Estimating Derivatives of a Function at a Point
Estimate a derivative from nearby data or a tangent, and retain approximation and unit labels.
College Board Topic 2.3 · CED p. 57 (PDF p. 62) · CED effective Fall 2020; project course outline 2026–27.
With a table, estimate a derivative using a secant through nearby inputs. A centered difference [f(a+h)−f(a−h)]/(2h) samples both sides. A one-sided difference is useful near a domain endpoint. These are approximations unless the structure of the model makes them exact.
| t (minutes) | 1.9 | 2.0 | 2.1 |
|---|---|---|---|
| T(t) (°C) | 18.61 | 19.00 | 19.41 |
The centered slope at 2 is (19.41−18.61)/0.2=4°C/min. The one-sided slopes are 3.9 and 4.1°C/min; agreement suggests a derivative near 4. The actual derivative is not proven by three measurements, and noisy measurements can make smaller intervals less reliable.
On a graph, estimate the slope of the tangent line using two well-spaced points on that tangent, rather than two far-apart points on the curve. Technology can compute a numerical derivative, but compare several step sizes or windows. Extremely small steps can amplify rounding or cancellation, and a numerical command may return a misleading value at a corner.
Let's work through it
- Use data at 1.9 and 2.1 to estimate T′(2). Reason: they straddle the desired input.
- Compute (19.41−18.61)/(2.1−1.9)=4. Reason: this is the centered secant slope.
- Report approximately 4°C/min, not an exact proved derivative. Reason: finitely many table entries cannot establish the limiting rate.
Try it from memory
Pause & retrieve
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Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
At x=3, nearby outputs are f(2.9)=7.7 and f(3.1)=8.3. Estimate f′(3).
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
At x=3, nearby outputs are f(2.9)=7.7 and f(3.1)=8.3. Estimate f′(3).
Worked answer and reason
Approximately 3.
The centered difference is 0.6/0.2.
Retrieval 2
Does decreasing h forever guarantee a better calculator estimate?
Worked answer and reason
No.
Rounding, cancellation and measurement noise can spoil tiny differences.
Catch the tempting mistake
- A symmetric secant proves a derivative exists.
- A corner can have symmetric secants while its one-sided limiting slopes disagree.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 2 problem 8 · No calculator · numerical
- Unit 2 problem 9 · No calculator · graphical
- Unit 2 problem 10 · Graphing calculator required · numerical
For the teacher
- Compare left, right and centered estimates at both a smooth point and |x| at 0.
- Ask students to distinguish numerical approximation, exact value and existence.
Glossary and deeper reading · 2 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Tangent line
A tangent line at a differentiable point passes through the point and has slope equal to the derivative there. It describes the function’s local linear behavior. For example, y=x² has tangent line y=4+4(x−2) at x=2. A tangent can cross the curve, so touching at exactly one point is not its general definition.