivative as rate of change and as a function

  • Intuition: the derivative measures the rate of change of $f$ with respect to $x$. If $|f'|$ is large, the function changes rapidly; if $f'$ is small in magnitude, the function changes slowly.
  • In a graph, the derivative at a point equals the slope of the tangent line there.
    • A larger derivative means a steeper tangent line.
    • Small (near-zero) derivatives mean flatter tangents.
  • The derivative is itself a function, denoted $f'$ (the derivative of $f$). To study it across all $x$, we treat $f'$ as a new function: $f'(x) = ext{(slope of tangent to $f$ at }x)$.
  • The