Quadratic Approximation Cheatsheet

1. Definition

When $x$ is close to some reference value $x_0$ (in other words $\Delta x \approx 0$), you can approximate $f(x)$ as the parabola that matches the curve at that reference value.

form	formula	analog
point–slope form	$f(x) \approx f(x_0) + f'(x_0)(x - x_0) + \frac{f''(x_0)}{2}(x - x_0)^2$	$y = a + b(x - x_0) + c(x - x_0)^2$
slope–intercept form	$f(x_0 + \Delta x) \approx f(x_0) + f'(x_0)\,\Delta x + \frac{f''(x_0)}{2}\Delta x^2$	$y = a + b\,\Delta x + c\,\Delta x^2$

A line can only match the height and the slope. It stays straight while the curve bends, so it falls behind the moment you leave the reference point. The $x^2$ term lets your guess bend too, and $f''(x_0)$ says how hard.

The teal line below is the linear approximation from the linear cheatsheet; the violet curve is the quadratic one. Drag $\Delta x$ out and watch the violet curve stay with the grey one for much longer.

Set $f''(x_0) = 0$ and the whole $x^2$ term disappears, leaving exactly the linear approximation. So quadratic approximation is never worse than linear — at worst the curvature is zero and the two agree.

2. Cheatsheet

Each entry gives you the general formula, then lets you pick the reference point. Every table compares the linear approximation $\tilde f_1$ against the quadratic one $\tilde f_2$, with both errors side by side so you can see how much the extra term buys you.

Watch the error columns as $\Delta x$ shrinks. The linear error falls like $\Delta x^2$; the quadratic error falls like $\Delta x^3$. At $\Delta x = 1/100$ that is already a difference of roughly a hundredfold.

2.1 Trigonometric

2.1.1$\sin x$

$\sin x$

$\approx$

$\sin x_0 + \cos x_0\left(x - x_0\right) - \frac{\sin x_0}{2}\left(x - x_0\right)^2$

reference$0$$\frac{\pi}{6}$$\frac{\pi}{4}$$\frac{\pi}{2}$

at $0$

$\sin x$

$\approx$

$\sin 0 + \cos 0x - \frac{\sin 0}{2}x^2$

$\approx$

$x$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$0.000000$	$0.000000$	$0.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$0.010000$	$0.010000$	$0.010000$	$1.67 \times 10^{-7}$	$1.67 \times 10^{-7}$
$0 + \frac{1}{10}$	$0.099833$	$0.100000$	$0.100000$	$0.000167$	$0.000167$
$0 + \frac{1}{2}$	$0.479426$	$0.500000$	$0.500000$	$0.020574$	$0.020574$
$0 + 1$	$0.841471$	$1.000000$	$1.000000$	$0.158529$	$0.158529$

$\sin x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{6}$

$\sin x$

$\approx$

$\sin \frac{\pi}{6} + \cos \frac{\pi}{6}\left(x - \frac{\pi}{6}\right) - \frac{\sin \frac{\pi}{6}}{2}\left(x - \frac{\pi}{6}\right)^2$

$\approx$

$\frac{1}{2} + \frac{\sqrt{3}}{2}\left(x - \frac{\pi}{6}\right) - \frac{1}{4}\left(x - \frac{\pi}{6}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{6} + 0$	$0.500000$	$0.500000$	$0.500000$	$0.000000$	$0.000000$
$\frac{\pi}{6} + \frac{1}{100}$	$0.508635$	$0.508660$	$0.508635$	$2.51 \times 10^{-5}$	$1.44 \times 10^{-7}$
$\frac{\pi}{6} + \frac{1}{10}$	$0.583960$	$0.586603$	$0.584103$	$0.002642$	$0.000142$
$\frac{\pi}{6} + \frac{1}{2}$	$0.853986$	$0.933013$	$0.870513$	$0.079027$	$0.016527$
$\frac{\pi}{6} + 1$	$0.998886$	$1.366025$	$1.116025$	$0.367139$	$0.117139$

$\sin x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{4}$

$\sin x$

$\approx$

$\sin \frac{\pi}{4} + \cos \frac{\pi}{4}\left(x - \frac{\pi}{4}\right) - \frac{\sin \frac{\pi}{4}}{2}\left(x - \frac{\pi}{4}\right)^2$

$\approx$

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}\left(x - \frac{\pi}{4}\right) - \frac{\sqrt{2}}{4}\left(x - \frac{\pi}{4}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{4} + 0$	$0.707107$	$0.707107$	$0.707107$	$0.000000$	$0.000000$
$\frac{\pi}{4} + \frac{1}{100}$	$0.714142$	$0.714178$	$0.714142$	$3.55 \times 10^{-5}$	$1.18 \times 10^{-7}$
$\frac{\pi}{4} + \frac{1}{10}$	$0.774167$	$0.777817$	$0.774282$	$0.003650$	$0.000115$
$\frac{\pi}{4} + \frac{1}{2}$	$0.959550$	$1.060660$	$0.972272$	$0.101111$	$0.012722$
$\frac{\pi}{4} + 1$	$0.977061$	$1.414214$	$1.060660$	$0.437152$	$0.083599$

$\sin x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{2}$

$\sin x$

$\approx$

$\sin \frac{\pi}{2} + \cos \frac{\pi}{2}\left(x - \frac{\pi}{2}\right) - \frac{\sin \frac{\pi}{2}}{2}\left(x - \frac{\pi}{2}\right)^2$

$\approx$

$1 - \frac{1}{2}\left(x - \frac{\pi}{2}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{2} + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$\frac{\pi}{2} + \frac{1}{100}$	$0.999950$	$1.000000$	$0.999950$	$5.00 \times 10^{-5}$	$4.17 \times 10^{-10}$
$\frac{\pi}{2} + \frac{1}{10}$	$0.995004$	$1.000000$	$0.995000$	$0.004996$	$4.17 \times 10^{-6}$
$\frac{\pi}{2} + \frac{1}{2}$	$0.877583$	$1.000000$	$0.875000$	$0.122417$	$0.002583$
$\frac{\pi}{2} + 1$	$0.540302$	$1.000000$	$0.500000$	$0.459698$	$0.040302$

$\sin x$$\tilde f_1$$\tilde f_2$

At 0 the quadratic term vanishes, so it matches the linear one. At π/2 it is the other way round: the linear term dies and the curvature carries everything.

