$$\text { Find all subgroups of order } 3 \text { in } Z _ { 9 } \oplus Z _ { 3 }$$

An initial value problem and its exact solution $y(x)$ is given. Apply Euler’s method twice to approximate to this solution on the interval $\left[ 0 , \frac { 1 } { 2 } \right]$, first with step size $h = 0.25$, then with step size $h = 0.1$. Compare the three-decimal-place values of the two approximations at $x = \frac { 1 } { 2 }$ with the value $y\left(\frac{1}{2}\right)$ of the actual solution.

$$y ^ { \prime } = - 2 x y , y ( 0 ) = 2 ; y ( x ) = 2 e ^ { - x ^ { 2 } }$$

If you have a parametric equation grapher, graph the equations over the given intervals. x = sec t (enter as 1/cos(t)), y = tan t (enter as sin(t)/cos(t)), over

$$\left. \begin{array} { l } a.{ - 1.5 \leq t \leq 1.5 } \\ b.{ - 0.5 \leq t \leq 0.5 } \\ c.{ - 0.1 \leq t \leq 0.1 } \end{array} \right.$$

$$\text { List the elements of the subgroups } \langle 3 \rangle \text { and } \langle 7 \rangle \text { in } U ( 20 )$$

6x + 2y = -12 4x + 3y = 7

Let G be an abelian group of order 72. a. Can you say how many subgroups of order 8 G has? Why, or why not? b. Can you say how many subgroups of order 4 G has? Why, or why not?

Show directly that the given function is linearly dependent on the real line. That is, find a nontrivial linear combination of the given function that vanishes identically.

$$f ( x ) = 2 x , g ( x ) = 3 x ^ { 2 } , h ( x ) = 5 x - 8 x ^ { 2 }$$

An investment of $\$5000$ doubles in value every decade. The function $f(x)=5000\cdot 2^x$, where $x$ is the number of decades, models the growth of the value of the investment. How much is the investment worth after $30$ yr?

An equation is given in spherical coordinates. Express the equation in rectangular coordinates and sketch the graph. θ = π/3

Factor the expression, if possible. Factor out any GCF first (including -1 if the leading coefficient is negative).

$$25 m ^ { 8 } - 60 m ^ { 4 } n + 36 n ^ { 2 }$$

In $Z _ { 24 } ,$ list all generators for the subgroup of order $8 .$ Let $G = \langle a \rangle$ and let $| a | = 24 .$ List all generators for the subgroup of order $8 .$

Find the general solution of the given system.

$$\mathbf{X}^{\prime}=\left(\begin{array}{rr}{12} & {-9} \\ {4} & {0}\end{array}\right) \mathbf{X}$$

Given that f(x) is a quadratic function, is its inverse a function? Explain.

Solve each problem, and express the answer in scientific notation. a. $\left(4 \times 10^{2}\right) \times\left(1 \times 10^{8}\right)$ b. $\left(2 \times 10^{-4}\right) \times\left(3 \times 10^{2}\right)$ c. $\left(6 \times 10^{2}\right) \div\left(2 \times 10^{\prime}\right)$ d. $\left(8 \times 10^{4}\right) \div\left(4 \times 10^{1}\right)$