In each of the following sentences, underline the correct spelling of the word in parentheses.

(truly, truely)

Carry out the solution process indicated in the text to derive the variation of parameters formula in $y_{p}(x)=-y_{1}(x) \int \frac{y_{2}(x) f(x)}{W(x)} d x+y_{2}(x) \int \frac{y_{1}(x) f(x)}{W(x)} d x$ from Egs. $u_{1}^{\prime} y_{1}+u_{2}^{\prime} y_{2}=0, u_{1}^{\prime} y_{1}^{\prime}+u_{2}^{\prime} y_{2}^{\prime}=f(x)$ and $y_{p}=u_{1} y_{1}+u_{2} y_{2}$ .

Show the following parts of Theorem $1.12$. Part (2)

Use $A$ to find the image of the vector $\bf{v}$.

$T$ is the reflection in the $x y$-coordinate plane in $R^3: T(x, y, z)=(x, y,-z), \mathbf{v}=(3,2,2)$.

Determine whether or not each of the given equations is exact; solve those that are exact.

$$\left(\frac{2 s-1}{t}\right) d s+\left(\frac{s-s^{2}}{t^{2}}\right) d t=0$$

Determine whether the set is a basis for R$^3$.

$$S=\{(1,0,0),(0,1,0),(0,0,1),(2,-1,0)\}$$

Work with a partner. Simplify each expression using algebra tiles. Draw algebra tile models to represent each expression. x+1+3x=

Use the properties of the derivative to find the following.

$$\frac{d}{d t}[4 \mathrm{u}(t)]$$

$$\mathbf{r}(t)=\langle 3 \sin t, 2 \cos t\rangle, \mathbf{u}(t)=\langle\sin t, 4 \cos t\rangle$$

Using your software package and the result of the previous part to determine $a d j(A) A$.

Goblin Green Paint is made by mixing blue and yellow paint in the ratio $4$ to $3$. Show and discuss at least four different ways to represent varying quantities of blue and yellow paint that will mix to make Goblin Green Paint. (These can include ways discussed in previous sections.) Discuss which aspects of proportional relationships are easier to see in some ways than in others.

Let $A$ and $B$ be $n \times n$ matrices. Prove that $(A+B)^2=A^2+A B+B A+B^2$.

Find the exact value of each expression. $\arccos (0)$

A rod of length L coincides with the interval [0, L] on the x-ax is. Set up the boundary-value problem for the temperature $u(x, t)$. The left end is held at temperature 100, and there is heat transfer from the right end into the surrounding medium at temperature zero. The initial temperature is f(x ) throughout.

Find the general solution of the linear system.

$$\begin{array}{l}{x^{\prime}=2 x+3 y} \\ {y^{\prime}=-x-2 y}\end{array}$$