2.1.2$\cos x$

$\cos x$

$\approx$

$\cos x_0 - \sin x_0\left(x - x_0\right) - \frac{\cos x_0}{2}\left(x - x_0\right)^2$

reference$0$$\frac{\pi}{6}$$\frac{\pi}{3}$$\frac{\pi}{2}$

at $0$

$\cos x$

$\approx$

$\cos 0 - \sin 0x - \frac{\cos 0}{2}x^2$

$\approx$

$1 - \frac{1}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$0.999950$	$1.000000$	$0.999950$	$5.00 \times 10^{-5}$	$4.17 \times 10^{-10}$
$0 + \frac{1}{10}$	$0.995004$	$1.000000$	$0.995000$	$0.004996$	$4.17 \times 10^{-6}$
$0 + \frac{1}{2}$	$0.877583$	$1.000000$	$0.875000$	$0.122417$	$0.002583$
$0 + 1$	$0.540302$	$1.000000$	$0.500000$	$0.459698$	$0.040302$

$\cos x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{6}$

$\cos x$

$\approx$

$\cos \frac{\pi}{6} - \sin \frac{\pi}{6}\left(x - \frac{\pi}{6}\right) - \frac{\cos \frac{\pi}{6}}{2}\left(x - \frac{\pi}{6}\right)^2$

$\approx$

$\frac{\sqrt{3}}{2} - \frac{1}{2}\left(x - \frac{\pi}{6}\right) - \frac{\sqrt{3}}{4}\left(x - \frac{\pi}{6}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{6} + 0$	$0.866025$	$0.866025$	$0.866025$	$0.000000$	$0.000000$
$\frac{\pi}{6} + \frac{1}{100}$	$0.860982$	$0.861025$	$0.860982$	$4.32 \times 10^{-5}$	$8.37 \times 10^{-8}$
$\frac{\pi}{6} + \frac{1}{10}$	$0.811782$	$0.816025$	$0.811695$	$0.004243$	$8.69 \times 10^{-5}$
$\frac{\pi}{6} + \frac{1}{2}$	$0.520296$	$0.616025$	$0.507772$	$0.095729$	$0.012524$
$\frac{\pi}{6} + 1$	$0.047180$	$0.366025$	$-0.066987$	$0.318845$	$0.114167$

$\cos x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{3}$

$\cos x$

$\approx$

$\cos \frac{\pi}{3} - \sin \frac{\pi}{3}\left(x - \frac{\pi}{3}\right) - \frac{\cos \frac{\pi}{3}}{2}\left(x - \frac{\pi}{3}\right)^2$

$\approx$

$\frac{1}{2} - \frac{\sqrt{3}}{2}\left(x - \frac{\pi}{3}\right) - \frac{1}{4}\left(x - \frac{\pi}{3}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{3} + 0$	$0.500000$	$0.500000$	$0.500000$	$0.000000$	$0.000000$
$\frac{\pi}{3} + \frac{1}{100}$	$0.491315$	$0.491340$	$0.491315$	$2.49 \times 10^{-5}$	$1.45 \times 10^{-7}$
$\frac{\pi}{3} + \frac{1}{10}$	$0.411044$	$0.413397$	$0.410897$	$0.002354$	$0.000146$
$\frac{\pi}{3} + \frac{1}{2}$	$0.023597$	$0.066987$	$0.004487$	$0.043391$	$0.019109$
$\frac{\pi}{3} + 1$	$-0.458584$	$-0.366025$	$-0.616025$	$0.092559$	$0.157441$

$\cos x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{2}$

$\cos x$

$\approx$

$\cos \frac{\pi}{2} - \sin \frac{\pi}{2}\left(x - \frac{\pi}{2}\right) - \frac{\cos \frac{\pi}{2}}{2}\left(x - \frac{\pi}{2}\right)^2$

$\approx$

$-\left(x - \frac{\pi}{2}\right)$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{2} + 0$	$6.12 \times 10^{-17}$	$6.12 \times 10^{-17}$	$6.12 \times 10^{-17}$	$0.000000$	$0.000000$
$\frac{\pi}{2} + \frac{1}{100}$	$-0.010000$	$-0.010000$	$-0.010000$	$1.67 \times 10^{-7}$	$1.67 \times 10^{-7}$
$\frac{\pi}{2} + \frac{1}{10}$	$-0.099833$	$-0.100000$	$-0.100000$	$0.000167$	$0.000167$
$\frac{\pi}{2} + \frac{1}{2}$	$-0.479426$	$-0.500000$	$-0.500000$	$0.020574$	$0.020574$
$\frac{\pi}{2} + 1$	$-0.841471$	$-1.000000$	$-1.000000$	$0.158529$	$0.158529$

$\cos x$$\tilde f_1$$\tilde f_2$

The famous one: near 0, cos x ≈ 1 − x²/2. At π/2 the curvature vanishes instead, so there the quadratic adds nothing.

2.1.3$\tan x$

$\tan x$

$\approx$

$\tan x_0 + \sec^2 x_0\left(x - x_0\right) + \sec^2 x_0 \tan x_0\left(x - x_0\right)^2$

reference$0$$\frac{\pi}{6}$$\frac{\pi}{4}$

at $0$

$\tan x$

$\approx$

$\tan 0 + \sec^2 0x + \sec^2 0 \tan 0x^2$

$\approx$

$x$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$0.000000$	$0.000000$	$0.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$0.010000$	$0.010000$	$0.010000$	$3.33 \times 10^{-7}$	$3.33 \times 10^{-7}$
$0 + \frac{1}{10}$	$0.100335$	$0.100000$	$0.100000$	$0.000335$	$0.000335$
$0 + \frac{1}{2}$	$0.546302$	$0.500000$	$0.500000$	$0.046302$	$0.046302$
$0 + 1$	$1.557408$	$1.000000$	$1.000000$	$0.557408$	$0.557408$

