4.3 Approximation of functions using differentials

Differential Approximation Fundamentals

Differential approximation lets you estimate how a function's value changes without computing it exactly. The core idea: if you know a function's value and its derivatives at a point, you can predict its value at nearby points using differentials. This is especially useful in multivariable calculus, where exact computation can get unwieldy fast.

Understanding Differentials and Approximations

For a function $y = f(x)$, the differential is defined as:

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

This approximates the actual change $\Delta y$ when $x$ changes by a small amount $dx$. The key relationship is:

$\Delta y \approx dy \quad \text{for small } dx$

The approximation works because $dy$ captures the linear (tangent-line) behavior of the function near the point. The smaller $dx$ is, the better the estimate.

For multivariable functions $z = f(x, y)$, the differential generalizes to:

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

This is the multivariable version you'll use most often in this course, and it connects directly to the tangent plane from earlier sections. The tangent plane is the linear approximation, and $dz$ gives you the approximate change in $z$ along that plane.

Higher-order differentials improve accuracy when $dx$ isn't tiny. The second-order differential adds a curvature correction:

$d^2y = f''(x)\,dx^2$

Including this term gives a quadratic approximation, which stays accurate over a wider range of $dx$ values.

Absolute and Relative Error

Once you use a differential to approximate a value, you need to know how good that approximation is. Two measures handle this:

• Absolute error: $|{\text{Actual}} - {\text{Approximation}}|$. This tells you the raw size of the error in the same units as the output. For example, if the actual value is 5.2 and your approximation gives 5.1, the absolute error is $|5.2 - 5.1| = 0.1$.
• Relative error: $\frac{|\text{Actual} - \text{Approximation}|}{|\text{Actual}|}$. This gives a scale-independent measure, often expressed as a percentage. A 0.1 error on a value of 100 (0.1% relative error) is far less concerning than a 0.1 error on a value of 0.5 (20% relative error).

Relative error is what you typically care about when deciding whether an approximation is "good enough" for a given application.

Error Analysis

Propagation of Errors

In practice, you rarely measure a quantity directly. Instead, you compute it from other measured quantities, each with its own uncertainty. Error propagation tells you how those input uncertainties combine to affect the output.

For a function $f(x, y)$ where $x$ and $y$ each have small errors $dx$ and $dy$, the propagated error in $f$ is approximated by the total differential:

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

Here, the partial derivatives act as sensitivity coefficients. A large $\frac{\partial f}{\partial x}$ means $f$ is very sensitive to errors in $x$, so even a small $dx$ can produce a significant error in $f$.

To find the worst-case (maximum) error, you take absolute values:

$|df| \leq \left|\frac{\partial f}{\partial x}\right||dx| + \left|\frac{\partial f}{\partial y}\right||dy|$

Example: Suppose you're computing the area of a rectangle, $A = xy$, where $x = 10 \pm 0.1$ cm and $y = 5 \pm 0.2$ cm. Then:

$dA = y\,dx + x\,dy = (5)(0.1) + (10)(0.2) = 0.5 + 2.0 = 2.5 \text{ cm}^2$

So the area is approximately $50 \pm 2.5$ cm$^2$. Notice that the error in $y$ contributes more because it gets multiplied by the larger dimension $x$.

Chain Rule for Differentials

When your function involves composition (a function of a function), the chain rule extends to differentials. For a composite function $f(g(x))$:

$df = f'(g(x)) \cdot dg$

where $dg = g'(x)\,dx$. This lets you track errors through multi-step calculations by chaining differentials at each stage.

For multivariable compositions, the same principle applies. If $z = f(u, v)$ where $u = u(x, y)$ and $v = v(x, y)$, you first find $du$ and $dv$ in terms of $dx$ and $dy$, then substitute into the expression for $dz$.

Applications

Differential approximation and error propagation show up constantly in applied work:

• Physics: Propagating measurement uncertainties through formulas (e.g., computing the uncertainty in a pendulum's period from uncertainties in length and timing measurements). Perturbation theory uses differentials to analyze how small parameter changes affect a system's behavior.
• Engineering: Tolerance analysis in manufacturing determines how variations in component dimensions affect the final assembly. If a part's radius has tolerance $\pm 0.01$ mm, the differential tells you how much the volume or surface area can vary.
• Numerical methods: Linear approximation underlies many computational algorithms. Understanding the differential error helps you choose step sizes and assess the reliability of numerical results.

The common thread across all these applications: differentials turn a hard "exact change" problem into a manageable linear estimate, and error analysis tells you when that estimate is trustworthy.