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

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

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work.

$\sin 3 t(0<t<\pi)$

Using the figure as a guide, transform Cartesian velocity components $(u, v, w)$ into cylindrical velocity components $\left(u_r, u_\theta, u_z\right)$. (Hint: Since the $z$-component of velocity remains the same in such a transformation, we need only to consider the $x y$-plane.)

find the Laplace transform of the given function. assume that term-by-term integration of the infinite series is permissible. f(t)=1−u1(t)+···+u2n(t)−u2n+1(t)=1+2n+1Σk=1(−1)kuk(t)

Consider the signal

$$x(t)=e^{-5t} u(t)+e^{-βt} u(t)$$

, and denote its Laplace transform by X(s). What are the constructs placed on the real and imaginary parts of β if the region of convergence of X(s) is Re(s)>-3?

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. Where appropriate, write f in terms of unit step functions. $y^{\prime \prime}+16 y=1, \quad y(0)=1, y^{\prime}(0)=2$

Use the linearity property (7) along with the transforms found in Example 2,

$$\mathcal{L}\left\{e^{a t}\right\}=\frac{1}{s-a}, \quad s>a \quad \text { and } \quad \mathcal{L}\{t\}=\frac{1}{s^2}, \quad s>0,$$

to calculate the Laplace transform $R(s)=\mathcal{L}\{r(t)\}$ of the given function $r(t)$. For what values $s$ does the Laplace transform exist?

$$r(t)=5 e^{-7 t}+t+2 e^{2 t}$$

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $y^{\prime \prime}+y=\delta(t-\pi / 2)+\delta(t-3 \pi / 2), \quad y(0)=0, y^{\prime}(0)=0$

Which of the following is NOT true about matter in the biosphere?

A Matter is recycled in the biosphere.

B Biogeochemical cycles transform and reuse molecules.

C The total amount of matter decreases over time.

D Water and nutrients pass between organisms and the environment.

Prove that if F($\omega$) is the Fourier transform of f(t),

$$\mathcal{F}\left[f(t) \sin \omega_{0} t\right]=\frac{j}{2}\left[F\left(\omega+\omega_{0}\right)-F\left(\omega-\omega_{0}\right)\right]$$

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.

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.

Find the Laplace transform of $f(t).$

$$f(t)=\left\{\begin{array}{ll}{t^{2}} & {\text { for } 0 \leq t<2} \\ {1-t-4 t^{2}} & {\text { for } t \geq 2}\end{array}\right.$$

Using the Laplace transform, solve the following problem. (Show the details.) $y^{\prime \prime}+3 y^{\prime}+2 y=4 t$ if $0<t<1$ and 8 if $t>1 ; \quad y(0)=0, \quad y^{\prime}(0)=0$