$\tan x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{6}$

$\tan x$

$\approx$

$\tan \frac{\pi}{6} + \sec^2 \frac{\pi}{6}\left(x - \frac{\pi}{6}\right) + \sec^2 \frac{\pi}{6} \tan \frac{\pi}{6}\left(x - \frac{\pi}{6}\right)^2$

$\approx$

$\frac{1}{\sqrt{3}} + \frac{4}{3}\left(x - \frac{\pi}{6}\right) + \frac{4\sqrt{3}}{9}\left(x - \frac{\pi}{6}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{6} + 0$	$0.577350$	$0.577350$	$0.577350$	$0.000000$	$0.000000$
$\frac{\pi}{6} + \frac{1}{100}$	$0.590761$	$0.590684$	$0.590761$	$7.79 \times 10^{-5}$	$8.97 \times 10^{-7}$
$\frac{\pi}{6} + \frac{1}{10}$	$0.719356$	$0.710684$	$0.718382$	$0.008672$	$0.000974$
$\frac{\pi}{6} + \frac{1}{2}$	$1.641346$	$1.244017$	$1.436467$	$0.397329$	$0.204879$
$\frac{\pi}{6} + 1$	$21.171805$	$1.910684$	$2.680484$	$19.261121$	$18.491321$

$\tan x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{4}$

$\tan x$

$\approx$

$\tan \frac{\pi}{4} + \sec^2 \frac{\pi}{4}\left(x - \frac{\pi}{4}\right) + \sec^2 \frac{\pi}{4} \tan \frac{\pi}{4}\left(x - \frac{\pi}{4}\right)^2$

$\approx$

$1 + 2\left(x - \frac{\pi}{4}\right) + 2\left(x - \frac{\pi}{4}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{4} + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$\frac{\pi}{4} + \frac{1}{100}$	$1.020203$	$1.020000$	$1.020200$	$0.000203$	$2.70 \times 10^{-6}$
$\frac{\pi}{4} + \frac{1}{10}$	$1.223049$	$1.200000$	$1.220000$	$0.023049$	$0.003049$
$\frac{\pi}{4} + \frac{1}{2}$	$3.408223$	$2.000000$	$2.500000$	$1.408223$	$0.908223$
$\frac{\pi}{4} + 1$	$-4.588038$	$3.000000$	$5.000000$	$7.588038$	$9.588038$

$\tan x$$\tilde f_1$$\tilde f_2$

tan is odd, so at 0 the curvature is zero and the quadratic collapses back to x.

2.1.4$\sec x$

$\sec x$

$\approx$

$\sec x_0 + \sec x_0 \tan x_0\left(x - x_0\right) + \frac{\sec x_0\tan^2 x_0 + \sec^3 x_0}{2}\left(x - x_0\right)^2$

reference$0$$\frac{\pi}{6}$$\frac{\pi}{4}$

at $0$

$\sec x$

$\approx$

$\sec 0 + \sec 0 \tan 0x + \frac{\sec 0\tan^2 0 + \sec^3 0}{2}x^2$

$\approx$

$1 + \frac{1}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.000050$	$1.000000$	$1.000050$	$5.00 \times 10^{-5}$	$2.08 \times 10^{-9}$
$0 + \frac{1}{10}$	$1.005021$	$1.000000$	$1.005000$	$0.005021$	$2.09 \times 10^{-5}$
$0 + \frac{1}{2}$	$1.139494$	$1.000000$	$1.125000$	$0.139494$	$0.014494$
$0 + 1$	$1.850816$	$1.000000$	$1.500000$	$0.850816$	$0.350816$

$\sec x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{6}$

$\sec x$

$\approx$

$\sec \frac{\pi}{6} + \sec \frac{\pi}{6} \tan \frac{\pi}{6}\left(x - \frac{\pi}{6}\right) + \frac{\sec \frac{\pi}{6}\tan^2 \frac{\pi}{6} + \sec^3 \frac{\pi}{6}}{2}\left(x - \frac{\pi}{6}\right)^2$

$\approx$

$\frac{2}{\sqrt{3}} + \frac{2}{3}\left(x - \frac{\pi}{6}\right) + \frac{5\sqrt{3}}{9}\left(x - \frac{\pi}{6}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{6} + 0$	$1.154701$	$1.154701$	$1.154701$	$0.000000$	$0.000000$
$\frac{\pi}{6} + \frac{1}{100}$	$1.161464$	$1.161367$	$1.161463$	$9.70 \times 10^{-5}$	$7.86 \times 10^{-7}$
$\frac{\pi}{6} + \frac{1}{10}$	$1.231858$	$1.221367$	$1.230990$	$0.010490$	$0.000868$
$\frac{\pi}{6} + \frac{1}{2}$	$1.921983$	$1.488034$	$1.728596$	$0.433949$	$0.193386$
$\frac{\pi}{6} + 1$	$21.195408$	$1.821367$	$2.783618$	$19.374041$	$18.411791$

$\sec x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{4}$

$\sec x$

$\approx$

$\sec \frac{\pi}{4} + \sec \frac{\pi}{4} \tan \frac{\pi}{4}\left(x - \frac{\pi}{4}\right) + \frac{\sec \frac{\pi}{4}\tan^2 \frac{\pi}{4} + \sec^3 \frac{\pi}{4}}{2}\left(x - \frac{\pi}{4}\right)^2$

$\approx$

$\sqrt{2} + \sqrt{2}\left(x - \frac{\pi}{4}\right) + \frac{3\sqrt{2}}{2}\left(x - \frac{\pi}{4}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{4} + 0$	$1.414214$	$1.414214$	$1.414214$	$0.000000$	$0.000000$
$\frac{\pi}{4} + \frac{1}{100}$	$1.428570$	$1.428356$	$1.428568$	$0.000215$	$2.63 \times 10^{-6}$
$\frac{\pi}{4} + \frac{1}{10}$	$1.579825$	$1.555635$	$1.576848$	$0.024191$	$0.002977$
$\frac{\pi}{4} + \frac{1}{2}$	$3.551899$	$2.121320$	$2.651650$	$1.430579$	$0.900249$
$\frac{\pi}{4} + 1$	$-4.695752$	$2.828427$	$4.949747$	$7.524180$	$9.645500$

$\sec x$$\tilde f_1$$\tilde f_2$

Mirror of cos: flat at 0, but curving upward instead of down, so the x² term flips sign.

