The average rate of change of a function $f(x)$ over the interval $[a, b]$ is:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} = \frac{\Delta y}{\Delta x}$

Interpretation

• Geometrically: Slope of the secant line connecting $(a, f(a))$ and $(b, f(b))$
• Physically: Average velocity (if $f(x)$ represents position)
• Generally: How fast $f$ changes on average between $x = a$ and $x = b$

Example

For $f(x) = x^2$ on $[1, 3]$: $\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = \frac{8}{2} = 4$

The function increases by 4 units per unit of $x$, on average.

Instantaneous Rate of Change

The instantaneous rate of change of $f(x)$ at $x = a$ is:

$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

Interpretation

• Geometrically: Slope of the tangent line at $(a, f(a))$
• Physically: Instantaneous velocity at time $t = a$
• Generally: How fast $f$ is changing right at $x = a$
• In Calculus: This limit is called the derivative, written $f'(a)$

Difference Quotient

The expression $\frac{f(a + h) - f(a)}{h}$ is called the difference quotient.

As $h \to 0$, the secant line approaches the tangent line.

Key Formulas

Average Rate of Change (over interval $[a, b]$): $\frac{f(b) - f(a)}{b - a}$

Instantaneous Rate of Change (at point $x = a$): $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

Alternative form (using $x$ instead of $h$): $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

Real-World Applications

Position and Velocity

• If $s(t)$ = position at time $t$
• Average velocity = $\frac{s(t_2) - s(t_1)}{t_2 - t_1}$
• Instantaneous velocity = $\lim_{h \to 0} \frac{s(t + h) - s(t)}{h}$

Cost and Marginal Cost

• If $C(x)$ = total cost to produce $x$ items
• Average cost per item = $\frac{C(x)}{x}$
• Marginal cost (cost of one more item) ≈ instantaneous rate of change

Connection to Derivatives

In calculus, the instantaneous rate of change is the derivative: $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

Precalculus focuses on:

• Computing average rates of change
• Understanding the concept of instantaneous rate of change
• Setting up (but not necessarily evaluating) limit expressions

where $a = 1$ and $b = 4$.

Step 1: Find $f(1)$: $f(1) = 1^2 + 3(1) = 1 + 3 = 4$

Step 2: Find $f(4)$: $f(4) = 4^2 + 3(4) = 16 + 12 = 28$

Step 3: Calculate the average rate of change: $\frac{f(4) - f(1)}{4 - 1} = \frac{28 - 4}{3} = \frac{24}{3} = 8$

Answer: The average rate of change is $8$.

Interpretation: On average, the function increases by 8 units for every 1 unit increase in $x$ over the interval $[1, 4]$.