Approximating Values Using Local Linearity and Linearization
Introduced| English | Meaning |
|---|---|
| tangent line/ˈtændʒənt laɪn/ | A straight line through a point on a curve with the same gradient as the curve at that point. |
| local linearity/ˈləʊkl lɪˈnɪərɪti/ | The tendency of a differentiable function to be well approximated by its tangent line close to the point of tangency. |
| linearization/ˌlɪnɪəraɪˈzeɪʃn/ | Approximating a differentiable function near a point a by the tangent-line expression f(a) + f'(a)(x - a). |
Zoom in far enough and a curve looks straight
- Pick any smooth curve, zoom in on one point, and it flattens into (almost) a straight line.
- That line is the tangent line 切线, and near the point it hugs the curve tightly.
- So the tangent is a great approximation to the function nearby — this is local linearity 局部线性性.
- It lets you estimate hard function values with just a slope and a point.
The tangent hugs the curve
y = √x
Near the point of tangency the line and curve nearly coincide — that closeness is why $L(x)$ approximates $f(x)$.
The idea that a smooth curve looks straight up close is called local ____.
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Local linearity underlies the tangent-line approximation.
Build the tangent line
- The tangent at $x=a$ passes through $\big(a,f(a)\big)$ with slope $f'(a)$.
- Point-slope form gives the linearization 线性化:
-
$$L(x)=f(a)+f'(a)(x-a)$$
- This $L(x)$ is just the tangent line, renamed as an approximation formula.
Near x = a the tangent line L(x) approximates f(x)
The linearization of $f$ at $a$ is $L(x)=$
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Point-slope of the tangent: $f(a)+f'(a)(x-a)$.
Approximate a nearby value
- To estimate $f(x)$ for $x$ near $a$, compute $L(x)$ instead — it's usually easy arithmetic.
- Choose $a$ to be a nearby point where $f(a)$ and $f'(a)$ are simple (a "nice" number).
- The closer $x$ is to $a$, the better the approximation.
- Example: to estimate $\sqrt{4.1}$, use $f(x)=\sqrt x$ at $a=4$, where $f(4)=2$ is clean.
Using $L(x)=2+\tfrac14(x-4)$, estimate $\sqrt{4.1}=L(4.1)$.
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$2+\tfrac14(0.1)=2.025$.
Over- or underestimate? Ask concavity
- The tangent line lies on one side of a curved graph, so $L(x)$ leans one way.
- Concave up (curve bends upward, $f''>0$): the tangent is below the curve → $L$ is an underestimate.
- Concave down ($f''<0$): the tangent is above the curve → $L$ is an overestimate.
- Checking the sign of $f''$ tells you which way your approximation errs.
If $f$ is concave down near $a$, the tangent-line estimate $L(x)$ is an...
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Concave down → tangent above the curve → overestimate.
A linearization is most accurate for inputs far from the point of tangency.
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It is most accurate near the point; accuracy drops as you move away.
Select all correct over/under conclusions.
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Up → tangent below → under; down → tangent above → over; both set by $f''$.
Linearization is only accurate near the point of tangency — far from $a$ the straight line drifts away from the curve. And the over/under call depends on concavity: concave up → tangent below → underestimate; concave down → overestimate. Don't guess the direction; check the sign of $f''$.
Estimate $\sqrt{4.1}$ using linearization at $a=4$.
- $f(x)=\sqrt x=x^{1/2}$, so $f'(x)=\tfrac{1}{2\sqrt x}$; $\;f(4)=2$, $\;f'(4)=\tfrac14$.
- $L(x)=2+\tfrac14(x-4)$, so $L(4.1)=2+\tfrac14(0.1)=2.025$.
- Since $\sqrt x$ is concave down ($f''<0$), this is a slight overestimate (true value $\approx2.0248$).
Local linearity lets the tangent line approximate a function near a point. The linearization is $L(x)=f(a)+f'(a)(x-a)$; use it to estimate $f(x)$ for $x$ near $a$. Concavity sets the direction of error: concave up → underestimate, concave down → overestimate.