Differentiability

Best linear approximation of derivative at a point:

Let $D \subseteq \mathbb{R}$ , $f : D \to \mathbb{R}$ , and let $x_0 \in D$ be an accumulation point of $D$ .
Then the following statements are equivalent:

i) The function $f$ is differentiable in $x_0$ .

ii) There exists an $m \in \mathbb{R}$ and a function $R : D \to \mathbb{R}$ such that for all $x \in D$ it holds that

f(x) = f(x_0) + m \cdot (x - x_0) + R(x), \qquad\text{where}\qquad \lim_{x \to x_0} \frac{R(x)}{x - x_0} = 0.

If one of these conditions is fulfilled (and hence both of them), then the parameter $m$
in Item ii) is uniquely determined and we have $m = f'(x_0)$ .

Tangent and Approx error:

The he tangent $t$ touching the graph of $f$ at $x_0$ is given by:

t : \mathbb{R} \to \mathbb{R}, \qquad t(x) = f(x_0) + f'(x_0)(x - x_0)

and can be interpreted as the best linear approximation of $f$ at $x_0$ .

The error or remainder function we make when approximating $f$ by its tangent $t$ at $x_0$ :

R : D \to \mathbb{R}, \qquad x \mapsto R(x) := f(x) - t(x) = f(x) - f(x_0) - f'(x_0)(x - x_0).

The error becomes “arbitrarily small” when $x$ is near $x_0$ , i.e., $\lim_{x \to x_0} R(x) = 0$ . However, this condition can also be fulfilled by a “bad” linear approximation of $f$ . Therefore, it is more interesting to consider the relative error

\frac{R(x)}{x - x_0}

as a measure for the quality of a linear approximation, and to call a linear approximation “good” if $\lim_{x \to x_0} \frac{R(x)}{x - x_0} = 0$ .

How to use the applet

Drag the blue point $x_0$ on the curve to choose the base point $P=(x_0,f(x_0))$ .
Move the slider $m$ to choose the slope of a candidate linear approximation

L(x)=f(x_0)+m(x-x_0).

Drag the orange point $x$ on the curve to choose a comparison point $Q=(x,f(x))$ .
The pink/green line is the linear function $L(x)$ through $P$ with slope $m$ .
The green dashed vertical segment labeled $R(x)$ shows the remainder (approximation error) at $x$ :

R(x)=f(x)-L(x).

The faint gray dashed line is the secant through $P$ and $Q$ (visual aid).
If $m$ is chosen close to the true derivative $f'(x_0)$ , then $L(x)$ is a good local approximation near $x_0$ , and $R(x)$ becomes small when $x$ is close to $x_0$ .
The applet highlights this situation by drawing $L(x)$ in green (and thicker) when $m$ is close to $f'(x_0)$ .

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Hold Shift + scroll to zoom
Hold Shift + drag to move
Points shaped like <> act as sliders

Description

Link to code

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Figure6.3 and Theorem6.1 page 94

This interactive applet illustrates when a linear approximation of a function at a point can be considered a “good” one.

The tangent line at $x_0$ is the best linear approximation of $f$ at that point.
The theorem above states that if $f$ is differentiable at $x_0$ , then $f(x) = f(x_0) + m(x - x_0) + R(x)$ ,
where the remainder term $R(x)$ satisfies $\lim_{x \to x_0} \frac{R(x)}{x - x_0} = 0$ .

A key insight is that the condition $R(x) \to 0$ alone does not guarantee a good approximation: even a poorly chosen linear function can have a remainder that vanishes in absolute terms. What matters is the behaviour of the relative remainder.

In the applet, the student can adjust the slope $m$ using a slider. When $m$ differs from $f'(x_0)$ , the linear approximation is bad. By dragging the point $x$ toward $x_0$ , the sutudent can notice that, the remainder $R(x)$ approaches $0$ as $x \to x_0$ , but the relative error $\frac{R(x)}{x - x_0}$ does not. When the chosen line coincides with the true tangent (i.e. $m = f'(x_0)$ ), it is highlighted in green, and both the absolute and relative remainders tend to zero as $x$ approaches $x_0$ .

    const isBestApprox = () => Math.abs(getM() - df(getX0())) < 0.2;

Thus the relative error is the difference of twp slopes: (Slope of Secant Line) - (Slope $m$ )

\frac{R(x)}{x - x_0} = \frac{f(x) -t(x)}{x - x_0} = \frac{f(x) -[f(x_0) +m(x -x_0)]}{x - x_0} = \frac{f(x) -f(x_0)}{x - x_0} - m

The height of the purple line mantian this ratio, since start from the secant up to the linear approximation.