A curve carries far more information than meets the eye. On the AP® Calculus exam, questions often pair a graph of a function with its derivative or second derivative and ask students to match them quickly. Mastering the graph of function and derivative relationship unlocks an efficient shortcut: shapes reveal slopes, and slopes reveal concavity. This lesson explains the visual links among f, f', and f'', shows how to sketch a derivative function graph from a curve, and demonstrates how to find the second derivative based on a function graph.
What We Review
Refresher: What a Derivative Tells Us
A derivative measures an “instantaneous rate of change,” which is simply the slope of the tangent line at each point.
- Verbal view – How fast?
- Algebraic view – Limit definition f'(x)=\displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
- Graphical view – Tilt of the curve
Therefore, a steep uphill segment has positive derivative values, while a steep downhill segment has negative ones. Importantly, matching graphs is often faster than grinding through formulas on test day, especially when calculator use is restricted.
From f to f′: Translating Shape into Slope
Key Feature Map
- Increasing → f' > 0
- Decreasing → f' < 0
- Local maximum or minimum (peak/valley) → f' = 0 and the sign of f' changes
- Horizontal tangent with no sign change → f' = 0 but not an extremum
Because of these links, a quick scan of one curve’s hills and valleys allows creation of its derivative function graph within seconds.
Example 1: Sketching the Derivative Function Graph
Prompt: The solid curve below resembles a roller-coaster: it rises from x=-4 to x=-1, falls sharply to x=1, rises gently to x=3, then levels off. Sketch f'.
Step-by-Step Solution
| Interval | Shape of f | Sign of f' | Relative Size |
| –4 to –1 | Increasing steeply | + | Large positive |
| –1 | Peak | 0 | |
| –1 to 1 | Decreasing steeply | – | Large negative |
| 1 | Valley | 0 | |
| 1 to 3 | Increasing gently | + | Small positive |
| 3 | Horizontal plateau | 0 | near zero slope |
- Plot zero points of f' at x=-1,1,3.
- Between –4 and –1, draw f' positive and high.
- Between –1 and 1, draw it negative, dipping low around x=0.
- Between 1 and 3, keep f' slightly positive.
- Approach zero at x=3 and beyond.
Consequently, the derivative function graph crosses the x-axis at each turning point of f and mirrors how “steep” each section feels.
From f′ to f″: Concavity and Inflection
Core Ideas
- Concave up (like a cup) → f'' > 0 because f' is rising
- Concave down (like a frown) → f'' < 0 because f' is falling
- Inflection point → f'' = 0 and a sign change in f''
Thus, reading the slopes of f' helps in graphing the second derivative.
Example 2: Building the Second Derivative Graph
Prompt: The red graph below represents the derivative of a function. Sketch a graph of the second derivative.
Solution
- Note where f' itself increases or decreases:
- –4 < x < –2: f' decreasing ⇒ f'' < 0
- –2 < x < 1: f' increasing ⇒ f'' > 0
- x>1: f' decreasing again ⇒ f'' < 0
- Mark x=-2 and x=1 as potential inflection points because f'' changes sign.
- For the second derivative graph, draw bars above the x-axis on intervals where f'' > 0 and below where f'' < 0. The result is a simple piecewise step-like sketch.
This exercise highlights how to find second derivative based on function graph—or in this case, the derivative’s graph—without writing formulas.
Putting It All Together: Matching f, f′, and f″
Multiple-choice items often display three unlabeled curves. The task: identify which is f, which is f', and which is f''. Follow this checklist:
- First, look for the curve with peaks and valleys; that one is usually f.
- Next, locate the curve crossing the x-axis at the same x-values as those peaks/valleys. That curve is probably f'.
- Finally, the remaining curve must be the second derivative graph.
Example 3: Three Unlabeled Graphs
Graph A has turning points at x=-2 and x=1. Graph B crosses the x-axis at exactly those x-values. Graph C changes sign at x=-3, -2, 1.
Solution outline:
- Graph A ≈ f because of visible extrema.
- Graph B ≈ f' since zeros align with extrema of Graph A.
- Graph C therefore must be f''.
Always justify by noting first-derivative sign changes and concavity evidence; the AP® FRQ rubric demands that logical link.
Because graphing the second derivative is sometimes tricky, quickly confirming sign-change points anchors the identification.
Common Pitfalls & Speed Tips
- Horizontal tangents are not always inflection points; verify a concavity change.
- Open intervals matter. Endpoints may lack derivatives; therefore, avoid assigning f' = 0 there.
- Sketch tiny sign charts next to the graph. The visual cue accelerates multiple-choice work.
Quick Reference Chart
| Term | Meaning |
| Critical point | f'(x)=0 or undefined |
| Increasing interval | f'(x) > 0 |
| Decreasing interval | f'(x) < 0 |
| Inflection point | f''(x)=0 with sign change |
| Concave up | f''(x) > 0 |
| Concave down | f''(x) < 0 |
Conclusion
Understanding how a graph of function and derivative interact transforms a complicated sketch into clear information: sign, slope, and curvature. With steady practice, students can glance at a curve and outline its derivative function graph or second derivative graph in moments, earning quick points on exam day. Practice regularly with both paper sketches and graphing technology to lock in these visual reflexes.
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