Add or subtract.
−4 + 3 − (−5)

Determine the interval(s) on which the vector-valued function is continuous.

$$\mathbf{r}(t)=\left\langle t, \frac{1}{t}\right\rangle$$

Show that

$$\mathbb{Z}_{18} / M \cong \mathbb{Z}_{6},$$

where M is the cyclic subgroup $\langle 6 \rangle$.

Determine which argument is correct. Explain your reasoning.

Argument 1: If it is noon on Monday, then the children are in school. The children are in school. So, it is noon on Monday.

Argument 2: If it is noon on Monday, then the children are in school. It is noon on Monday. So, the children are in school.

Consider the following position functions r and R for two objects.

$$\mathbf { r } ( t ) = \left\langle 4 + t ^ { 2 } , 3 - 2 t ^ { 4 } , 1 + 3 t ^ { 6 } \right\rangle , [ a , b ] = [ 0,6 ]$$

$$\mathbf { R } ( t ) = \left\langle 4 + \ln t , 3 - 2 \ln ^ { 2 } t , 1 + 3 \ln ^ { 3 } t \right\rangle \text { on } [ c , d ]$$

Find the interval [c, d] over which the R trajectory is the same as the r trajectory over [a, b].

Cole had $\frac{7}{8}$ of a tank of gas in the lawn mower. After mowing the grass, he had $\frac{1}{4}$ of a tank. What fraction of a tank did Cole use mowing the lawn?

For each real symmetric matrix $A_1$ i. determine an orthogonal matrix $Q$ such that $Q^T A Q$ is diagonal; ii. use the result of part $i$ to find $A^5$.

(a)

$$\mathrm{A}=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]~~~~~~$$

(b)

$$A=\left[\begin{array}{rrr}3 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 3\end{array}\right]$$

(c)

$$A=\left[\begin{array}{lll}3 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 3\end{array}\right]$$

(d)

$$A=\left[\begin{array}{rrrr}-1 & -4 & 4 & 0 \\ -4 & -5 & 0 & -4 \\ 4 & 0 & -5 & -4 \\ 0 & -4 & -4 & -1\end{array}\right]$$

.

Round

$$4 \frac { 3 } { 10 }$$

to the nearest whole number.

Refer to $c(x)=x^{3}-4 x^{2}-2 x+8$ and $d(x)=x^{3}+3 x^{2}-4$. Find the horizontal asymptote or slant asymptote of the graph of

$$g(x)=\frac{c(x)}{d(x)}=\frac{x^{3}-4 x^{2}-2 x+8}{x^{3}+3 x^{2}-4}$$

Prove:

Let $\langle a\rangle$ be a cyclic group of order $n$. If $n$ and $m$ are relatively prime, then the function $f(x)=x^m$ is an automorphism of $\langle a\rangle$.

Add.

$$78.87 + 54 + 32.9569 + 0.0043$$

Latisha had graphed the polynomial $y = x ^ { 3 } - 5 x ^ { 2 } + 8 x – 6$ during class and remembered that it had one real root at x = 3. Explain how you could use division to find the other roots, and then find them.

What would happen to the organisms in a river with a fast current if a dam was built? Explain your answer.

Write each expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. $\sin ^{2} \theta+\tan ^{2} \theta+\cos ^{2} \theta$