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Here's an explanation of the parametric curve drawing: Consider two functions $$f(x)$$ and $$g(x)$$ continuous on the interval $$[a,b]$$ and differentiable on $$(a,b)$$.

For every $$x \in [a,b]$$, we consider the point $$(f(x),g(x))$$. If we trace out the points $$(f(x),g(x))$$ over every $$x \in [a,b]$$, we get a curve in two dimensions, as shown in the graph.

In the drawing, the slope of the red line is $$\frac{g(b)-g(a)}{f(b)-f(a)}$$. (This is because $$\frac{\Delta y}{\Delta x}=\frac{g(b)-g(a)}{f(b)-f(a)}$$, assuming that the vertical axis, which contains the value of $$g(x)$$, is the $$y$$-axis.)

The slope of the green line is $$\frac{g'(c)}{f'(c)}$$. (Why? Because $$\frac{\text{d}g}{\text{d}f}\Big|_{x=c} = \frac{\text dg / \text dx}{\text df / \text dx}\Big|_{x=c} = \frac{g'(c)}{f'(c)}$$.) The drawing illustrates that for the value of $$c$$ chosen in the pictures, the slopes of the red line and green line are the same, i.e. $$\frac{g(b)-g(a)}{f(b)-f(a)} = \frac{g'(c)}{f'(c)}$$.

sid-kap
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