Question

Use Linear Approximation to estimate deltaf=f(3.02)-f(3) for f(x)=sin (pix/5)

Answer 1

The increment in $f$ is approximately $-3.883\times 10^{-3}$.

How to estimate an increment in $f(x)$ by linear approximation

The linear approximation is derived from definition of tangent, that is to say:

$\Delta f \approx m_{x}\cdot \Delta x$ (1)

Where:

• $m_{x}$ - Slope of the function evaluated at $x$.
• $\Delta x$ - Increment in $x$.
• $\Delta f$ - Increment in $f$.

The slope is found by derivatives:

$m_{x} = \frac{\pi}{5} \cdot \cos \frac{\pi\cdot x}{5}$    (2)

If we know that $x = 3$ and $\Delta x = 0.02$, then the increment in $f$ is:

$\Delta f \approx \left(\frac{\pi}{5}\cdot \cos \frac{3\pi}{5} \right)\cdot (0.02)$

$\Delta f \approx -3.883\times 10^{-3}$

The increment in $f$ is approximately $-3.883\times 10^{-3}$. $\blacksquare$

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