2.1.5$\cot x$

$\cot x$

$\approx$

$\cot x_0 - \csc^2 x_0\left(x - x_0\right) + \csc^2 x_0 \cot x_0\left(x - x_0\right)^2$

reference$\frac{\pi}{4}$$\frac{\pi}{2}$

at $\frac{\pi}{4}$

$\cot x$

$\approx$

$\cot \frac{\pi}{4} - \csc^2 \frac{\pi}{4}\left(x - \frac{\pi}{4}\right) + \csc^2 \frac{\pi}{4} \cot \frac{\pi}{4}\left(x - \frac{\pi}{4}\right)^2$

$\approx$

$1 - 2\left(x - \frac{\pi}{4}\right) + 2\left(x - \frac{\pi}{4}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{4} + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$\frac{\pi}{4} + \frac{1}{100}$	$0.980197$	$0.980000$	$0.980200$	$0.000197$	$2.63 \times 10^{-6}$
$\frac{\pi}{4} + \frac{1}{10}$	$0.817629$	$0.800000$	$0.820000$	$0.017629$	$0.002371$
$\frac{\pi}{4} + \frac{1}{2}$	$0.293408$	$0.000000$	$0.500000$	$0.293408$	$0.206592$
$\frac{\pi}{4} + 1$	$-0.217958$	$-1.000000$	$1.000000$	$0.782042$	$1.217958$

$\cot x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{2}$

$\cot x$

$\approx$

$\cot \frac{\pi}{2} - \csc^2 \frac{\pi}{2}\left(x - \frac{\pi}{2}\right) + \csc^2 \frac{\pi}{2} \cot \frac{\pi}{2}\left(x - \frac{\pi}{2}\right)^2$

$\approx$

$-\left(x - \frac{\pi}{2}\right)$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{2} + 0$	$6.12 \times 10^{-17}$	$6.12 \times 10^{-17}$	$6.12 \times 10^{-17}$	$0.000000$	$0.000000$
$\frac{\pi}{2} + \frac{1}{100}$	$-0.010000$	$-0.010000$	$-0.010000$	$3.33 \times 10^{-7}$	$3.33 \times 10^{-7}$
$\frac{\pi}{2} + \frac{1}{10}$	$-0.100335$	$-0.100000$	$-0.100000$	$0.000335$	$0.000335$
$\frac{\pi}{2} + \frac{1}{2}$	$-0.546302$	$-0.500000$	$-0.500000$	$0.046302$	$0.046302$
$\frac{\pi}{2} + 1$	$-1.557408$	$-1.000000$	$-1.000000$	$0.557408$	$0.557408$

$\cot x$$\tilde f_1$$\tilde f_2$

Still nothing at 0 — cot blows up there. At π/2 it has an inflection, so the quadratic term is zero.

2.1.6$\csc x$

$\csc x$

$\approx$

$\csc x_0 - \csc x_0 \cot x_0\left(x - x_0\right) + \frac{\csc x_0\cot^2 x_0 + \csc^3 x_0}{2}\left(x - x_0\right)^2$

reference$\frac{\pi}{4}$$\frac{\pi}{2}$

at $\frac{\pi}{4}$

$\csc x$

$\approx$

$\csc \frac{\pi}{4} - \csc \frac{\pi}{4} \cot \frac{\pi}{4}\left(x - \frac{\pi}{4}\right) + \frac{\csc \frac{\pi}{4}\cot^2 \frac{\pi}{4} + \csc^3 \frac{\pi}{4}}{2}\left(x - \frac{\pi}{4}\right)^2$

$\approx$

$\sqrt{2} - \sqrt{2}\left(x - \frac{\pi}{4}\right) + \frac{3\sqrt{2}}{2}\left(x - \frac{\pi}{4}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{4} + 0$	$1.414214$	$1.414214$	$1.414214$	$0.000000$	$0.000000$
$\frac{\pi}{4} + \frac{1}{100}$	$1.400281$	$1.400071$	$1.400284$	$0.000210$	$2.56 \times 10^{-6}$
$\frac{\pi}{4} + \frac{1}{10}$	$1.291711$	$1.272792$	$1.294005$	$0.018919$	$0.002295$
$\frac{\pi}{4} + \frac{1}{2}$	$1.042156$	$0.707107$	$1.237437$	$0.335049$	$0.195281$
$\frac{\pi}{4} + 1$	$1.023477$	$-2.22 \times 10^{-16}$	$2.121320$	$1.023477$	$1.097843$

$\csc x$$\tilde f_1$$\tilde f_2$

at $\frac{\pi}{2}$

$\csc x$

$\approx$

$\csc \frac{\pi}{2} - \csc \frac{\pi}{2} \cot \frac{\pi}{2}\left(x - \frac{\pi}{2}\right) + \frac{\csc \frac{\pi}{2}\cot^2 \frac{\pi}{2} + \csc^3 \frac{\pi}{2}}{2}\left(x - \frac{\pi}{2}\right)^2$

$\approx$

$1 + \frac{1}{2}\left(x - \frac{\pi}{2}\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$\frac{\pi}{2} + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$\frac{\pi}{2} + \frac{1}{100}$	$1.000050$	$1.000000$	$1.000050$	$5.00 \times 10^{-5}$	$2.08 \times 10^{-9}$
$\frac{\pi}{2} + \frac{1}{10}$	$1.005021$	$1.000000$	$1.005000$	$0.005021$	$2.09 \times 10^{-5}$
$\frac{\pi}{2} + \frac{1}{2}$	$1.139494$	$1.000000$	$1.125000$	$0.139494$	$0.014494$
$\frac{\pi}{2} + 1$	$1.850816$	$1.000000$	$1.500000$	$0.850816$	$0.350816$

$\csc x$$\tilde f_1$$\tilde f_2$

At π/2 csc bottoms out, so the linear part is flat and the whole shape comes from the x² term.

