Solve each equation. $\frac{x}{3}-7=6-\frac{3 x}{4}$

Find the Laplace transform of the given piecewise defined function.

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

$$2 x + 3 y = 12$$

$$x - y = 1$$

Find the solution of the initial value problem. Discuss the interval of existence and provide a sketch of your solution. $(2 x+3) y^{\prime}=y+(2 x+3)^{1 / 2}$, y(-1)=0

Let $f(x)=x^{2}+1.$ Find (a) $f((1,3]).$ (b) $f([-1,0] \cup[2,4]).$ (c) $f^{-1}([-1,1])$ (d) $f^{-1}([-2,3])$ (e) $f^{-1}([5,10])$ (f) $f^{-1}([-1,5] \cup[17,26])$

Determine the values of a and b that satisfy the equation. 5 - 4i = (a + 3) + (b - 1)i

Determine whether each of the following arguments is valid or invalid. If it is valid, give a proof. If it is invalid, give an assignment of truth values to the variables that makes the premises true and the conclusion false. The Yankees will be in the World Series or the Phillies won't be there. If the Phillies are not in the World Series, the National League cannot win. In fact, the National League wins. Therefore, the Yankees are in the World Series.

Solve the equation and check your solution. $\sqrt{y}=7$

Given the function $\phi$, place the ordinary differential equation $\phi\left(t, y, y^{\prime}\right)=0$ in normal form. $\phi(x, y, z)=x z-2 y-x^{2}$

A soup kitchen makes 16 gallons of soup. Each day, a quarter of the soup is served and the rest is saved for the next day. Write the first five terms of the sequence of the number of fluid ounces of soup left each day. Write an equation that represents the nth term of the sequence. When is all the soup gone?

Find the general solution of the given differential equation. In each case, sketch at least six members of the family of solution curves. $y^{\prime}=t^{2} e^{3 t}$

Rationalize the denominator and simplify further, if possible.

$$\frac{21 x^{2}}{\sqrt{7 x}}$$

A local used-car dealer gets complaints about his cars as shown in the table below. Find the expected number of complaints per day.

$$\begin{array}{l|ccccccc} \text { Number of } & & & & & & & \\ \text { Complaints } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \text { per Day } & & & & & & & \\ \hline \text { Probability } & 0.02 & 0.06 & 0.16 & 0.25 & 0.32 & 0.13 & 0.06 \end{array}$$

The manufacturer of a popular personal computer has orders from two dealers. Dealer $D_{1}$ wants at least 32 computers, and dealer $D_{2}$ wants at least 20 computers. The manufacturer can fill the orders from either of two warehouses, $\mathrm{W}_{1}$ or $\mathrm{W}_{2}.$ There are 25 computers on hand at $\mathrm{W}_{1},$ and 30 at $\mathrm{W}_{2}.$ The costs (in dollars) to ship one computer to each dealer from each warehouse are given in the following table.

$$\begin{array}{lccc} & & {\text { Dealer }} \\ \hline & & D_{1} & D_{2} \\ \text {Warehouse}& W_{1} & \$ 14 & \$ 12 \\ & W_{2} & \$ 12 & \$ 10 \end{array}$$

(a) How should the orders be filled to minimize shipping costs? (b) Find and interpret the values of any nonzero slack or surplus variables.