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Calculus I · Unit 2B · lesson

Local Linearity

Concept

Learning objectives

Explain local linearity and construct a linearization.

Linear Approximation, Differentials, and Newton's Method

A Smooth Curve Looks Like Its Tangent Up Close

Explanation

Before the formulas

The approximation in Local Linearity replaces a nonlinear function with its best local line. The known value supplies the starting height, and the derivative supplies the rate of change. The estimate is useful because lines are easy to calculate with, not because the original curve has become exactly linear.

Always identify the center aa and the target xx. Write Δx=x−a\Delta x=x-a, calculate the predicted change f′(a)Δxf'(a)\Delta x, and then add it to f(a)f(a). Use concavity or a numerical comparison to understand the direction and size of the error.

Explanation

A differentiable curve has a best local line

The tangent line is not merely a line that touches the graph. Near the contact point, it reproduces the function's value and first-order change. As you zoom in, the remaining difference between curve and line becomes small compared with the horizontal displacement.

This is why derivatives support approximation. Instead of evaluating a difficult function exactly at a nearby input, evaluate the tangent line, whose arithmetic is simple. The approximation is local, so distance from the base point matters.

A differentiable curve resembles a line when viewed closely enough. This is not merely a visual curiosity; it is the operational meaning of differentiability. The tangent line captures the function's first-order response to a small input change.

Local linearity explains why derivatives are useful even when exact formulas are complicated. Near a known input, the function can be replaced temporarily by a much simpler line, with an error that becomes relatively smaller as the input change shrinks.

Differentiability means more than possessing a tangent slope. Near a differentiable point, the function is well approximated by its tangent line. If xx is close to aa, then

f(x)≈f(a)+f′(a)(x−a).f(x)\approx f(a)+f'(a)(x-a).
Definition

Linearization

The linearization of ff at x=ax=a is

L(x)=f(a)+f′(a)(x−a).\boxed{L(x)=f(a)+f'(a)(x-a)}.

It is exactly the tangent-line function at aa.

Guided walkthrough

Approximate a square root

Use linearization to estimate 4.1\sqrt{4.1}.

Answer reveal

Worked solution

Write a real attempt before opening the supplied answer.

Near x=4, the graph of x is nearly indistinguishable from its tangent line.
Read this graph as text

square-root curve and its linearization. Near (x=4 ), the graph of ( x ) is nearly indistinguishable from its tangent line. See the adjacent lesson prose and the detailed editorial brief.

Every relationship in square-root curve and its linearization 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: Make local linearity visible by comparing a function with its tangent approximation while the viewing window and target point change.

Visual study

Near x=4x=4, the graph of x\sqrt x is nearly indistinguishable from its tangent line.

Near x=4, the graph of x is nearly indistinguishable from its tangent line.

Optional advanced note

The derivative as the best first-order approximation

The rigorous statement of local linearity is

lim⁡h→0f(a+h)−f(a)−f′(a)hh=0.\lim_{h\to0}\frac{f(a+h)-f(a)-f'(a)h}{h}=0.

The numerator is the approximation error. Dividing by hh and obtaining zero means the error shrinks faster than the input displacement itself. This is the one-variable prototype of the Frechet derivative used in advanced analysis.

After the explanation

Use the section idea

Reading lens

A differentiable curve behaves like its tangent line over a small enough neighborhood, with concavity explaining the direction of error.

Mental model

Linearization, differentials, and Newton's method reuse one local line for estimation, uncertainty, or a better root guess.

Decision

Choose a nearby easy input, record the local slope, and state why the requested change is small enough for the model.

Common trap

Presenting a tangent estimate as exact or running Newton iterations without checking residuals and failure modes.

Check yourself

Can you give the estimate, error direction, units, and a reason the local line is trustworthy here?

Interactive checklinear-extra-01

Use the tangent to x\sqrt{x} at x=4x=4 to estimate 4.04\sqrt{4.04}.

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Show hint

L(x)=2+(x−4)/4L(x)=2+(x-4)/4.

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