Change in tandem (function behavior) — AP Precalculus
1. Average and Instantaneous Rates of Change ★★☆☆☆ ⏱ 4 min
The foundation of analyzing change in tandem (co-variation) is quantifying how much a function's output changes for a given change in input. This connects graphical, algebraic, and numerical descriptions of function behavior.
\text{AROC} = \frac{f(b) - f(a)}{b-a}
AROC equals the slope of the secant line connecting the points $(a, f(a))$ and $(b, f(b))$, describing average behavior over the entire interval.
\text{IROC} = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = f'(a)
2. Increasing/Decreasing Behavior and Local Extrema ★★★☆☆ ⏱ 4 min
We use the sign of the instantaneous rate of change (first derivative) to describe whether a function is growing or shrinking over an interval.
- A function is **increasing** on interval $I$ if $f'(x) > 0$ for all $x \in I$: for any $x_1 < x_2$, $f(x_1) < f(x_2)$.
- A function is **decreasing** on interval $I$ if $f'(x) < 0$ for all $x \in I$: for any $x_1 < x_2$, $f(x_1) > f(x_2)$.
For a degree $n$ polynomial, the maximum number of local extrema (turning points) is $n-1$.
3. Concavity and Inflection Points ★★★☆☆ ⏱ 4 min
Concavity describes how the rate of change itself changes as input changes, making it a core concept for change in tandem: it is change in the rate of change.
- A function is **concave up** on interval $I$ if $f''(x) > 0$: the slope of the tangent line increases as $x$ increases, and the graph curves upward.
- A function is **concave down** on interval $I$ if $f''(x) < 0$: the slope of the tangent line decreases as $x$ increases, and the graph curves downward.
A zero of $f''(x)$ is not automatically an inflection point. For a degree $n$ polynomial, the maximum number of inflection points is $n-2$.
4. AP-Style Practice ★★★★☆ ⏱ 6 min
Common Pitfalls
Why: Students assume any zero of the second derivative is automatically an inflection point, without checking for a sign change. For $f(x)=x^4$, $f''(x)=12x^2$ which is positive on both sides of $x=0$, so concavity does not change.
Why: Students confuse interval-by-interval increasing behavior with global behavior, ignoring that $f(1) = -1 < f(3) = 1$, which violates the definition of an increasing function.
Why: Students mix up the order of subtraction in the numerator, leading to a sign error.
Why: Students forget that $x=2$ is a hole, so the function is not defined at that point and cannot be an extremum.
Why: Students confuse concavity (change in the rate of change) with increasing/decreasing behavior (the sign of the rate of change itself).