To approximate the derivative of the function \( f(x) = \sin(x) \) at \( x = \frac{\pi}{4} \) using the Forward Difference method, we calculate the slope between the points \( \sin\left(\frac{\pi}{4}\right) \) and \( \sin\left(\frac{\pi}{4} + h\right) \), where \( h \) is a small value. The formula for the Forward Difference approximation is given by:
\[ f'(x) \approx \frac{f(x + h) - f(x)}{h} \]
We will use this formula to compute the derivative and then plot the slope.
The Forward Difference method is a numerical technique used to approximate the derivative of a function at a given point. The derivative of a function \( f(x) \) at a point \( x \) can be approximated using the formula:
\[
f'(x) \approx \frac{f(x + h) - f(x)}{h}
\]
where \( h \) is a small value. In this problem, we are given \( f(x) = \sin(x) \) and need to approximate its derivative at \( x = \frac{\pi}{4} \).
First, we calculate the sine values at the points \( x = \frac{\pi}{4} \) and \( x = \frac{\pi}{4} + h \).
\[
f\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]
\[
f\left(\frac{\pi}{4} + h\right) = \sin\left(\frac{\pi}{4} + h\right)
\]
Using the Forward Difference formula, we approximate the derivative:
\[
f'\left(\frac{\pi}{4}\right) \approx \frac{\sin\left(\frac{\pi}{4} + h\right) - \sin\left(\frac{\pi}{4}\right)}{h}
\]
Substitute \( h = 1 \times 10^{-5} \):
\[
f'\left(\frac{\pi}{4}\right) \approx \frac{\sin\left(\frac{\pi}{4} + 1 \times 10^{-5}\right) - \frac{\sqrt{2}}{2}}{1 \times 10^{-5}}
\]
Using a calculator or software to find the sine value:
\[
\sin\left(\frac{\pi}{4} + 1 \times 10^{-5}\right) \approx \sin\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{4}\right) \times 1 \times 10^{-5}
\]
Since \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\), we have:
\[
\sin\left(\frac{\pi}{4} + 1 \times 10^{-5}\right) \approx \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \times 1 \times 10^{-5}
\]
Substitute the values back into the Forward Difference formula:
\[
f'\left(\frac{\pi}{4}\right) \approx \frac{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \times 1 \times 10^{-5}\right) - \frac{\sqrt{2}}{2}}{1 \times 10^{-5}}
\]
Simplify:
\[
f'\left(\frac{\pi}{4}\right) \approx \frac{\frac{\sqrt{2}}{2} \times 1 \times 10^{-5}}{1 \times 10^{-5}} = \frac{\sqrt{2}}{2}
\]
The approximate derivative of \( f(x) = \sin(x) \) at \( x = \frac{\pi}{4} \) using the Forward Difference method is:
\[
\boxed{\frac{\sqrt{2}}{2}}
\]
Answered
