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The formula for the inverse Fourier transform shows that a function \(f(x)\) can be written in terms of its Fourier transform via \begin{equation} f(x)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \tilde{f}(k)\, e^{ikx}\, dk \end{equation} Take the derivative of both sides of this equation with respect to \(x\) and simplify. Interpret your expression as the inverse Fourier transform of something.
Students will need a short lecture giving the definition of the inverse Fourier Transform \begin{equation} {\cal{F}}^{-1}(\tilde{f}) =f(x)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(k)\, e^{ikx}\, dk \end{equation}
assignment Homework
group Small Group Activity
5 min.
group Small Group Activity
30 min.
assignment Homework
Find the differential of each of the following expressions; zap each of the following with \(d\):
\[f=3x-5z^2+2xy\]
\[g=\frac{c^{1/2}b}{a^2}\]
\[h=\sin^2(\omega t)\]
\[j=a^x\]
group Small Group Activity
10 min.
format_list_numbered Sequence
assignment Homework
Set up the integrals for the Fourier series for this state.
Which terms will have the largest coefficients? Explain briefly.
Are there any coefficients that you know will be zero? Explain briefly.
Using the technology of your choice or by hand, calculate the four largest coefficients. With screen shots or otherwise, show your work.