AP Calculus AB: Increasing and Decreasing Functions

Understanding where a function increases and decreases is fundamental to calculus and its applications. It allows you to visualize the shape of a graph, optimize real-world systems, and solve problems in physics, engineering, and economics. Mastering this concept transforms the derivative from an abstract slope formula into a powerful tool for analyzing behavior.

The Core Link: Derivative as a Behavior Indicator

The foundation of this analysis is a direct and powerful connection: The sign of a function's first derivative tells you whether the function is increasing or decreasing.

A function $f$ is increasing on an interval if, as $x$ moves from left to right, the $y$-values get larger. Visually, the graph rises. Mathematically, this happens when the slope of the tangent line is positive. Since the derivative $f^{\prime }(x)$ gives the slope of the tangent line, we can state:

• If $f^{\prime }(x)>0$ for all $x$ in an interval, then $f$ is increasing on that interval.
• If $f^{\prime }(x)<0$ for all $x$ in an interval, then $f$ is decreasing on that interval.

Think of driving a car: a positive velocity (derivative of position) means you're moving forward, increasing your distance from the start. A negative velocity means you're moving backward, decreasing that distance. The derivative gives the instantaneous rate of change; its sign tells you the direction of that change.

Identifying Critical Points: Where Behavior Can Change

A function can only change from increasing to decreasing (or vice versa) at points where its derivative is zero or where its derivative does not exist. These points are called critical points (or critical numbers).

To find all critical points of a function $f(x)$:

1. Compute the first derivative, $f^{\prime }(x)$.
2. Find all $x$-values where $f^{\prime }(x)=0$.
3. Find all $x$-values where $f^{\prime }(x)$ is undefined (provided $f(x)$ itself is defined at those points).

These $x$-values partition the domain of the function into potential intervals of increase or decrease. For example, consider $f(x)=x^3-3x^2$. First, find the derivative: $f^{\prime }(x)=3x^2-6x$. Set it equal to zero: $3x^2-6x=3x(x-2)=0$. This gives $x=0$ and $x=2$ as critical points where $f^{\prime }(x)=0$. The derivative is a polynomial, so it is defined everywhere. Therefore, the critical points are $x=0$ and $x=2$. These two points divide the number line into three intervals: $(-\infty ,0)$, $(0,2)$, and $(2,\infty )$.

Constructing the First Derivative Sign Chart

A sign chart is an organized tool that uses test points to determine the sign of $f^{\prime }(x)$ on each interval created by the critical points. It provides a clear visual summary of the function's behavior.

Continuing with our example $f(x)=x^3-3x^2$ and $f^{\prime }(x)=3x(x-2)$:

The logic is straightforward: pick any number within the interval (avoid the endpoints), plug it into the factored form of the derivative, and determine if the result is positive or negative. Since the derivative is continuous on these intervals and never zero inside them, its sign must be constant across the entire interval. The chart tells us $f(x)$ increases on $(-\infty ,0)$, decreases on $(0,2)$, and increases again on $(2,\infty )$.

Determining Precise Intervals of Increase and Decrease

The final step is to translate the sign chart into a precise, written answer using interval notation. You must decide whether to include the critical points themselves in the intervals. By definition, a function is increasing or decreasing on an open interval (parentheses, not brackets). At the exact critical point, the instantaneous rate of change is zero or undefined, so the function is neither increasing nor decreasing at that single point.

Therefore, for $f(x)=x^3-3x^2$:

• $f$ is increasing on the intervals: $(-\infty ,0)$ and $(2,\infty )$.
• $f$ is decreasing on the interval: $(0,2)$.

This analysis also reveals that at $x=0$, the function changes from increasing to decreasing, meaning $f(0)$ is a local maximum. At $x=2$, it changes from decreasing to increasing, so $f(2)$ is a local minimum. This is the foundation for the First Derivative Test for local extrema.

Applying the Concept: An Engineering Scenario

Imagine you are modeling the height $h(t)$ of a roller coaster car above the ground, where $t$ is time in seconds. The derivative $h^{\prime }(t)$ represents vertical velocity. By finding when $h^{\prime }(t)>0$, you identify precise time intervals where the car is climbing a hill. By finding when $h^{\prime }(t)<0$, you identify intervals where the car is plunging down a drop. The critical points (where $h^{\prime }(t)=0$) correspond to the brief instants at the very peak of a hill or the very bottom of a valley, where the vertical motion changes direction. This application shows how interval analysis translates directly into understanding system behavior.

Common Pitfalls

1. Ignoring the Domain: Always identify the domain of $f(x)$ first. Critical points can only occur within the domain. If $f(x)$ has a discontinuity, like a vertical asymptote, it also splits the number line and must be included when creating test intervals. Forgetting this can lead to incorrect interval conclusions.
2. Misidentifying Critical Points: Remember, a critical point requires that $f(x)$ itself is defined. If $f^{\prime }(x)$ is undefined at an $x$-value but $f(x)$ is also undefined there, it is not a critical point. Also, a point where $f^{\prime }(x)=0$ is only critical if it's in the domain.
3. Using the Unfactored Derivative for the Sign Test: Always factor $f^{\prime }(x)$ completely before choosing test values. The factored form makes it much easier and faster to evaluate the sign. Plugging a test value into an unfactored, complicated derivative is error-prone.
4. Including Endpoints in the Intervals: The standard results are stated for open intervals $(a,b)$. Do not write that the function is increasing on $(-\infty ,0]$. It increases up to $x=0$, but is not increasing at $x=0$ because the derivative there is zero.

Summary

• The first derivative is a direct indicator of function behavior: $f^{\prime }(x)>0$ implies increasing, $f^{\prime }(x)<0$ implies decreasing.
• Behavior can only change at critical points, where $f^{\prime }(x)=0$ or $f^{\prime }(x)$ is undefined (and $f(x)$ is defined).
• A sign chart for $f^{\prime }(x)$, using test points on intervals divided by critical points, provides a clear, visual method for analysis.
• The final answer specifies open intervals—using parentheses—where the function is increasing or decreasing.
• This process is the essential first step in curve sketching and optimization, linking algebraic calculation to graphical behavior.