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CalculusBy Educator Today

C09 · Unit C2 Derivative foundations

Derivatives from numerical and graphical evidence

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 2.1–2.3, 2.8–2.9
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

A derivative can be estimated from nearby measurements using a secant slope. A symmetric estimate [f(a+h) minus f(a minus h)]/(2h) uses both sides but is still an approximation without further information. A graph of f minutes encodes slope, not height: positive f minutes means f increases, and zeros of f minutes mark horizontal tangent candidates.

Worked example

A table gives f(1.9) equals 7.61 and f(2.1) equals 8.41. Estimate f minutes (2) is approximately (8.41 minus 7.61)/0.2 equals 4. If f measures distance in metres and x seconds, this is approximately 4 m over s. The samples do not uniquely determine the exact derivative.

Explore the mathematics

Derivatives from numerical and graphical evidence

Bring two points together and compare secant and tangent slopes.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict how the secant changes as the two points get closer.xy-1123At x=1, h=0.125; secant slope=2.125

Predict before continuing

Predict how the secant changes as the two points get closer.

Read the complete explanation
  1. Predict how the secant changes as the two points get closer.
  2. The orange secant joins two points: its slope is Δy/Δx.
  3. The green tangent has slope 2. Reduce h to see the secant approach it.
  4. The derivative is a limit, not substitution of h=0 into a quotient with zero denominator. Here f′(1)=2.
See the idea: Derivatives from numerical and graphical evidence
See the idea: Derivatives from numerical and graphical evidenceThe line joining the two measured points has slope (8.41−7.61)/(2.1−1.9)=4. It is a secant estimate: the unknown curve between the samples need not be this straight line.xy1.87.21.97.6282.18.42.28.8(1.9,7.61)(2.1,8.41)

The line joining the two measured points has slope (8.41−7.61)/(2.1−1.9)=4. It is a secant estimate: the unknown curve between the samples need not be this straight line.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Divide by the full input separation, not by h when using samples a minus h and a+h.

Investigate and explain

Sketch two functions with the same derivative graph but different vertical positions. Identify which information the derivative loses.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.