4.2 Linear Approximations and Differentials

Linear Approximations and Differentials

Linear approximations and differentials let you estimate function values without computing them exactly. The core idea: zoom in close enough on any smooth curve, and it starts to look like a straight line. That straight line (the tangent line) becomes your estimation tool.

These techniques show up constantly in physics, engineering, and later math courses whenever exact answers are impractical and a close estimate will do.

Linear Approximation at a Point

If a function is differentiable at a point, its tangent line hugs the curve closely in a small neighborhood around that point. This means you can swap out the actual (possibly complicated) function for the much simpler tangent line and get a solid estimate, as long as you stay near the point of tangency.

The further you move from that point, the worse the approximation gets. This is strictly a local tool.

Construction of Function Linearization

The linearization of $f(x)$ at $x = a$ is just the equation of the tangent line at that point:

$L(x) = f(a) + f'(a)(x - a)$

Each piece has a clear role:

• $f(a)$: the known function value at the base point (your starting height)
• $f'(a)$: the slope of the tangent line (how fast $f$ is changing at $a$)
• $(x - a)$: how far your input is from the base point

How to use it (step by step):

1. Choose a base point $a$ where $f(a)$ and $f'(a)$ are easy to compute.

2. Calculate $f(a)$ and $f'(a)$.

3. Plug into $L(x) = f(a) + f'(a)(x - a)$.

4. Substitute the $x$-value you want to estimate.

Example: Estimate $\sin(0.1)$ using linearization at $a = 0$.

1. $f(x) = \sin(x)$, so $f(0) = \sin(0) = 0$

2. $f'(x) = \cos(x)$, so $f'(0) = \cos(0) = 1$

3. $L(x) = 0 + 1 \cdot (x - 0) = x$

4. $L(0.1) = 0.1$

The actual value is $\sin(0.1) \approx 0.0998$, so the estimate of $0.1$ is off by only about $0.0002$. That's the power of staying close to the base point.

Example: Estimate $\sqrt{4.1}$ using linearization at $a = 4$.

1. $f(x) = \sqrt{x}$, so $f(4) = 2$

2. $f'(x) = \frac{1}{2\sqrt{x}}$, so $f'(4) = \frac{1}{4}$

3. $L(x) = 2 + \frac{1}{4}(x - 4)$

4. $L(4.1) = 2 + \frac{1}{4}(0.1) = 2.025$

The actual value is $\sqrt{4.1} \approx 2.0248$, so the estimate is very close.

Graphical Representation of Differentials

Differentials give you a way to talk about small changes in input and output separately.

For $y = f(x)$, define:

• $dx$: a small change in $x$ (this is an independent quantity you choose)
• $dy$: the corresponding estimated change in $y$, given by $dy = f'(x) \, dx$

The key distinction to keep straight:

• $dy$ is the change along the tangent line (your estimate)
• $\Delta y = f(x + dx) - f(x)$ is the actual change along the curve

Graphically, if you move $dx$ units to the right from a point on the curve, $dy$ is the vertical distance traveled along the tangent line, while $\Delta y$ is the vertical distance traveled along the actual curve. The gap between $dy$ and $\Delta y$ is the approximation error, and it shrinks as $dx$ gets smaller.

Example: For $f(x) = x^2$ at $x = 3$ with $dx = 0.01$:

• $dy = f'(3) \cdot dx = 6(0.01) = 0.06$
• $\Delta y = f(3.01) - f(3) = 9.0601 - 9 = 0.0601$

The differential $dy = 0.06$ is a quick, close estimate of the actual change $0.0601$.

Error Analysis in Differential Approximations

Since differentials give estimates, you need ways to measure how good those estimates are.

• Absolute error = $|\Delta y - dy|$, the raw size of the mistake
• Relative error = $\frac{|\Delta y - dy|}{|\Delta y|}$, the error as a fraction of the true change
• Percentage error = relative error $\times$ 100%

A small percentage error means the linear approximation did its job well. What counts as "acceptable" depends on context: engineering tolerances differ from back-of-the-envelope physics estimates.

The general pattern: the smaller $dx$ is, the smaller the error. And functions with less curvature near the base point produce better approximations, because the tangent line stays closer to the curve.

Continuity and Differentiability

Linear approximation only works where the function is differentiable. Differentiability requires continuity, but goes further: the function must also have a well-defined derivative (no sharp corners, cusps, or vertical tangents).

If $f$ is differentiable at $a$, then $f$ can be approximated by its tangent line near $a$. If $f$ is not differentiable at $a$, there's no tangent line to use, and the whole method breaks down at that point.