Calculus I · Unit 2B · lesson
Reading Function and Derivative Graphs
Learning objectives
Infer signs and trends of derivatives from graphs, and reconstruct plausible original functions from derivative graphs.
Translate Shape into Derivative Information
Before the formulas
The graphs in Reading Function and Derivative Graphs should be read vertically through a common input. A height on the derivative graph is a slope on the original graph. A zero of velocity is a horizontal tangent of position; a zero of acceleration is a horizontal tangent of velocity.
Do not match graphs by superficial shape. Translate one feature at a time: sign, zeros, increasing behavior, and concavity. This produces a defensible interpretation even when the graphs are unfamiliar or not drawn to a convenient scale.
Translate heights on one graph into slopes on another
To sketch from , begin where the slope information is clearest. Horizontal tangents on become zeros of . Rising intervals become positive regions; falling intervals become negative regions. Steeper parts of produce larger magnitudes on .
Do not copy the shape of . The derivative graph records slope, not height. A high point on can correspond to a zero on , and a low point on can do the same.
Graph translation is a language skill. A rising original graph corresponds to a positive derivative; a local maximum corresponds to a derivative that changes from positive to negative; and a straight segment corresponds to a constant derivative. The derivative graph should be constructed from slopes, not from the heights of the original graph.
Read from both directions. Given , infer ; given , reconstruct possible behavior of . The second task is less unique because many original functions can share the same derivative up to a vertical shift.
For a graph of :
• where increases; • where decreases; • at horizontal tangents; • large means a steep graph; • where slopes are increasing; • where slopes are decreasing.
Read a derivative graph as motion
Suppose is positive on , zero at , negative on , and increasing throughout. Describe .
Worked solution
Write a real attempt before opening the supplied answer.
Read this graph as text
translating between f and f'. The zeros and signs of (f'(x)=3x 2-3 ) encode the increasing and decreasing behavior of (f(x)=x 3-3x ). See the adjacent lesson prose and the detailed editorial brief.
Every relationship in translating between f and f' is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.
Why it matters: Train bidirectional translation between the local slopes of a function and the values and signs of its derivative.
The zeros and signs of encode the increasing and decreasing behavior of .
The zeros and signs of f'(x)=3x 2-3 encode the increasing and decreasing behavior of f(x)=x 3-3x.
Sketch a possible if is always positive and decreasing.
Sketch for a graph of with one local maximum and one local minimum.
If has a local maximum at , what can be said about when it exists?
After the explanation
Use the section idea
Use the sign, size, units, and zeros of derivatives to tell a time-aligned story about motion or another changing quantity.
Position, velocity, and acceleration are synchronized views: amount, rate of amount, and rate of the rate.
Separate direction from speed, and compare the signs of velocity and acceleration before describing speeding behavior.
Treating negative velocity as slowing down or confusing a function's height with the slope of its graph.
Can you interpret a first and second derivative at the same input without mixing their units or meanings?
Source & rights
Original instruction with traceable references.
BetterGrades-original composition declared by source handoff; owner provenance review required before public release
Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.