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.

Let

$$y[n]=(1/9)^nu[n]$$

. Determine two distinct signals sch that each has a z-transform X(z) which satisfies both of the following conditions: 1. [X(z)+X(-z)]/2=Y(z²). 2. X(z) has only one pole and only one zero in the z-plane.

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

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

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

Suppose that you have a double integral over a region R in the xy-plane and you want to transform that integral into an equivalent double integral over a region S in the uv-plane. Describe the procedure you would use.

Suppose you have used Gaussian elimination to transform the augmented matrix of a linear system into row-echelon form. How can you tell if the system has

(a) exactly one solution?

(b) no solution?

(c) infinitely many solutions?

Let $y=x^{7 / 3}$. Transform this equation into an implicit relation involving only integer exponents. Then differentiate implicitly to find $y^{\prime}$. Show that your answer agrees with the answer found directly, using the formula for the derivative of a power on the original equation.

Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. $y^{\prime \prime}+9 y=\cos 3 t, \quad y(0)=2, \quad y^{\prime}(0)=5$

describe how to transform the graph of an appropriate monomial function

$$f(x)=x^n$$

into the graph of the given polynomial function. Sketch the transformed graph by hand and support your answer with a grapher. Compute the location of the y-intercept as a check on the transformed graph.

$$g(x)=-(x+5)^3$$

Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $Y(s)=\frac{7 s^{2}+3 s+16}{(s+1)\left(s^{2}+4\right)}$

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.

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

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

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

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. $y^{\prime \prime}+16 y=1, \quad y(0)=1, y^{\prime}(0)=2$