Approximations

If $f(x)$ is a differentiable function of $x$, its derivative at $x=a$ is defined by the limit $f'(a)=\displaystyle\lim_{h\to0}\dfrac{f(a+h)-f(a)}{h}$. Using the symbol $\approx$ for "approximately equal to", for a sufficiently small $h$ we have

Multiplying both sides by $h$ and rearranging: $h\,f'(a)\approx f(a+h)-f(a)$, which gives the key approximation formula:

This says that near $x=a$, the curve $y=f(x)$ can be replaced by its tangent line at $a$ without much loss of accuracy, provided $h$ is small. It is used to estimate the value of $f$ at a point $a+h$ that is close to a "nice" reference point $a$ where $f(a)$ and $f'(a)$ can be computed exactly (a perfect square or cube for roots/powers, a standard angle for trig functions, $x=1$ for $e^x$/logarithms, and so on).

Worked Example 1 — Square root. Approximate $\sqrt{64.1}$. Let $f(x)=\sqrt x$, $a=64$, $h=0.1$. $f(64)=8$, $f'(x)=\dfrac{1}{2\sqrt x}$ so $f'(64)=\dfrac{1}{16}=0.0625$. So $\sqrt{64.1}\approx8+(0.1)(0.0625)=8.00625$.

Worked Example 2 — Cube. Approximate $(3.98)^3$. Let $f(x)=x^3$, $a=4$, $h=-0.02$. $f(4)=64$, $f'(4)=3(16)=48$. So $(3.98)^3\approx64+(-0.02)(48)=63.04$.

Worked Example 3 — Trigonometric, with a degree/minute offset. Approximate $\sin(30°30')$, given $1°=0.0175^c$ and $\cos30°=0.866$. Convert the offset: $30°30'=30°+\dfrac12°$, i.e. $a=\dfrac{\pi}{6}$, $h=\dfrac12(0.0175)=0.00875$. $f(x)=\sin x$: $f\left(\dfrac{\pi}{6}\right)=\dfrac12=0.5$, $f'(x)=\cos x$ so $f'\left(\dfrac{\pi}{6}\right)=0.866$. So $\sin(30°30')\approx0.5+(0.00875)(0.866)=0.5075775$.

Worked Example 4 — Inverse trigonometric. Approximate $\tan^{-1}(0.99)$, given $\pi\approx3.1416$. Let $f(x)=\tan^{-1}x$, $a=1$, $h=-0.01$. $f(1)=\dfrac{\pi}{4}$, $f'(x)=\dfrac{1}{1+x^2}$ so $f'(1)=0.5$. So $\tan^{-1}(0.99)\approx\dfrac{\pi}{4}-(0.01)(0.5)=0.7854-0.005=0.7804$.

Worked Example 5 — Exponential. Approximate $e^{1.005}$, given $e=2.7183$. Let $f(x)=e^x$, $a=1$, $h=0.005$. $f(1)=f'(1)=2.7183$ (since $e^x$ is its own derivative). So $e^{1.005}\approx2.7183+(0.005)(2.7183)\approx2.73189$. …