Linear Approximations and Differentials

Definition Let $f$ be differentiable at $a$. The linearization of $f$ at $a$ is the function

Its graph is the tangent line to the graph of $f$ at $(a, f(a))$.

Definition The tangent line approximation, or linear approximation, of $f$ near $a$ is

used for $x$ close to $a$. The error of the approximation is $f(x) - L(x)$.

Definition Let $y = f(x)$ with $f$ differentiable at $x$. The differential $dx$ is an independent variable, any real number, and the differential $dy$ is defined by

For an increment $\Delta x = dx$, the actual change in $y$ is

The differential $dy$ is the change along the tangent line, and $\Delta y \approx dy$ when $\Delta x$ is small.

Definition A function $f$ defined on an open interval containing $a$ is differentiable at $a$ if and only if there is a number $m$ such that the error $E(h) = f(a + h) - f(a) - mh$ satisfies

and in that case $m = f'(a)$. So for a differentiable function the error of the linear approximation goes to $0$ faster than the increment $h$ itself.

Definition Let $E$ and $\varphi$ be defined for $h$ near $0$, with $h \neq 0$. As $h \to 0$, write

where $\varphi(h) \neq 0$ for small $h \neq 0$, and write $E(h) = O(\varphi(h))$ if there are constants $C > 0$ and $\delta > 0$ with $\lvert E(h) \rvert \le C\lvert \varphi(h) \rvert$ whenever $0 < \lvert h \rvert < \delta$. In this notation, $f$ is differentiable at $a$ exactly when

Definition Let $f$ be twice differentiable on an open interval containing $a$ and $x$.

1. If $\lvert f'(t) \rvert \le M_1$ for every $t$ between $a$ and $x$, then $\lvert f(x) - f(a) \rvert \le M_1 \lvert x - a \rvert$. This follows from the mean value theorem.
2. If $\lvert f''(t) \rvert \le M_2$ for every $t$ between $a$ and $x$, then the error of the linear approximation satisfies

The second bound comes from Taylor’s theorem, which says that

for some $c$ between $a$ and $x$. Applying the plain mean value theorem twice only gives the weaker bound $M_2 (x - a)^2$, without the factor $1/2$.