Unit 5: Analytical Applications of Differentiation
Increasing and Decreasing Intervals; The First Derivative Test
Welcome to one of the most powerful parts of Calculus! Up until now, you have learned how to find derivatives. Now, we are going to use those derivatives as a "GPS" to map out exactly how a function behaves. By looking at the first derivative, we can tell if a graph is climbing uphill, sliding downhill, or hitting a peak or a valley—even if we can't see the graph itself!
Quick Review: Remember that the derivative \( f'(x) \) represents the slope of the tangent line at any point \( x \). This simple fact is the key to everything in this chapter.
1. Determining Intervals of Increase and Decrease
Imagine you are walking along the graph of a function from left to right. Your "altitude" is the value of \( f(x) \). The derivative \( f'(x) \) tells us exactly what is happening to that altitude:
- Increasing Intervals: If \( f'(x) > 0 \) on an interval, the slope is positive. This means the function \( f(x) \) is increasing (going up).
- Decreasing Intervals: If \( f'(x) < 0 \) on an interval, the slope is negative. This means the function \( f(x) \) is decreasing (going down).
- Constant Intervals: If \( f'(x) = 0 \) for every \( x \) in an interval, the function is constant (a flat horizontal line).
Analogy: Think of \( f'(x) \) as your "velocity" while climbing a mountain. If your velocity is positive, you are moving upward. If it is negative, you are moving downward.
Key Takeaway: To find where a function is going up or down, we just need to find where its derivative is positive or negative.
2. Critical Points: The "Turning Points"
Before a function can change from increasing to decreasing, it usually has to "pause" or "break." We call these locations critical points.
A value \( x = c \) in the domain of \( f \) is a critical point if:
1. \( f'(c) = 0 \) (the tangent line is horizontal).
2. \( f'(c) \) is undefined (there is a sharp corner, a cusp, or a vertical tangent).
Note: For a point to be a critical point, the original function \( f(c) \) must actually exist there!
3. The First Derivative Test
The First Derivative Test is the formal process we use to identify relative (local) extrema—which is just a fancy way of saying "local high points" (maximums) and "local low points" (minimums).
How to identify a Relative Maximum:
If \( f'(x) \) changes from positive to negative at a critical point \( x = c \), then \( f(c) \) is a relative maximum.
Visualization: The graph was going up \( (\nearrow) \), hit a peak, and started going down \( (\searrow) \).
How to identify a Relative Minimum:
If \( f'(x) \) changes from negative to positive at a critical point \( x = c \), then \( f(c) \) is a relative minimum.
Visualization: The graph was going down \( (\searrow) \), hit a valley, and started going up \( (\nearrow) \).
What if the sign doesn't change?
If \( f'(x) \) is positive on both sides (or negative on both sides) of the critical point, then there is no relative extremum at that point. The graph might just be "taking a shelf break" before continuing in the same direction.
4. Step-by-Step: Solving the Problems
Don't worry if this seems like a lot of steps; it follows a very logical rhythm. Follow these steps for any "find the intervals of increase/decrease" or "find the relative extrema" problem:
- Find the derivative: Calculate \( f'(x) \).
- Find the critical points: Set \( f'(x) = 0 \) and find where \( f'(x) \) is undefined.
- Create a Sign Chart:
- Draw a number line and mark your critical points. This divides the line into intervals.
- Pick a "test value" from each interval and plug it into the derivative \( f'(x) \).
- Record whether the result is positive \( (+) \) or negative \( (-) \).
- Interpret the results: Use the signs to determine where the function increases/decreases and where the peaks and valleys are.
Did you know? On the AP Exam Free Response Questions (FRQs), a "sign chart" is a great tool for you, but it is not considered a sufficient justification. You must write out your conclusion in words!
5. Writing a Perfect AP Justification
When the exam asks you to "Justify your answer," use these specific templates to ensure full credit:
- For Increasing: "\( f(x) \) is increasing on the interval \( (a, b) \) because \( f'(x) > 0 \) on that interval."
- For a Relative Max: "There is a relative maximum at \( x = c \) because \( f'(x) \) changes from positive to negative at \( x = c \)."
- For a Relative Min: "There is a relative minimum at \( x = c \) because \( f'(x) \) changes from negative to positive at \( x = c \)."
6. Common Pitfalls to Avoid
Mistake 1: Plugging test values into the wrong function.
Always plug your test values into the derivative \( f'(x) \) to check for increasing/decreasing. If you plug them into the original \( f(x) \), you're just finding the height of the point, not the direction of the curve.
Mistake 2: Forgetting "Undefined" points.
Critical points aren't just where the derivative is zero. If you have a function like \( f(x) = x^{2/3} \), the derivative is undefined at \( x = 0 \). That is still a critical point and a potential location for a minimum!
Mistake 3: Assuming signs always alternate.
While signs often go \( +, -, +, - \), they don't always. For example, in \( f(x) = x^3 \), the derivative \( f'(x) = 3x^2 \) is positive on both sides of \( x = 0 \). Always test every interval!
Key Takeaway: The First Derivative Test is your primary tool for finding local high and low points. In later chapters, you will learn the Second Derivative Test and the Candidates Test, which provide different ways to look at extrema, but the First Derivative Test remains the most versatile "all-purpose" tool in your kit.
Quick Tip: If you're on a calculator-permitted section of the exam, you can graph \( f'(x) \) and see where it crosses the x-axis to find your critical points and signs very quickly!