The linear approximation of a function at one point.

Definition 6.2.1   Let $f$ be a function defined in a neighborhood of the real $x_0$ . The function $f$ has a linear approximation at $x_0$ if there exist a real number $A$ and a function $\varepsilon$ defined on a neighborhood of 0, such that

$$\begin{cases}f(x_0+h)=f(x_0)+Ah+h\; \varepsilon(h) \underset{h \rightarrow 0}{\lim} \varepsilon (h)=0 \end{cases}$$

Example 6.2.2   Let $f(x)=x^2$ and $x_0=1$ . Then we have:

$$f(x_0+h)=f(1+h)=(1+h)^2=1+2h+h^2=1+\underbrace{2}_{=f'(1)}h+h \cdot \underbrace{h}_{\varepsilon (h)}$$

Example 6.2.3   Let $f(x)=x^3$ and $x_0=1$ . Then we have:

$$f(x_0+h)=f(1+h)=(1+h)^3=1+3h+3h^2+h^3=1+\underbrace{3}_{=f'(1)}h+h \cdot \underbrace{3h+h^3}_{\varepsilon (h)}$$

With the notations of Definition 1.4, we have:

$$f'(x_0)=\frac {df}{dx} \bracevert_{x=x_0} \Longleftrightarrow df \bracevert_{x=x_0} = f'(x_0) \; dx$$

Definition 6.2.4   The expression $df \bracevert_{x=x_0} = f'(x_0) \; dx$ is called the differential of $f$ at $x_0$ .

Figure 1: The graph of a function and a chord.

An important use of differentials is the approximation of the value of a function $f$ at a point $x_0$ . Actually, we can rewrite Definition 2.1 as follows:

$$\Delta y=f(x_0+\Delta x)=f(x_0)+ \frac {df}{dx} \bracevert_{x_0} \Delta x + \Delta x \varepsilon (\Delta x ).$$

When $\Delta x$ is arbitrarily close to 0, we replace $\Delta x$ by $dx$ and $\Delta x$ by $dy$ or $df$ (these notations have the same meaning). Then we have:

$$f(x) \approx f(x_0)+df \bracevert_{x_0}$$

Example 6.2.5   With the settings of Example 2.3, we wish to compute $1.023^3$ . Let $x_0=1$ and $dx=0.23$ ; we have:

$$1.023 = 1 + 3 \cdot 0.023 = 1.069.$$

Compare this with the ``actual value'', which is 1.070599167. Not so far! We will learn later how to have an estimation of the error, when computing an approximation with the differential instead of the actual value. Remember that if an scientist makes measurements providing data with an accuracy of $10^{-n}$ , for some positive integer $n$ , then a computation leading to results with a greater accuracy (i.e. $10^{-m}$ , with $m>n$ ) is meaningless.