Linear Approximations and Differentials

Zoom in on a point of a differentiable curve and the graph flattens until it is indistinguishable from its tangent line. A value $f (x)$ that is hard to compute can therefore be replaced, for $x$ near a convenient point $a$ , by the value of the tangent line, a linear expression. Written in the language of differentials, the same tangent-line estimate measures both a function value and the error that a measurement error propagates into a computed quantity.

The linear approximation

Near $(a, f (a))$ the tangent line stays close to the curve. Its equation is $y = f (a) + f^{'} (a) (x - a)$ , and using it in place of $f$ gives the linear approximation.

The linearization L(x) is the tangent at (a, f(a)); near a it hugs the curve, so f(x) is well approximated by L(x) = f(a) + f'(a)(x - a).

Example. Approximate $3.98$ and $4.05$ using $f (x) = x + 3$ at $a = 1$ . Here $f (1) = 2$ and $f^{'} (x) = \frac{1}{2} (x + 3)^{- 1/2}$ , so $f^{'} (1) = \frac{1}{4}$ . The linearization is

L (x) = 2 + \frac{1}{4} (x - 1) = \frac{7}{4} + \frac{x}{4}, so x + 3 \approx \frac{7}{4} + \frac{x}{4} (x near 1) .

Then $3.98 = 0.98 + 3 \approx \frac{7}{4} + \frac{0.98}{4} = 1.995$ and $4.05 \approx \frac{7}{4} + \frac{1.05}{4} = 2.0125$ . Both overestimate, because the square-root graph is concave down and lies below its tangent. The advantage over a calculator is that the approximation is a formula valid across a whole interval, not a single evaluation.

The approximation degrades as $x$ moves away from $a$ .

$x$	argument	$L (x)$	actual value
$3.9$	$0.9$	$1.975$	$1.97484 \dots$
$3.98$	$0.98$	$1.995$	$1.99499 \dots$
$4$	$1$	$2$	$2.00000 \dots$
$4.05$	$1.05$	$2.0125$	$2.01246 \dots$
$5$	$2$	$2.25$	$2.23607 \dots$
$6$	$3$	$2.5$	$2.44949 \dots$

Close to $a = 1$ the estimate is accurate to several decimals; by $x = 3$ it is off in the first. How wide an interval is acceptable depends on the tolerance: requiring $∣ x + 3 - L (x) ∣ < 0.5$ holds on roughly $- 2.6 < x < 8.6$ , while tightening to $0.1$ shrinks the interval to about $- 1.1 < x < 3.9$ .

The base point $a$ need not be $0$ ; it need only be a value where $f$ and $f^{'}$ are easy to evaluate and close to the target.

Small-angle approximations in physics

Linear approximations let a physicist replace an awkward function with its tangent line inside a derivation. The linearization of $sin x$ at $a = 0$ is $L (x) = x$ , since $sin 0 = 0$ and the slope there is $cos 0 = 1$ . Likewise $cos x$ linearizes to $1$ at $0$ . For $x$ near zero,

sin x \approx x, cos x \approx 1, tan x \approx x .

Small-angle approximation. Near x = 0 the line y = x is tangent to y = sin x; they agree to within about 2 percent out to roughly 0.35 radians.

The pendulum period is derived by replacing $sin θ$ with $θ$ in the equation of motion, valid because the swing angle is small; the two agree to within about $2%$ out to roughly $2 0^{\circ}$ . Paraxial (Gaussian) optics rests on the same two substitutions, which turn the lens equations linear and make lens design tractable.

Differentials

The same idea, written with differentials, separates the change predicted by the tangent line from the true change in the function.

Here $d y$ depends on both $x$ and $d x$ . Dividing (when $d x \neq = 0$ ) recovers $d y / d x = f^{'} (x)$ , and now the left side is a genuine ratio of the two differentials, not merely a symbol for the derivative.

The geometry distinguishes $d y$ from the actual change. Let $P = (x, f (x))$ and $Q = (x + Δ x, f (x + Δ x))$ , and set $d x = Δ x$ . Along the run $d x$ , the tangent line rises by $d y = f^{'} (x) d x$ , reaching a point $R$ , while the curve rises by the true increment $Δ y = f (x + Δ x) - f (x)$ , reaching $Q$ . So $d y$ is the change in the linearization and $Δ y$ is the change in the function.

Differentials against the true change. Over a run dx the tangent rises by dy = f'(x) dx to R, while the curve rises by the actual change to Q; their gap shrinks faster than dx.

Example. Compare $Δ y$ and $d y$ for $y = x^{3} + x^{2} - 2 x + 1$ as $x$ moves from $2$ . The differential is $d y = (3 x^{2} + 2 x - 2) d x$ , and at $x = 2$ this is $d y = 14 d x$ .

Step $d x = Δ x$	$Δ y$ (exact)	$d y = 14 d x$
$0.05$	$0.717625$	$0.7$
$0.01$	$0.140701$	$0.14$

The approximation $Δ y \approx d y$ improves as $d x$ shrinks, and $d y$ is far easier to compute. For complicated $f$ , computing $Δ y$ exactly may be impossible, and the differential is the only practical estimate. In this notation the linear approximation reads $f (a + d x) \approx f (a) + d y$ .

Estimating propagated error

Differentials estimate how a small error in a measured input propagates into a computed output. If a quantity $y = f (x)$ is computed from a measured $x$ whose error is $d x$ , the resulting error in $y$ is approximately $d y = f^{'} (x) d x$ .

Example. A sphere's radius is measured as $r = 21$ cm with possible error at most $d r = 0.05$ cm. The volume $V = \frac{4}{3} π r^{3}$ has differential

d V = 4 π r^{2} d r = 4 π (21)^{2} (0.05) \approx 277 cm^{3} .

The factor $4 π r^{2}$ is the surface area, so $d V$ is the volume of a thin shell of thickness $d r$ wrapped around the sphere: surface area times thickness.

Error propagation. A radius error dr thickens the sphere by a shell of volume 4 pi r^2 dr, the surface area times the thickness.

An absolute error of $277$ cm³ sounds large, but the relative error is more informative. Dividing by the volume,

\frac{d V}{V} = \frac{4 π r ^{2} d r}{\frac{4}{3} π r ^{3}} = 3 \frac{d r}{r},

so the relative error in the volume is about three times the relative error in the radius. Here $d r / r = 0.05/21 \approx 0.0024$ gives a volume relative error of about $0.007$ , i.e. $0.24%$ in the radius becomes $0.7%$ in the volume. The same reasoning shows a power law $y = k x^{n}$ multiplies relative error by $∣ n ∣$ .

Beyond the tangent line

The linearization is the best first-degree approximation: $L$ matches $f$ in value and first derivative at $a$ . A closer fit uses a parabola $P (x) = f (a) + f^{'} (a) (x - a) + \frac{1}{2} f^{''} (a) (x - a)^{2}$ matching value, slope, and concavity. Continuing to match higher derivatives produces the Taylor polynomials, whose limit represents the whole function as an infinite series. The tangent line is the degree-one member of that family, and the error it leaves is what the higher-degree terms correct.