Question: User: (a)(i) Find the Fourier integral representation of the function
f(x) = \begin{cases} 1 & \text{if } |x| < 1 \\ 0 & \text{if } |x| > 1 \end{cases}
(ii) Prove that
$$\int_0^\infty \frac{\cos w \sin w}{w} \, dw = \frac{\pi}{4}.$$
(b) Find the Fourier sine transform of
$$f(x) = e^{-ax}, a > 0.$$
(c) Find the directional derivative of
$$f(x,y) = 3x^3 - 4xy$$
along the parabola
$$y = 2x^2 - x + 2$$
at the point
$$(1,3).$
1. **Problem statement:**
Find the Fourier integral representation of the function
$$f(x) = \begin{cases} 1 & \text{if } |x| < 1 \\ 0 & \text{if } |x| > 1 \end{cases}$$
2. **Fourier integral representation formula:**
For a function $f(x)$, the Fourier integral representation is given by
$$f(x) = \frac{1}{\pi} \int_0^\infty \left[ A(w) \cos(wx) + B(w) \sin(wx) \right] dw,$$
where
$$A(w) = \int_{-\infty}^\infty f(t) \cos(wt) dt, \quad B(w) = \int_{-\infty}^\infty f(t) \sin(wt) dt.$$
3. **Calculate $A(w)$ and $B(w)$:**
Since $f(t) = 1$ for $|t| < 1$ and $0$ otherwise,
$$A(w) = \int_{-1}^1 \cos(wt) dt = \left[ \frac{\sin(wt)}{w} \right]_{-1}^1 = \frac{\sin w - (-\sin w)}{w} = \frac{2 \sin w}{w}.$$
Because $f(t)$ is even, $B(w) = 0$ (since sine is odd).
4. **Fourier integral representation:**
$$f(x) = \frac{1}{\pi} \int_0^\infty \frac{2 \sin w}{w} \cos(wx) dw = \frac{2}{\pi} \int_0^\infty \frac{\sin w}{w} \cos(wx) dw.$$
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5. **Prove that**
$$\int_0^\infty \frac{\cos w \sin w}{w} dw = \frac{\pi}{4}.$$
Rewrite the integral:
$$\int_0^\infty \frac{\cos w \sin w}{w} dw = \int_0^\infty \frac{\sin(2w)}{2w} dw$$
using the identity $\sin(2w) = 2 \sin w \cos w$.
6. **Use substitution:**
Let $u = 2w$, then $du = 2 dw$, so $dw = \frac{du}{2}$.
The integral becomes
$$\int_0^\infty \frac{\sin u}{u} \frac{du}{2} = \frac{1}{2} \int_0^\infty \frac{\sin u}{u} du.$$
7. **Known integral:**
$$\int_0^\infty \frac{\sin u}{u} du = \frac{\pi}{2}.$$
Therefore,
$$\int_0^\infty \frac{\cos w \sin w}{w} dw = \frac{1}{2} \times \frac{\pi}{2} = \frac{\pi}{4}.$$
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8. **Find the Fourier sine transform of**
$$f(x) = e^{-ax}, a > 0.$$
The Fourier sine transform is defined as
$$F_s(w) = \sqrt{\frac{2}{\pi}} \int_0^\infty f(x) \sin(wx) dx.$$
9. **Calculate:**
$$F_s(w) = \sqrt{\frac{2}{\pi}} \int_0^\infty e^{-ax} \sin(wx) dx.$$
Use the integral formula
$$\int_0^\infty e^{-px} \sin(qx) dx = \frac{q}{p^2 + q^2}, \quad p > 0.$$
Here, $p = a$, $q = w$.
10. **Result:**
$$F_s(w) = \sqrt{\frac{2}{\pi}} \frac{w}{a^2 + w^2}.$$
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11. **Find the directional derivative of**
$$f(x,y) = 3x^3 - 4xy$$
along the parabola
$$y = 2x^2 - x + 2$$
at the point
$$(1,3).$$
12. **Calculate gradient:**
$$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (9x^2 - 4y, -4x).$$
At $(1,3)$,
$$\nabla f(1,3) = (9(1)^2 - 4(3), -4(1)) = (9 - 12, -4) = (-3, -4).$$
13. **Find direction vector along parabola:**
Parametrize $y$ as a function of $x$:
$$y = 2x^2 - x + 2.$$
Derivative:
$$\frac{dy}{dx} = 4x - 1.$$
At $x=1$,
$$\frac{dy}{dx} = 4(1) - 1 = 3.$$
Direction vector is
$$\mathbf{u} = (1, 3).$$
14. **Normalize direction vector:**
$$|\mathbf{u}| = \sqrt{1^2 + 3^2} = \sqrt{10}.$$
Unit vector:
$$\hat{\mathbf{u}} = \left( \frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}} \right).$$
15. **Directional derivative formula:**
$$D_{\mathbf{u}} f = \nabla f \cdot \hat{\mathbf{u}} = (-3, -4) \cdot \left( \frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}} \right) = \frac{-3}{\sqrt{10}} + \frac{-12}{\sqrt{10}} = \frac{-15}{\sqrt{10}} = -\frac{15}{\sqrt{10}}.$$