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quarto/derivatives/linearization.qmd
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The plot shows the tangent line with slope $dy/dx$ and the actual change in $y$, $\Delta y$, for some specified $\Delta x$. The small gap above the sine curve is the error were the value of the sine approximated using the drawn tangent line. We can see that approximating the value of $\Delta y = \sin(c+\Delta x) - \sin(c)$ with the often easier to compute $(dy/dx) \cdot \Delta x = f'(c)\Delta x$ - for small enough values of $\Delta x$ - is not going to be too far off provided $\Delta x$ is not too large.
This approximation is known as linearization. It can be used both in theoretical computations and in pratical applications. To see how effective it is, we look at some examples.
This approximation is known as linearization. It can be used both in theoretical computations and in practical applications. To see how effective it is, we look at some examples.
##### Example