Sketching and connecting a function with its first and second derivatives

Welcome to the "Grand Finale" of Unit 5!

You’ve already learned about critical points, concavity, and those handy derivative tests. Now, it’s time to put all those pieces of the puzzle together. In this chapter, we focus on sketching graphs and making connections between a function \(f(x)\), its first derivative \(f'(x)\), and its second derivative \(f''(x)\). Think of this as "Calculus Detective Work"—you are using clues from one graph to reveal the secrets of another!

Don't worry if this seems like a lot to track at once. By the end of these notes, you’ll have a clear system for translating between these three mathematical layers.

The "Big Three" Connection

To succeed on the AP Exam, you need to be able to move fluidly between \(f\), \(f'\), and \(f''\). The most important thing to remember is that the y-value of a derivative tells you the slope or behavior of the function above it.

1. The First Derivative Connection: \(f(x)\) and \(f'(x)\)
- When \(f'(x) > 0\) (the graph of \(f'\) is above the x-axis), \(f(x)\) is increasing.
- When \(f'(x) < 0\) (the graph of \(f'\) is below the x-axis), \(f(x)\) is decreasing.
- When \(f'(x) = 0\) or is undefined, \(f(x)\) has a critical point (a possible relative maximum or minimum).

2. The Second Derivative Connection: \(f(x)\) and \(f''(x)\)
- When \(f''(x) > 0\), \(f(x)\) is concave up (it looks like a cup \(\cup\)).
- When \(f''(x) < 0\), \(f(x)\) is concave down (it looks like a frown \(\cap\)).
- When \(f''(x)\) changes sign, \(f(x)\) has a point of inflection.

3. The "Bridge" Connection: \(f'(x)\) and \(f''(x)\)
This is the one that often trips students up! Remember that \(f''(x)\) is just the derivative of \(f'(x)\).
- When \(f'(x)\) is increasing, its derivative \(f''(x)\) is positive, which means \(f(x)\) is concave up.
- When \(f'(x)\) is decreasing, its derivative \(f''(x)\) is negative, which means \(f(x)\) is concave down.
- When \(f'(x)\) has a relative extremum (a peak or valley on the \(f'\) graph), \(f(x)\) has a point of inflection.

Key Takeaway: The "height" (y-value) of a derivative graph tells you the "slope" of the graph before it!

Reading the Graph of \(f'(x)\) like a Pro

On the AP Exam, you will often be given a graph of \(f'(x)\) and asked questions about \(f(x)\). This requires a shift in mindset. You aren't looking at where the graph is "high" or "low," but where it is positive, negative, or changing direction.

How to find features of \(f(x)\) using only the graph of \(f'(x)\):

Relative Maximums of \(f(x)\): Look for where the graph of \(f'(x)\) crosses the x-axis from above to below (positive to negative).
Relative Minimums of \(f(x)\): Look for where the graph of \(f'(x)\) crosses the x-axis from below to above (negative to positive).
Points of Inflection of \(f(x)\): Look for the "peaks and valleys" (relative extrema) on the \(f'(x)\) graph.

Quick Review Tip: If you are looking at the graph of \(f'(x)\), the slope of that graph is your \(f''(x)\). If the slope is positive, \(f(x)\) is concave up!

Step-by-Step: Sketching \(f(x)\) from \(f'(x)\)

If you are asked to sketch a possible graph of the original function \(f(x)\) based on its derivative, follow these steps:

1. Identify Critical Points: Mark every x-value where \(f'(x) = 0\). These are where your sketch might have a "turn" or a horizontal tangent.
2. Determine Intervals of Increase/Decrease: Look at where the \(f'\) graph is above or below the x-axis. Lightly shade these areas on your paper to remind you which way your sketch should be heading.
3. Determine Concavity: Look at where the \(f'\) graph is going up (increasing) or going down (decreasing). This tells you how to "curve" your sketch.
4. Connect the Dots: Start at a given point (if provided) or any arbitrary y-value. Sketch a smooth curve that follows the directions you've gathered. Remember: unless told otherwise, functions in AP Calculus are usually continuous and differentiable (smooth).

Did you know? You can't know the exact "height" (y-intercept) of \(f(x)\) just by looking at \(f'(x)\). Your sketch can be shifted up or down unless the problem gives you a specific point like \(f(0) = 2\).

The "Summary Table" Cheat Sheet

This table is a lifesaver for matching graphs in Multiple Choice questions!

If \(f(x)\) is... \(\rightarrow\) Then \(f'(x)\) is... \(\rightarrow\) Then \(f''(x)\) is...
Increasing \(\rightarrow\) Positive (above x-axis) \(\rightarrow\) (Not enough info)
Decreasing \(\rightarrow\) Negative (below x-axis) \(\rightarrow\) (Not enough info)
Relative Extrema \(\rightarrow\) Zero (crosses x-axis) \(\rightarrow\) Non-zero (determines max/min)
Concave Up \(\rightarrow\) Increasing (positive slope) \(\rightarrow\) Positive
Concave Down \(\rightarrow\) Decreasing (negative slope) \(\rightarrow\) Negative
Inflection Point \(\rightarrow\) Relative Extrema \(\rightarrow\) Zero (or undefined) and changes sign

Common Mistakes to Avoid

Mistake 1: Confusing "Positive" with "Increasing."
Just because the graph of \(f'(x)\) is going up doesn't mean \(f(x)\) is going up. If \(f'(x)\) is going up but is still negative (below the x-axis), the original function \(f(x)\) is actually decreasing but concave up!

Mistake 2: Missing the Sign Change for Inflection Points.
A point of inflection only happens if \(f''(x)\) changes sign. On an \(f'(x)\) graph, this means the graph must change from increasing to decreasing (or vice-versa). If \(f'(x)\) just "touches" a peak and stays on one side, it's not an inflection point.

Mistake 3: Forgetting Units.
In contextual problems (Unit 5.10-5.11), if \(f(t)\) is in feet and \(t\) is in seconds, then \(f'(t)\) is in \(feet/sec\) and \(f''(t)\) is in \(feet/sec^2\). Always keep your units consistent!

Final Checklist for Success

- Can you identify where \(f(x)\) is increasing by looking at the graph of \(f'(x)\)?
- Can you find the x-coordinates of inflection points by looking at the "turns" of \(f'(x)\)?
- Do you remember that \(f''(x)\) being positive means the slope of \(f(x)\) is getting larger (concave up)?
- Can you justify your answers using the phrases: "Because \(f'(x)\) changes from positive to negative..." or "Because \(f'(x)\) is increasing..."?

Justification Tip: On Free-Response Questions, never say "the graph moves from up to down." Always use official language like "\(f'(x)\) changes from positive to negative" or "the derivative is increasing."