2.2 Exponents and logarithms

2.2.1$e^x$

$e^x$

$\approx$

$e^{x_0} + e^{x_0}\left(x - x_0\right) + \frac{e^{x_0}}{2}\left(x - x_0\right)^2$

reference$0$$1$$2$

at $0$

$e^x$

$\approx$

$e^{0} + e^{0}x + \frac{e^{0}}{2}x^2$

$\approx$

$1 + x + \frac{1}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.010050$	$1.010000$	$1.010050$	$5.02 \times 10^{-5}$	$1.67 \times 10^{-7}$
$0 + \frac{1}{10}$	$1.105171$	$1.100000$	$1.105000$	$0.005171$	$0.000171$
$0 + \frac{1}{2}$	$1.648721$	$1.500000$	$1.625000$	$0.148721$	$0.023721$
$0 + 1$	$2.718282$	$2.000000$	$2.500000$	$0.718282$	$0.218282$

$e^x$$\tilde f_1$$\tilde f_2$

at $1$

$e^x$

$\approx$

$e^{1} + e^{1}\left(x - 1\right) + \frac{e^{1}}{2}\left(x - 1\right)^2$

$\approx$

$e + e\left(x - 1\right) + \frac{e}{2}\left(x - 1\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$1 + 0$	$2.718282$	$2.718282$	$2.718282$	$0.000000$	$0.000000$
$1 + \frac{1}{100}$	$2.745601$	$2.745465$	$2.745601$	$0.000136$	$4.54 \times 10^{-7}$
$1 + \frac{1}{10}$	$3.004166$	$2.990110$	$3.003701$	$0.014056$	$0.000465$
$1 + \frac{1}{2}$	$4.481689$	$4.077423$	$4.417208$	$0.404266$	$0.064481$
$1 + 1$	$7.389056$	$5.436564$	$6.795705$	$1.952492$	$0.593352$

$e^x$$\tilde f_1$$\tilde f_2$

at $2$

$e^x$

$\approx$

$e^{2} + e^{2}\left(x - 2\right) + \frac{e^{2}}{2}\left(x - 2\right)^2$

$\approx$

$e^2 + e^2\left(x - 2\right) + \frac{e^2}{2}\left(x - 2\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$2 + 0$	$7.389056$	$7.389056$	$7.389056$	$0.000000$	$0.000000$
$2 + \frac{1}{100}$	$7.463317$	$7.462947$	$7.463316$	$0.000371$	$1.23 \times 10^{-6}$
$2 + \frac{1}{10}$	$8.166170$	$8.127962$	$8.164907$	$0.038208$	$0.001263$
$2 + \frac{1}{2}$	$12.182494$	$11.083584$	$12.007216$	$1.098910$	$0.175278$
$2 + 1$	$20.085537$	$14.778112$	$18.472640$	$5.307425$	$1.612897$

$e^x$$\tilde f_1$$\tilde f_2$

Every derivative is the same, so all three coefficients share a factor of e^{x₀}.

2.2.2$\ln(1+x)$

$\ln(1+x)$

$\approx$

$\ln(1+x_0) + \dfrac{1}{1+x_0}\left(x - x_0\right) - \dfrac{1}{2(1+x_0)^2}\left(x - x_0\right)^2$

reference$0$$1$$2$

at $0$

$\ln(1+x)$

$\approx$

$\ln(1+0) + \dfrac{1}{1+0}x - \dfrac{1}{2(1+0)^2}x^2$

$\approx$

$x - \frac{1}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$0.000000$	$0.000000$	$0.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$0.009950$	$0.010000$	$0.009950$	$4.97 \times 10^{-5}$	$3.31 \times 10^{-7}$
$0 + \frac{1}{10}$	$0.095310$	$0.100000$	$0.095000$	$0.004690$	$0.000310$
$0 + \frac{1}{2}$	$0.405465$	$0.500000$	$0.375000$	$0.094535$	$0.030465$
$0 + 1$	$0.693147$	$1.000000$	$0.500000$	$0.306853$	$0.193147$

$\ln(1+x)$$\tilde f_1$$\tilde f_2$

at $1$

$\ln(1+x)$

$\approx$

$\ln(1+1) + \dfrac{1}{1+1}\left(x - 1\right) - \dfrac{1}{2(1+1)^2}\left(x - 1\right)^2$

$\approx$

$\ln 2 + \frac{1}{2}\left(x - 1\right) - \frac{1}{8}\left(x - 1\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$1 + 0$	$0.693147$	$0.693147$	$0.693147$	$0.000000$	$0.000000$
$1 + \frac{1}{100}$	$0.698135$	$0.698147$	$0.698135$	$1.25 \times 10^{-5}$	$4.15 \times 10^{-8}$
$1 + \frac{1}{10}$	$0.741937$	$0.743147$	$0.741897$	$0.001210$	$4.02 \times 10^{-5}$
$1 + \frac{1}{2}$	$0.916291$	$0.943147$	$0.911897$	$0.026856$	$0.004394$
$1 + 1$	$1.098612$	$1.193147$	$1.068147$	$0.094535$	$0.030465$

$\ln(1+x)$$\tilde f_1$$\tilde f_2$

at $2$

$\ln(1+x)$

$\approx$

$\ln(1+2) + \dfrac{1}{1+2}\left(x - 2\right) - \dfrac{1}{2(1+2)^2}\left(x - 2\right)^2$

$\approx$

$\ln 3 + \frac{1}{3}\left(x - 2\right) - \frac{1}{18}\left(x - 2\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$2 + 0$	$1.098612$	$1.098612$	$1.098612$	$0.000000$	$0.000000$
$2 + \frac{1}{100}$	$1.101940$	$1.101946$	$1.101940$	$5.54 \times 10^{-6}$	$1.23 \times 10^{-8}$
$2 + \frac{1}{10}$	$1.131402$	$1.131946$	$1.131390$	$0.000544$	$1.20 \times 10^{-5}$
$2 + \frac{1}{2}$	$1.252763$	$1.265279$	$1.251390$	$0.012516$	$0.001373$
$2 + 1$	$1.386294$	$1.431946$	$1.376390$	$0.045651$	$0.009904$

$\ln(1+x)$$\tilde f_1$$\tilde f_2$

Near 0 this is x − x²/2, the start of the alternating log series.

