Linear Approximation and Taylor Series

By the end you should be able to: Construct the linearisation of a function at a point, use it to approximate values, extend it to the Taylor series, and explain why accuracy falls away from the centre.

Near a point, a smooth curve looks like its tangent line. That observation, taken seriously, is one of the most useful tools in applied mathematics.

The linearisation

The tangent to $f$ at $x = a$ is

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

Used as an approximation, $L$ is the linearisation of $f$ at $a$ . For $x$ near $a$ :

f(x) \approx f(a) + f'(a)(x-a)

It matches $f$ in two respects at $x=a$ : same value, same slope. No line can do better.

A worked instance

Take $f(x) = x^3 + 2x + 5$ about $a = 5$ .

f(5) = 125 + 10 + 5 = 140, \qquad f'(x) = 3x^2+2, \qquad f'(5) = 77

L(x) = 140 + 77(x-5) = 77x - 245

How good is it? Compare against the exact values:

$x$	$f(x)$ exact	$L(x)$	Error
5	140	140	0
5.1	147.851	147.7	0.151
5.5	182.375	178.5	3.875
6	233	217	16

The pattern is the point: excellent nearby, degrading fast as you move away. Ten times the distance costs far more than ten times the error, because the leading error term grows with $(x-a)^2$ .

Taylor series: more terms, more agreement

A line matches value and slope. Why stop there? Match the second derivative too, and the third:

f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots

The linearisation is simply this series truncated after two terms.

The two you should know cold

About $a = 0$ (the Maclaurin case):

e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots

\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots

Note $\sin$ has only odd powers — it is an odd function, and the series inherits that symmetry. Similarly $\cos$ has only even powers.

Watching the terms earn their keep

Approximating $e^{0.1}$ , whose true value is $1.1051709\ldots$ :

Terms used	Approximation	Error
$1 + x$	1.1	$0.0051709$
$1 + x + x^2/2$	1.105	$0.0001709$

One extra term cuts the error by roughly thirty times — for $x = 0.1$ . The gain shrinks as $x$ grows, which is the same distance effect as before.

Where this is used

Physics: $\sin\theta \approx \theta$ for small angles — the pendulum equation is only solvable because of it
Computing: how libraries actually evaluate $e^x$ , $\sin x$ , $\ln x$
Optimisation: Newton's method is repeated linearisation
Error analysis: propagating small measurement errors through a formula

The honest caveat

An approximation without an error estimate is a guess with good manners. The tables above exist because "approximately" is not a claim until you say how approximately, and over what range.