Secant Tangent and Derivative at a point

Derivative at a point:

Let $D \subseteq \mathbb{R}$ and $f : D \to \mathbb{R}$ . The function $f$ is called differentiable in $x_0 \in D$ , if $x_0$ is an accumulation point of $D$ and if the limit $f'(x_0) := \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}$ exists. In this case, $f'(x_0)$ is called the derivative of $f$ in $x_0$ .

How to use the applet

Drag the green point $(a,f(a))$ along the curve to choose the base point $a$ .
Move the slider $h$ to choose a second point $B=(a+h,f(a+h))$ .
The red line is the secant through $(a,f(a))$ and $(a+h,f(a+h))$ . Its slope is

\frac{f(a+h)-f(a)}{(a+h)-a}=\frac{f(a+h)-f(a)}{h}.

The orange arrows highlight the horizontal change $h$ and the vertical change $\Delta f=f(a+h)-f(a)$ .
When $h$ is very small, the applet switches to the purple tangent line at $(a,f(a))$ , whose slope equals the derivative $f'(a)$ .
As $h\to 0$ , the secant slope approaches the tangent slope:

\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}=f'(a).

For the example $f(x)=x^2$ , the applet displays $f'(a)=2a$ . Try moving $a$ and check how the tangent steepness changes.

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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 ) Figure 6.1 page 91, Figure 6.2 and Definition 6.1

The example is an geometric interpretation of the derivative, specifically showing the connetcion between the slope of the secant between two points and the slope of the tangent at a point. Here a link where i took inspiration from.

The slope of the secant is given by:

    () => `Secant: slope = ${((pointB.Y() - pointA.Y()) / (pointB.X() - pointA.X())).toFixed(3)}`

Meaning:

\frac{ f(a+h) - f (a)}{(a+h) - a} = \frac{ f(a+h) - f (a)}{h}

Where pointB = (a+h, f(a+h)), and the value h is controlled by a slider.

When h approaches zero (the slider sets a minimum value of 0.001), the secant slope corresponds to the slope of the tangent at pointA, showing that the tangent line is the best approximation of a derivative at a point.

The tangent is calculated using the derivative of f since:

\lim_{h \to 0} \frac{ f(a+h) - f (a)}{h} = f'(a)