2.2.3$(1+x)^r$

r =1/2−1/2−123

$\sqrt{1+x}$

$\approx$

$(1+x_0)^r + r(1+x_0)^{r-1}\left(x - x_0\right) + \frac{r(r-1)}{2}(1+x_0)^{r-2}\left(x - x_0\right)^2$

reference$0$$1$

at $0$

$(1+x)^r$

$\approx$

$(1+0)^r + r(1+0)^{r-1}x + \frac{r(r-1)}{2}(1+0)^{r-2}x^2$

$\approx$

$1 + rx + \frac{r(r-1)}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.004988$	$1.005000$	$1.004987$	$1.24 \times 10^{-5}$	$6.21 \times 10^{-8}$
$0 + \frac{1}{10}$	$1.048809$	$1.050000$	$1.048750$	$0.001191$	$5.88 \times 10^{-5}$
$0 + \frac{1}{2}$	$1.224745$	$1.250000$	$1.218750$	$0.025255$	$0.005995$
$0 + 1$	$1.414214$	$1.500000$	$1.375000$	$0.085786$	$0.039214$

$(1+x)^r$$\tilde f_1$$\tilde f_2$

at $1$

$(1+x)^r$

$\approx$

$(1+1)^r + r(1+1)^{r-1}\left(x - 1\right) + \frac{r(r-1)}{2}(1+1)^{r-2}\left(x - 1\right)^2$

$\approx$

$2^r + r\,2^{r-1}\left(x - 1\right) + r(r-1)2^{r-3}\left(x - 1\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$1 + 0$	$1.414214$	$1.414214$	$1.414214$	$0.000000$	$0.000000$
$1 + \frac{1}{100}$	$1.417745$	$1.417749$	$1.417745$	$4.41 \times 10^{-6}$	$1.10 \times 10^{-8}$
$1 + \frac{1}{10}$	$1.449138$	$1.449569$	$1.449127$	$0.000431$	$1.07 \times 10^{-5}$
$1 + \frac{1}{2}$	$1.581139$	$1.590990$	$1.579942$	$0.009851$	$0.001197$
$1 + 1$	$1.732051$	$1.767767$	$1.723573$	$0.035716$	$0.008478$

$(1+x)^r$$\tilde f_1$$\tilde f_2$

$\dfrac{1}{\sqrt{1+x}}$

$\approx$

$(1+x_0)^r + r(1+x_0)^{r-1}\left(x - x_0\right) + \frac{r(r-1)}{2}(1+x_0)^{r-2}\left(x - x_0\right)^2$

reference$0$$1$

at $0$

$(1+x)^r$

$\approx$

$(1+0)^r + r(1+0)^{r-1}x + \frac{r(r-1)}{2}(1+0)^{r-2}x^2$

$\approx$

$1 + rx + \frac{r(r-1)}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$0.995037$	$0.995000$	$0.995038$	$3.72 \times 10^{-5}$	$3.10 \times 10^{-7}$
$0 + \frac{1}{10}$	$0.953463$	$0.950000$	$0.953750$	$0.003463$	$0.000287$
$0 + \frac{1}{2}$	$0.816497$	$0.750000$	$0.843750$	$0.066497$	$0.027253$
$0 + 1$	$0.707107$	$0.500000$	$0.875000$	$0.207107$	$0.167893$

$(1+x)^r$$\tilde f_1$$\tilde f_2$

at $1$

$(1+x)^r$

$\approx$

$(1+1)^r + r(1+1)^{r-1}\left(x - 1\right) + \frac{r(r-1)}{2}(1+1)^{r-2}\left(x - 1\right)^2$

$\approx$

$2^r + r\,2^{r-1}\left(x - 1\right) + r(r-1)2^{r-3}\left(x - 1\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$1 + 0$	$0.707107$	$0.707107$	$0.707107$	$0.000000$	$0.000000$
$1 + \frac{1}{100}$	$0.705346$	$0.705339$	$0.705346$	$6.60 \times 10^{-6}$	$2.75 \times 10^{-8}$
$1 + \frac{1}{10}$	$0.690066$	$0.689429$	$0.690092$	$0.000636$	$2.65 \times 10^{-5}$
$1 + \frac{1}{2}$	$0.632456$	$0.618718$	$0.635291$	$0.013737$	$0.002836$
$1 + 1$	$0.577350$	$0.530330$	$0.596621$	$0.047020$	$0.019271$

$(1+x)^r$$\tilde f_1$$\tilde f_2$

$\dfrac{1}{1+x}$

$\approx$

$(1+x_0)^r + r(1+x_0)^{r-1}\left(x - x_0\right) + \frac{r(r-1)}{2}(1+x_0)^{r-2}\left(x - x_0\right)^2$

reference$0$$1$

at $0$

$(1+x)^r$

$\approx$

$(1+0)^r + r(1+0)^{r-1}x + \frac{r(r-1)}{2}(1+0)^{r-2}x^2$

$\approx$

$1 + rx + \frac{r(r-1)}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$0.990099$	$0.990000$	$0.990100$	$9.90 \times 10^{-5}$	$9.90 \times 10^{-7}$
$0 + \frac{1}{10}$	$0.909091$	$0.900000$	$0.910000$	$0.009091$	$0.000909$
$0 + \frac{1}{2}$	$0.666667$	$0.500000$	$0.750000$	$0.166667$	$0.083333$
$0 + 1$	$0.500000$	$0.000000$	$1.000000$	$0.500000$	$0.500000$

$(1+x)^r$$\tilde f_1$$\tilde f_2$

at $1$

$(1+x)^r$

$\approx$

$(1+1)^r + r(1+1)^{r-1}\left(x - 1\right) + \frac{r(r-1)}{2}(1+1)^{r-2}\left(x - 1\right)^2$

$\approx$

$2^r + r\,2^{r-1}\left(x - 1\right) + r(r-1)2^{r-3}\left(x - 1\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$1 + 0$	$0.500000$	$0.500000$	$0.500000$	$0.000000$	$0.000000$
$1 + \frac{1}{100}$	$0.497512$	$0.497500$	$0.497513$	$1.24 \times 10^{-5}$	$6.22 \times 10^{-8}$
$1 + \frac{1}{10}$	$0.476190$	$0.475000$	$0.476250$	$0.001190$	$5.95 \times 10^{-5}$
$1 + \frac{1}{2}$	$0.400000$	$0.375000$	$0.406250$	$0.025000$	$0.006250$
$1 + 1$	$0.333333$	$0.250000$	$0.375000$	$0.083333$	$0.041667$

