Calculus I · 2B · lesson
Local Linearity
Learn local linearity with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Local linearity, differentials, and Newton's methodWhat this section is building
Learn local linearity with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Linearization, differentials, and Newton's method reuse one local line for estimation, uncertainty, or a better root guess.
Choose a nearby easy input, record the local slope, and state why the requested change is small enough for the model.
Presenting a tangent estimate as exact or running Newton iterations without checking residuals and failure modes.
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
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 and the target . Write , calculate the predicted change , and then add it to . Use concavity or a numerical comparison to understand the direction and size of the error.
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 is close to , then
Linearization
The linearization of at is
It is exactly the tangent-line function at .
Approximate a square root
Use linearization to estimate .
Worked solution
Write a real attempt before opening the supplied answer.
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.
Near , the graph of is nearly indistinguishable from its tangent line.
Near x=4, the graph of x is nearly indistinguishable from its tangent line.
The derivative as the best first-order approximation
The rigorous statement of local linearity is
The numerator is the approximation error. Dividing by 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.
linear-extra-01Use the tangent to at to estimate .
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