📏Linear Approximation & Differentials

Stand on a beach and the horizon is a straight line, even though you are standing on a sphere. Zoom in on any smooth curve, far enough, and the same thing happens: the bend runs out of room to show itself and what is left is a straight line. That line is the tangent, and the whole of this craft is the permission it gives you — near the point of contact you may work with the line instead of the curve, because the two are almost the same thing.

Written as a function, the tangent line at x = a is called the linearization of f at a:

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

Read it as an instruction rather than a formula. *Start at the height you know. Then walk along at the slope you know, for as far as you have moved.* Only two numbers are needed, and they are the two numbers calculus always hands you: the height f(a) and the slope f'(a).

Three things are worth noticing straight away.

• ◆At x = a the bracket collapses. (a - a) = 0, so L(a) = f(a) exactly. The line and the curve always agree at the anchor, and that is the check to run on any answer you write.
• ◆The bracket is (x - a), never x. The slope multiplies the *change* in x. If you write f'(a)·x instead, your line has the right steepness but starts in the wrong place.
• ◆a is chosen, not given by fate. You pick the anchor, and you pick it to be a value where f(a) is easy and which sits close to the value you actually want. Both conditions matter: 25 is a fine anchor for sqrt(26) and a poor one for sqrt(70).

The same idea travels under several names — linear approximation, local linearity, the tangent line approximation, first-order approximation. They are all L(x) = f(a) + f'(a)(x - a).

The classic use is a number your head cannot do: sqrt(26), (1.02)⁵, the cube root of 28, 1/4.1, sin(0.1), ln(1.1).

The recipe is always the same four moves.

1. Name the function. "Estimate sqrt(26)" means f(x) = sqrt(x). "Estimate (1.02)^5" means f(x) = x⁵. Choosing the function is a real decision: for (1.02)⁵ you could also take f(x) = (1 + x)⁵ and anchor at 0. Either works; write down which one you chose.

2. Choose the anchor. The nearest value where f is exact. For square roots that is the nearest perfect square, for cube roots the nearest perfect cube, for powers usually 1 or a small whole number, for sin, cos and e^x usually 0, and for ln usually 1.

3. Get the two numbers. f(a), and f'(a) from the derivative rules you already have.

4. Add the correction. estimate = f(a) + f'(a)(x - a). The second term is small, so if your answer is far from f(a) something has gone wrong.

Worked in miniature: sqrt(26). f(x) = sqrt(x), a = 25, f(25) = 5, f'(x) = 1/(2*sqrt(x)) so f'(25) = 1/10. The change is 26 - 25 = 1.

sqrt(26) is about 5 + (1/10)(1) = 5.1

The true value is 5.0990..., so the estimate is out by about one part in five thousand — from one multiplication.

The special ones worth memorising. Anchored at 0, three linearizations turn up everywhere in mathematics and physics:

sin(x) is about x (x in radians) e^x is about 1 + x ln(1 + x) is about x

Each is just L(x) = f(0) + f'(0)x with the numbers filled in, and each is the reason a physicist writes "for small angles, sin(x) = x" without blinking.

Why the anchor has to be close. The gap between a curve and its tangent grows like the SQUARE of the distance from the anchor. Move twice as far and the gap is roughly four times as big; move ten times as far and it is a hundred times as big. That is why 25 beats 36 as an anchor for sqrt(26), even though both give exact roots, and why no linear approximation is any good far from home.

The same idea, written in the language of small changes, becomes the differential. If x moves by a small amount dx, then the tangent line predicts that y moves by

dy = f'(x) dx

and that is all a differential is: a slope times a step. Nothing new has happened — dy is the rise of the tangent line, exactly the correction term from the linearization wearing shorter clothes.

Alongside it sits the true change, written delta y:

delta y = f(x + dx) - f(x)

dy and delta y are not the same number, and telling them apart is most of what this section is for. dy travels along the straight line; delta y travels along the curve. They start together and drift apart, and the drift is small when dx is small.

