Sketch the given function and represent it as indicated. If you have a CAS, graph approximate curves obtained by replacing ∞ with finite limits; also look for Gibbs phenomena. f(x)=x+1 if 0<x<1 and 0 otherwise; by the Fourier sine transform

So far, we have always used base 10 for a logarithmic transformation. The reason for this is that our number system is based on base 10 and it is therefore easy to logarithmically transform numbers of the form . . . , 0.01, 0.1, 1, 10, 100, 1000, . . . when we use base 10. Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship in a coordinate system whose axes are accordingly transformed so that a straight line results. $y=3^{x};$ base 3

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.

$$\sinh t(0<t<2)$$

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 $t>1 ; \quad y(0)=0, \quad y^{\prime}(0)=0$

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.

$e^t(0<t<2)$

When Jefferson assumed the presidency following the election of $1800$ , he expected to transform the national government. Describe President Jefferson's republican vision and his successes and failures in implementing it. Did subsequent Republican presidents advance the same objectives?

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.)

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).

Transform the given initial value problem

$$\begin{gather*} t \ U'' + U' + t \ U = 0, \,\,\, U(1) = 1, \,\,\, U'(1) = 0 \end{gather*}$$

into an initial value problem for the two first order equations. Then write the system in matrix form.

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.

Solve the following initial value problem by the Laplace transform. (Show the details of your work.)

$$y^{\prime \prime}+2 y^{\prime}-3 y=6 e^{-2 t}, \quad y(0)=2, \quad y^{\prime}(0)=-14$$

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. $\frac{d y}{d t}+2 y=t, \quad y(0)=-1$

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

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

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?