Compute y = F₈c by the three steps of the Fast Fourier Transform if c = (1, 0, 1, 0, 1, 0, 1, 0). Repeat the computation with c = (0, 1, 0, 1, 0, 1, 0, 1).

Given the exponential function f(x) = 1.2$^x$ and the logistic function $g(x)=\frac{1.2^x}{1.2^x+1}$.

Transform the equation of the logistic function so that an exponential term appears only once. Show numerically that the resulting equation is equivalent to $g(x)$ as given.

Use the given row transformation to transform each matrix. Do not use a calculator.

$$\left[\begin{array}{rr}-1 & 4 \\ 7 & 0\end{array}\right] \begin{array}{r}7 R_1\\\\\end{array}$$

The transformation of a figure into its image is described. Describe the transformations that will transform the image back into the original figure. Then write them algebraically. The figure is dilated by a scale factor of 0.5 and translated 6 units left and 3 units up.

Use the integral definition to calculate the Laplace transform of the function

$$f(t)=\left\{\begin{array}{ll}{t-1} & {\text { if } 0 \leq t<2} \\ {1} & {\text { if } t \geq 2}\end{array}\right.$$

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Solve the following initial value problems by the Laplace transform. (If necessary, use partial fraction expansion. Show all details.)

$y^{\prime \prime}+2 y^{\prime}+2 y=0, \quad y(0)=1$, $y^{\prime}(0)=-3$

Use the time-shift property to calculate the Fourier transform of the double pulse defined by

$$f(t)=\left\{\begin{array}{ll}{1} & {(1 \leqslant|t| \leqslant 2)} \\ {0} & {(\text { otherwise })}\end{array}\right.$$

Find the Fourier sine transform of the function in the indicated problem, and write f(x) as a Fourier integral. Verify that the sine integral for f(x) is the same as the exponential integral found previously.

$$f(x)=\left\{\begin{array}{ll} x, & |x|<1 \\ 0, & |x|>1 \end{array}\right.$$

For the following problem, use the composite argument properties to transform the left side of the equation to a single function of a composite argument. Then solve the equation algebraically to get

The general solution for x or $\theta$

$$\cos 3 x \cos x+\sin 3 x \sin x=-1$$

Find the Fourier transform of $f(x)=e^{-x^{2} /\left(2 \sigma^{2}\right)}$. Hint: Complete the square in the x terms in the exponent and make the change of variable $y=x+\sigma^{2} i \alpha$. Use tables or computer to evaluate the definite integral.

Find the Laplace transform of the given function

$$\begin{gather*} \begin{cases} 0, & 0 \leq t \leq 1, \\ 1, & 1 < t \leq 2 \\ 0, & 2 < t. \end{cases} \end{gather*}$$

Solve the following initial value problems by the Laplace transform. (If necessary, use partial fraction expansion. Show all details.)

$y^{\prime \prime}-4 y^{\prime}+4 y=0, \quad y(0)=2.1$, $y^{\prime}(0)=3.9$

Derive the quadratic formula by using the change in variable $x=y-\frac{1}{2}\left(\frac{b}{a}\right)$ to transform the quadratic equation $x^2+\frac{b}{a} x+\frac{c}{a}=0$ into one involving the difference of two squares and solve the resulting equation.

Use the linearity of ${L^{-1}}$, partial fraction expansions, and Table (5.3.1) to find the inverse Laplace transform of the function ${\dfrac{s^3 - 2s^2 - 6s - 6}{(s^2 + 2s + 2) \ s^2}}$.