$(1+x)^r$$\tilde f_1$$\tilde f_2$

$(1+x)^2$

$\approx$

$(1+x_0)^r + r(1+x_0)^{r-1}\left(x - x_0\right) + \frac{r(r-1)}{2}(1+x_0)^{r-2}\left(x - x_0\right)^2$

reference$0$$1$

at $0$

$(1+x)^r$

$\approx$

$(1+0)^r + r(1+0)^{r-1}x + \frac{r(r-1)}{2}(1+0)^{r-2}x^2$

$\approx$

$1 + rx + \frac{r(r-1)}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.020100$	$1.020000$	$1.020100$	$1.00 \times 10^{-4}$	$0.000000$
$0 + \frac{1}{10}$	$1.210000$	$1.200000$	$1.210000$	$0.010000$	$2.22 \times 10^{-16}$
$0 + \frac{1}{2}$	$2.250000$	$2.000000$	$2.250000$	$0.250000$	$0.000000$
$0 + 1$	$4.000000$	$3.000000$	$4.000000$	$1.000000$	$0.000000$

$(1+x)^r$$\tilde f_1$$\tilde f_2$

at $1$

$(1+x)^r$

$\approx$

$(1+1)^r + r(1+1)^{r-1}\left(x - 1\right) + \frac{r(r-1)}{2}(1+1)^{r-2}\left(x - 1\right)^2$

$\approx$

$2^r + r\,2^{r-1}\left(x - 1\right) + r(r-1)2^{r-3}\left(x - 1\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$1 + 0$	$4.000000$	$4.000000$	$4.000000$	$0.000000$	$0.000000$
$1 + \frac{1}{100}$	$4.040100$	$4.040000$	$4.040100$	$1.00 \times 10^{-4}$	$8.88 \times 10^{-16}$
$1 + \frac{1}{10}$	$4.410000$	$4.400000$	$4.410000$	$0.010000$	$0.000000$
$1 + \frac{1}{2}$	$6.250000$	$6.000000$	$6.250000$	$0.250000$	$0.000000$
$1 + 1$	$9.000000$	$8.000000$	$9.000000$	$1.000000$	$0.000000$

$(1+x)^r$$\tilde f_1$$\tilde f_2$

$(1+x)^3$

$\approx$

$(1+x_0)^r + r(1+x_0)^{r-1}\left(x - x_0\right) + \frac{r(r-1)}{2}(1+x_0)^{r-2}\left(x - x_0\right)^2$

reference$0$$1$

at $0$

$(1+x)^r$

$\approx$

$(1+0)^r + r(1+0)^{r-1}x + \frac{r(r-1)}{2}(1+0)^{r-2}x^2$

$\approx$

$1 + rx + \frac{r(r-1)}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.030301$	$1.030000$	$1.030300$	$0.000301$	$1.00 \times 10^{-6}$
$0 + \frac{1}{10}$	$1.331000$	$1.300000$	$1.330000$	$0.031000$	$0.001000$
$0 + \frac{1}{2}$	$3.375000$	$2.500000$	$3.250000$	$0.875000$	$0.125000$
$0 + 1$	$8.000000$	$4.000000$	$7.000000$	$4.000000$	$1.000000$

$(1+x)^r$$\tilde f_1$$\tilde f_2$

at $1$

$(1+x)^r$

$\approx$

$(1+1)^r + r(1+1)^{r-1}\left(x - 1\right) + \frac{r(r-1)}{2}(1+1)^{r-2}\left(x - 1\right)^2$

$\approx$

$2^r + r\,2^{r-1}\left(x - 1\right) + r(r-1)2^{r-3}\left(x - 1\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$1 + 0$	$8.000000$	$8.000000$	$8.000000$	$0.000000$	$0.000000$
$1 + \frac{1}{100}$	$8.120601$	$8.120000$	$8.120600$	$0.000601$	$1.00 \times 10^{-6}$
$1 + \frac{1}{10}$	$9.261000$	$9.200000$	$9.260000$	$0.061000$	$0.001000$
$1 + \frac{1}{2}$	$15.625000$	$14.000000$	$15.500000$	$1.625000$	$0.125000$
$1 + 1$	$27.000000$	$20.000000$	$26.000000$	$7.000000$	$1.000000$

$(1+x)^r$$\tilde f_1$$\tilde f_2$

At r = 2 the quadratic approximation is exact — (1+x)² already is a quadratic, so there is no error at all.

2.2.4$a^x$

a =2310

$a^x$

$\approx$

$a^{x_0} + a^{x_0}\ln a\left(x - x_0\right) + \frac{a^{x_0}\ln^2 a}{2}\left(x - x_0\right)^2$

reference$0$$1$

at $0$

$a^x$

$\approx$

$a^{0} + a^{0}\ln ax + \frac{a^{0}\ln^2 a}{2}x^2$

$\approx$

$1 + \ln ax + \frac{\ln^2 a}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.006956$	$1.006931$	$1.006955$	$2.41 \times 10^{-5}$	$5.56 \times 10^{-8}$
$0 + \frac{1}{10}$	$1.071773$	$1.069315$	$1.071717$	$0.002459$	$5.65 \times 10^{-5}$
$0 + \frac{1}{2}$	$1.414214$	$1.346574$	$1.406630$	$0.067640$	$0.007583$
$0 + 1$	$2.000000$	$1.693147$	$1.933374$	$0.306853$	$0.066626$

$a^x$$\tilde f_1$$\tilde f_2$

at $1$

$a^x$

$\approx$

$a^{1} + a^{1}\ln a\left(x - 1\right) + \frac{a^{1}\ln^2 a}{2}\left(x - 1\right)^2$

