Would your answer change if your decision saved your company $\$ 1$ million?

Prove Corollary 58.1.

Corollary 58.1: Let P be a finite partially ordered set, and let x and y be distinct elements of P. If x$\lt$y in all linear extensions of P, then x$\lt$y in P. Conversely, if x$\lt$y in one linear extension, but x$\gt$y in another, then x and y are incomparable in P.

Without solving $x^2+10^{2.4}=0$, decide how many solutions there are to the equation.

Check your solutions after solving this equation.

$$\dfrac{11}{q}-3=\dfrac{1}{q}$$

Check your solutions after solving this equation.

$$\dfrac{4}{y}+\dfrac{1}{y}=2$$

What impact did Ezana's decision to become a Christian have on the kingdom of Aksum?

Find an expression for f ( x ) by solving for y and replacing y with f ( x ). Then find the domain of f and compute f(4). $x y - 3 = 2 y$

Graph each equation by making a table.

$$x=-2$$

In the following problem, find the Wronskian of two solutions of the given differential equation without solving the equation.

$$t^2 y^{\prime \prime}-t(t+2) y^{\prime}+(t+2) y=0$$

Work this exercise without pencil and paper. Do not use a calculator.

Consider the following problem. One number is three less than six times a second number. Their sum is 32. Find the numbers. If $x$ represents the second number, which equation is correct for solving this problem?

A. $32-(x+3)=6 x$

B. $(3-6 x)+x=32$

C. $32-(3-6 x)=x$

D. $(6 x-3)+x=32$

Are every two triangular prisms similar? Explain your decision.

Solve graphically and then check by solving algebraically.

$$4 x-1=3-2 x$$

Solve the problem $\operatorname{PDE} \quad u_{t}=u_{x x}-u+x \quad 0<x<1 \quad 0<t<\infty$

$$\text { BCs } \quad\left\{\begin{array}{l} {u(0, t)=0} \\ {u(1, t)=1} \end{array}\right. 0<t<\infty$$

$\text { IC } \quad u(x, 0)=0 \quad 0 \leqslant x \leqslant 1$ by
 (a) changing the nonhomogeneous BCs to homogeneous ones. (b) transforming into a new equation without the term -u. (c) solving the resulting problem.

Explain the difference in the techniques that are used when solving equations and verifying identities.