Welcome to Unit 5: Analyzing Functions!

Have you ever looked at a roller coaster and wondered exactly where it stops climbing and starts its thrilling drop? In calculus, we use the first derivative to pinpoint those exact moments. In this chapter, we will learn how to determine where a function is moving up (increasing), where it is moving down (decreasing), and how to identify those "peak" and "valley" points called relative extrema.

Don't worry if derivatives still feel a bit abstract. Think of the first derivative as a motion detector: it tells us if a function is rising, falling, or standing still.

1. Prerequisite: Finding Critical Points

Before we can find where a function increases or decreases, we need to find the "turning points." These are called critical points. A critical point occurs at \(x = c\) in the domain of \(f\) if:

  1. \(f'(c) = 0\) (The tangent line is horizontal)
  2. \(f'(c)\) is undefined (The graph might have a sharp corner or a vertical tangent)

Quick Review: You learned about these in Topic 5.2. We use them here as the "fences" that divide our graph into different sections or intervals.

2. Increasing and Decreasing Intervals (Topic 5.3)

The First Derivative Test for Monotonicity is a fancy way of saying: "Look at the sign of the slope."

  • If \(f'(x) > 0\) (positive) for all \(x\) in an interval, then \(f\) is increasing on that interval. (The graph goes up from left to right).
  • If \(f'(x) < 0\) (negative) for all \(x\) in an interval, then \(f\) is decreasing on that interval. (The graph goes down from left to right).

How to Find These Intervals (Step-by-Step)

Step 1: Find the derivative, \(f'(x)\).

Step 2: Set \(f'(x) = 0\) and find where \(f'(x)\) is undefined to get your critical points.

Step 3: Create a sign chart. Draw a number line and mark your critical points. This divides the line into intervals.

Step 4: Pick a "test value" inside each interval and plug it into the derivative \(f'(x)\). You only care if the result is positive or negative!

Step 5: Write your conclusion using interval notation.

Example: If \(f'(x) = 2x - 4\), the critical point is \(x = 2\).
Testing \(x = 0\): \(f'(0) = -4\) (Negative \(\implies\) Decreasing).
Testing \(x = 3\): \(f'(3) = 2\) (Positive \(\implies\) Increasing).

Key Takeaway

Positive derivative \(\implies\) Increasing function. Negative derivative \(\implies\) Decreasing function.

3. The First Derivative Test (Topic 5.4)

Now that we know where the function is rising and falling, we can find the Relative (Local) Extrema. These are the "high points" (maxima) and "low points" (minima) of the graph.

The Rule of Thumb:

  1. Relative Maximum: Occurs at a critical point \(c\) if \(f'(x)\) changes from positive to negative at \(x = c\). (The graph stops going up and starts going down).
  2. Relative Minimum: Occurs at a critical point \(c\) if \(f'(x)\) changes from negative to positive at \(x = c\). (The graph stops going down and starts going up).
  3. No Extrema: If \(f'(x)\) is positive on both sides (or negative on both sides) of a critical point, there is no relative maximum or minimum there. The graph just took a little "breather" and kept going the same way.

Analogy: Imagine hiking. If you are walking uphill and then suddenly start walking downhill, you must have passed a peak (Maximum). If you were walking downhill and then started walking uphill, you were just at the bottom of a valley (Minimum).

4. AP Exam Tips: Justification and Notation

On the AP Exam, Practice 3 (Justification) is worth a lot of points. Simply drawing a sign chart is not enough for a full-credit answer on a Free-Response Question (FRQ). You must write out your reasoning in sentences.

How to write a perfect justification:

For a Maximum: "\(f(x)\) has a relative maximum at \(x = c\) because \(f'(x)\) changes from positive to negative at \(x = c\)."

For a Minimum: "\(f(x)\) has a relative minimum at \(x = c\) because \(f'(x)\) changes from negative to positive at \(x = c\)."

For Intervals: "\(f(x)\) is increasing on the interval \((a, b)\) because \(f'(x) > 0\) on that interval."

Common Mistakes to Avoid:
  • The "It" Trap: Never say "It is increasing because it is positive." The graders don't know what "it" is! Always specify: "\(f(x)\) is increasing because \(f'(x)\) is positive."
  • Calculator Setup: If you use your calculator to find zeros of a derivative on the calculator-active section, you must write the equation you are solving (e.g., \(f'(x) = 0\)) before writing the answer.
  • Endpoints: The First Derivative Test identifies relative extrema. To find absolute extrema on a closed interval, you would use the Candidates Test (which you will cover in Topic 5.5).

5. Summary & Quick Review

Did you know? Even if a function is continuous, it might not have a derivative at a certain point (like the tip of a "V" shape). We still check these points because they are often where the minimum or maximum lives!

Checklist for Success:

  • Do I have the correct derivative?
  • Did I find all critical points (both \(f'(x) = 0\) and \(f'(x)\) = undefined)?
  • Did I test values in the derivative, not the original function?
  • Did I justify my answer using the specific language: "\(f'(x)\) changes from... to..."?

Don't worry if this seems tricky at first! The more sign charts you build, the more intuitive it becomes. You are learning to read the "hidden language" of graphs!

Next Step: You'll soon combine this with the "Candidates Test" to find the absolute highest and lowest points on a specific interval. Keep going!