$\approx$

$a + a\ln a\left(x - 1\right) + \frac{a\ln^2 a}{2}\left(x - 1\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$1 + 0$	$2.000000$	$2.000000$	$2.000000$	$0.000000$	$0.000000$
$1 + \frac{1}{100}$	$2.013911$	$2.013863$	$2.013911$	$4.82 \times 10^{-5}$	$1.11 \times 10^{-7}$
$1 + \frac{1}{10}$	$2.143547$	$2.138629$	$2.143434$	$0.004917$	$0.000113$
$1 + \frac{1}{2}$	$2.828427$	$2.693147$	$2.813260$	$0.135280$	$0.015167$
$1 + 1$	$4.000000$	$3.386294$	$3.866747$	$0.613706$	$0.133253$

$a^x$$\tilde f_1$$\tilde f_2$

$a^x$

$\approx$

$a^{x_0} + a^{x_0}\ln a\left(x - x_0\right) + \frac{a^{x_0}\ln^2 a}{2}\left(x - x_0\right)^2$

reference$0$$1$

at $0$

$a^x$

$\approx$

$a^{0} + a^{0}\ln ax + \frac{a^{0}\ln^2 a}{2}x^2$

$\approx$

$1 + \ln ax + \frac{\ln^2 a}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.011047$	$1.010986$	$1.011046$	$6.06 \times 10^{-5}$	$2.22 \times 10^{-7}$
$0 + \frac{1}{10}$	$1.116123$	$1.109861$	$1.115896$	$0.006262$	$0.000227$
$0 + \frac{1}{2}$	$1.732051$	$1.549306$	$1.700175$	$0.182745$	$0.031876$
$0 + 1$	$3.000000$	$2.098612$	$2.702087$	$0.901388$	$0.297913$

$a^x$$\tilde f_1$$\tilde f_2$

at $1$

$a^x$

$\approx$

$a^{1} + a^{1}\ln a\left(x - 1\right) + \frac{a^{1}\ln^2 a}{2}\left(x - 1\right)^2$

$\approx$

$a + a\ln a\left(x - 1\right) + \frac{a\ln^2 a}{2}\left(x - 1\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$1 + 0$	$3.000000$	$3.000000$	$3.000000$	$0.000000$	$0.000000$
$1 + \frac{1}{100}$	$3.033140$	$3.032958$	$3.033139$	$0.000182$	$6.65 \times 10^{-7}$
$1 + \frac{1}{10}$	$3.348370$	$3.329584$	$3.347688$	$0.018786$	$0.000682$
$1 + \frac{1}{2}$	$5.196152$	$4.647918$	$5.100524$	$0.548234$	$0.095628$
$1 + 1$	$9.000000$	$6.295837$	$8.106260$	$2.704163$	$0.893740$

$a^x$$\tilde f_1$$\tilde f_2$

$a^x$

$\approx$

$a^{x_0} + a^{x_0}\ln a\left(x - x_0\right) + \frac{a^{x_0}\ln^2 a}{2}\left(x - x_0\right)^2$

reference$0$$1$

at $0$

$a^x$

$\approx$

$a^{0} + a^{0}\ln ax + \frac{a^{0}\ln^2 a}{2}x^2$

$\approx$

$1 + \ln ax + \frac{\ln^2 a}{2}x^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$0 + 0$	$1.000000$	$1.000000$	$1.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.023293$	$1.023026$	$1.023291$	$0.000267$	$2.05 \times 10^{-6}$
$0 + \frac{1}{10}$	$1.258925$	$1.230259$	$1.256768$	$0.028667$	$0.002157$
$0 + \frac{1}{2}$	$3.162278$	$2.151293$	$2.814030$	$1.010985$	$0.348248$
$0 + 1$	$10.000000$	$3.302585$	$5.953534$	$6.697415$	$4.046466$

$a^x$$\tilde f_1$$\tilde f_2$

at $1$

$a^x$

$\approx$

$a^{1} + a^{1}\ln a\left(x - 1\right) + \frac{a^{1}\ln^2 a}{2}\left(x - 1\right)^2$

$\approx$

$a + a\ln a\left(x - 1\right) + \frac{a\ln^2 a}{2}\left(x - 1\right)^2$

	$f(x)$	$\tilde f_1(x)$	$\tilde f_2(x)$	linear error	quad. error
$1 + 0$	$10.000000$	$10.000000$	$10.000000$	$0.000000$	$0.000000$
$1 + \frac{1}{100}$	$10.232930$	$10.230259$	$10.232909$	$0.002671$	$2.05 \times 10^{-5}$
$1 + \frac{1}{10}$	$12.589254$	$12.302585$	$12.567680$	$0.286669$	$0.021574$
$1 + \frac{1}{2}$	$31.622777$	$21.512925$	$28.140298$	$10.109851$	$3.482478$
$1 + 1$	$100.000000$	$33.025851$	$59.535341$	$66.974149$	$40.464659$

$a^x$$\tilde f_1$$\tilde f_2$

Covers e^x too: set a = e, so ln a = 1 and the coefficients collapse to 1, 1, 1/2.

3. Algebra

You do not have to differentiate a messy function twice. You can build its approximation out of the entries above instead.

Write $Q(f)$ for the quadratic approximation of $f$ near $0$, and $L(f)$ for the linear one.

The useful fact is that you can approximate first and combine second. For a product:

$$Q\big(Q(f)\,Q(g)\big) = Q(fg)$$

So the $Q$ of a product is just the quadratic approximation of the product of the quadratic approximations. You never have to touch $fg$ itself.

Here is the full set:

	rule
sum	$Q(f+g) = Q(f) + Q(g)$
scale	$Q(kf) = k\,Q(f)$
product	$Q\big(Q(f)\,Q(g)\big) = Q(fg)$
composition, when $g(0)=0$	$Q\big(Q(f)\circ Q(g)\big) = Q(f\circ g)$

Applied to a polynomial, $Q$ does nothing but chop off everything above $x^2$. So the outer $Q$ on each left-hand side is the cut — a product of two quadratics runs up to $x^4$, and those high terms are not trustworthy, since the inputs were only accurate to second order.

The $g(0) = 0$ condition on the last line is not a deep restriction — it falls out of how $Q$ is defined here. $Q(f)$ only carries information about $f$ near $0$, so plugging in $g(x)$ is only valid where $g$ actually lands near $0$. Centre $Q$ at a different reference point and the condition moves with it: substitution is valid whenever the inside function's value at your reference point equals the outside function's reference point.