4.2 Linear approximations and differentials

Linear Approximation and Differentials

Linear approximation lets you estimate the value of a function near a known point without computing the exact answer. In single-variable calculus, you used tangent lines for this. In multivariable calculus, you use tangent planes instead, and the idea of differentials expands to account for changes in multiple inputs at once.

Tangent Plane Approximation

For a single-variable function, the linear approximation (or linearization) of $f(x)$ at a point $a$ is:

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

This is just the equation of the tangent line at $x = a$, used as a stand-in for the function nearby.

For a function of two variables $z = f(x, y)$, the tangent plane at $(a, b)$ plays the same role. The linear approximation becomes:

$L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$

This works well when $(x, y)$ is close to $(a, b)$ and $f$ is differentiable at that point. Differentiability is the key requirement: it guarantees the surface is smooth enough for the tangent plane to be a good local stand-in.

Differentials and Their Applications

The differential captures how much a function's output changes in response to small changes in its inputs.

For a single-variable function $y = f(x)$:

$dy = f'(x)\, dx$

Here $dx$ represents a small change in $x$, and $dy$ approximates the resulting change in $y$.

For a two-variable function $z = f(x, y)$, the total differential is:

$dz = \frac{\partial f}{\partial x}\, dx + \frac{\partial f}{\partial y}\, dy$

Each term accounts for the contribution of one variable's change to the overall change in $z$. This generalizes naturally to functions of three or more variables by adding one partial-derivative term per variable.

Example: Suppose $z = x^2 y$ and you want to estimate how $z$ changes when $(x, y)$ moves from $(2, 3)$ by $dx = 0.01$ and $dy = -0.02$.

1. Compute the partial derivatives: $f_x = 2xy = 12$ and $f_y = x^2 = 4$ at $(2, 3)$.

2. Apply the total differential: $dz = 12(0.01) + 4(-0.02) = 0.12 - 0.08 = 0.04$.

3. So $z$ increases by approximately $0.04$.

Error Estimation and Increments

Error Estimation

The error in a linear approximation is the difference between the true function value and the approximation:

$E(x) = f(x) - L(x)$

For single-variable functions, the magnitude of this error can be bounded using the second derivative. If $|f''(c)| \leq M$ for all $c$ between $x$ and $a$, then:

$|E(x)| \leq \frac{M}{2}(x - a)^2$

This tells you two things:

• The error shrinks quadratically as you get closer to $a$.
• Functions with large second derivatives (high curvature) produce less reliable linear approximations.

For multivariable functions, analogous bounds involve the second-order partial derivatives, but the single-variable bound captures the core idea.

Increments and Their Role

An increment $\Delta x$ is the actual change in a variable, and $\Delta y = f(x + \Delta x) - f(x)$ is the actual change in the function's output. The differential $dy$ is the approximate change predicted by the tangent line.

The key relationship:

$\Delta y \approx dy = f'(x)\, \Delta x$

This approximation improves as $\Delta x$ gets smaller. The difference between $\Delta y$ and $dy$ is exactly the error $E$. In practice, differentials are often easier to compute than exact increments, which is why they're so useful for quick estimates and error propagation.

Advanced Topics

Multivariable Taylor Series

The linear approximation is actually the first-order Taylor polynomial. The first-order Taylor polynomial of $f(x, y)$ about $(a, b)$ is:

$P_1(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$

This is identical to the tangent plane approximation from earlier. Higher-order Taylor polynomials improve accuracy by incorporating curvature information through second-order (and higher) partial derivatives. The second-order polynomial, for instance, adds terms involving $f_{xx}$, $f_{yy}$, and $f_{xy}$, which account for how the surface bends.

Applications and Limitations

Multivariable Taylor approximations show up throughout physics, engineering, and numerical methods wherever you need to replace a complicated function with a simpler polynomial near a point of interest.

A few things to keep in mind:

• Accuracy depends on distance. The farther you move from the expansion point, the worse the approximation gets.
• More terms help. Including higher-order terms improves accuracy near the expansion point, at the cost of more computation.
• Convergence isn't guaranteed. Taylor series don't converge for every function at every point. The region where the series converges depends on the specific function, and outside that region the approximation can be wildly wrong.