Worked in miniature: y = x² at x = 3, dx = 0.1.

dy = f'(3) dx = (6)(0.1) = 0.6 delta y = f(3.1) - f(3) = 9.61 - 9 = 0.61

The differential misses by 0.01, which for this function is exactly (dx)² — a hint of the general truth that the error of a linear approximation is second order in the step. Halve the step and the error quarters.

Which one you want depends on the question. If a problem says "estimate the change", "approximately how much", or hands you a rate and a small interval, it wants dy. If it says "find the change" or "how much did it actually change", it wants delta y, and you must evaluate f twice.

Signs matter and are easy to lose. dx can be negative — a step backwards in time, a cooling rather than a warming, a shrinking radius. Then dy is negative too if the slope is positive, and positive if the slope is negative. Work out the step as (new value) - (old value) every time and let the sign look after itself.

No measurement is exact. A radius measured as 10 mm is really 10 mm give or take some small dr, and every quantity computed from it inherits that uncertainty. Differentials are how the uncertainty is carried across — this is the single most practical use of the whole topic.

Absolute error. If V is computed from r, the possible error in V is

dV = (dV/dr) dr

with dr the measuring error. For a sphere, V = (4/3)*pi*r³ gives dV/dr = 4*pi*r², so dV = 4*pi*r² dr. Notice what that says: the error in the volume is the surface area times the error in the radius, which is exactly the thin shell you would paint on if the radius grew by dr.

Relative error divides by the quantity itself, and something lovely happens:

dV/V = (4*pi*r² dr) / ((4/3)*pi*r³) = 3 dr/r

Everything cancels except a number. The relative error in a volume is three times the relative error in the radius; in an area it is twice. The rule generalises: if Q depends on the power rⁿ, then dQ/Q = n dr/r. A 1% slip in a measured length becomes a 3% slip in a volume computed from it.

Percentage error is the relative error multiplied by 100, and nothing else. Keeping the two apart is worth a moment of care: 0.02 and 2% are the same number wearing different hats, and 2 is neither of them.

Running it backwards. "How accurately must the radius be measured if the volume is to be within 1.5%?" is the same equation solved the other way:

3 dr/r is at most 0.015, so dr is at most 0.005 r

For r = 20 mm that is 0.1 mm. Notice the tolerance is *tighter* than the target, by the same factor of 3 — the power that magnifies the error also shrinks the allowance.

An estimate is worth more when you know which way it leans, and the second derivative tells you at no extra cost.

• ◆f'' < 0 near a (concave down): the curve bends away downwards, the tangent line rides above it, and the estimate is an over-estimate.
• ◆f'' > 0 near a (concave up): the curve bends upwards, the tangent line runs underneath, and the estimate is an under-estimate.

So sqrt(26) is about 5.1, and because sqrt is concave down (f''(x) = -1/(4*x^3/2), always negative) we know 5.1 is a little too big before ever reaching for a calculator. And e⁰.05 is about 1.05, which is a little too small, because e^x is concave up everywhere.

Two edge cases complete the picture.

A straight line is its own linearization. If f(x) = 3x - 5 then f'' = 0 everywhere, the tangent line IS f, and the estimate is exact at every x. This is the only way a linear approximation is ever exact away from the anchor.

At an inflection point the test says nothing on its own. f(x) = x³ at a = 0 has f''(0) = 0, and f'' changes sign there: the curve crosses its tangent at the anchor. The estimate is too small for x > 0 and too big for x < 0. A zero second derivative is not a promise of exactness — it is a signal to look on each side separately.

Reading a tangent off a table or a graph. Often nobody hands you a formula: you are given a reading and a rate, or a graph with a tangent already drawn. It changes nothing. If T(3) = 21 degrees and T'(3) = 4 degrees per hour, then

T(3.5) is about 21 + (4)(0.5) = 23 degrees

and the same estimate run backwards gives T(2.8) about 21 + (4)(-0.2) = 20.2 degrees. Take the reading, take the rate, multiply the rate by the step, and mind the sign of the step.

The same trick even works on curves that are not functions. If x² + y² = 25 passes through (3, 4), implicit differentiation gives dy/dx = -x/y = -3/4 there, so at x = 3.1 the curve has y about 4 + (-3/4)(0.1) = 3.925. A point and a slope are all a tangent line ever needs.