Approximating Values of a Function Using Local Linearity and Linearization
Construct a nearby tangent approximation and use interval concavity to justify its error direction.
College Board Topic 4.6 · CED p. 87 (PDF p. 92) · CED effective Fall 2020; project course outline 2026–27.
Zoom close enough to a smooth curve and its tangent is a useful stand-in. At x = a, the linear model is L(x) = f(a) + f′(a)(x − a). It uses the exact value and tangent slope at the anchor. The estimate improves locally as x approaches a; there is no universal distance that makes every tangent approximation accurate.
For f(x) = √x near a = 9, f(9) = 3 and f′(9) = 1/6. Thus √9.12 ≈ 3 + 0.12/6 = 3.02. The true curve is concave down for positive x because f″(x) = −1/(4x^(3/2)) < 0. On a nearby interval where that concavity holds, the tangent sits above the curve, so the estimate is an overestimate.
Concave up gives a tangent underestimate; concave down gives an overestimate, provided the concavity holds between the anchor and target. If all you know is f″(a), you need suitable continuity/local behavior before extending that sign to an interval. A graph can suggest the error direction, while a derivative sign can justify it. Differententials express the same approximation: Δf ≈ df = f′(a)Δx. Both quantities use output units.
Let's work through it
For f(x) = ln x at a = 1, f(1) = 0 and f′(1) = 1.
Why: Pick an anchor whose value and derivative are easy to evaluate.
L(x) = x − 1, so ln(1.04) ≈ 0.04.
Why: A tangent model changes output by slope times input change.
Since f″(x) = −1/x² < 0 for positive x, 0.04 is an overestimate.
Why: The function is concave down throughout the interval from 1 to 1.04.
Do not reuse this as a high-accuracy estimate of ln 5.
Why: Local linearity gives a nearby approximation, not a globally accurate replacement.
Try it from memory
Pause & retrieve
Try it before you peek.
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
Write the linearization at a using f(a) and f′(a).
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
Write the linearization at a using f(a) and f′(a).
Worked answer and reason
L(x) = f(a) + f′(a)(x − a).
The model must pass through the anchor and have the same slope.
Retrieval 2
A curve is concave up between anchor and target. Is its tangent estimate high or low?
Worked answer and reason
Low: an underestimate.
An increasing slope lifts the curve above its tangent.
Retrieval 3
What does f′(a)Δx estimate?
Worked answer and reason
The output change Δf, not the new output f(a + Δx) itself.
Add f(a) when the question asks for the new value.
Catch the tempting mistake
- A tangent approximation is exact because the derivative is exact.
- The slope can be exact while the curve bends away from the line.
- An estimate is an underestimate whenever the slope is positive.
- The direction of error depends on concavity, not the sign of the slope.
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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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 4 problem 22 · No calculator · analytic
- Unit 4 problem 23 · No calculator · numerical
- Unit 4 problem 24 · No calculator · analytic
- Unit 4 problem 25 · Graphing calculator required · analytic
For the teacher
- Ask for a predicted error direction before permitting calculator comparison.
- Have learners choose a useful anchor themselves, and discuss why a convenient nearby value beats an inconvenient exact target.
Glossary and deeper reading · 3 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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Concavity
Concavity describes how a graph’s slope changes as the input increases. A graph is concave up where its derivative increases and concave down where its derivative decreases. For example, x² is concave up because its derivative 2x increases. A positive second derivative establishes concave-up behavior on an interval, while a negative one establishes concave-down behavior.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.