The increment in [tex]f[/tex] is approximately [tex]-3.883\times 10^{-3}[/tex].
The linear approximation is derived from definition of tangent, that is to say:
[tex]\Delta f \approx m_{x}\cdot \Delta x[/tex] (1)
Where:
The slope is found by derivatives:
[tex]m_{x} = \frac{\pi}{5} \cdot \cos \frac{\pi\cdot x}{5}[/tex] (2)
If we know that [tex]x = 3[/tex] and [tex]\Delta x = 0.02[/tex], then the increment in [tex]f[/tex] is:
[tex]\Delta f \approx \left(\frac{\pi}{5}\cdot \cos \frac{3\pi}{5} \right)\cdot (0.02)[/tex]
[tex]\Delta f \approx -3.883\times 10^{-3}[/tex]
The increment in [tex]f[/tex] is approximately [tex]-3.883\times 10^{-3}[/tex]. [tex]\blacksquare[/tex]
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