Parallelogram ABCD has vertices

$$A \left( - 2 , - 5 \frac { 1 } { 2 } \right) , B \left( - 4 , - 5 \frac { 1 } { 2 } \right)$$

, C(-3, -2), and D(-1, -2). Find the vertices of parallelogram

$$A ^ { \prime } B ^ { \prime } C D ^ { \prime }$$

after a transformation of

$$2 \frac { 1 } { 2 }$$

units down.

Computer algebra systems sometimes need a helping hand from human beings. Try to evaluate

$$∫(1+ln x)√1+(x ln x)^2 dx$$

with a computer algebra system. If it doesn’t return an answer, make a substitution that changes the integral into one that the CAS can evaluate.

Simplify.

$$\frac { 3 } { 8 } - \frac { 1 } { 2 } \cdot \frac { 2 } { 3 } \cdot \frac { 3 } { 4 }$$

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term,

$$a_1$$

and common ratio, r.

$$\text { Find } a _ { 8 } \text { when } a _ { 1 } = 1,000,000 , r = 0.1$$

Let S be the graph of

$$x ^ { 2 } + z ^ { 2 } + 6 z = 16$$

in 3-space. (a) The intersection of S with the xz-plane is a circle with center ___ and radius . (b) The intersection of S with the xy-plane is two lines, x= and x=___. (c)

a. Graph y = cos x and y = sec x together for

$$- 3 \pi / 2 \leq x\leq 3 \pi / 2.$$

Comment on the behavior of sec x in relation to the signs and values of cos x. b. Graph y = sin x and y = csc x together for

$$- \pi \leq x \leq 2 \pi.$$

Comment on the behavior of csc x in relation to the signs and values of sin x.

The area of a rectangular rug is given by the trinomial

$$r^2−3r−4.$$

What are the possible dimensions of the rug? Use factoring.

As measured in the inertial frame S, a proton has four-momentum p. Also as measured in S, an observer at rest in a frame S' has four-velocity u. Show that the proton's energy, as measured by this observer, is -u • p.

Find the exact value of the following under the given conditions.

$$a.\sin ( \alpha + \beta )\ b. \cos ( \alpha - \beta )\ c. \tan ( \alpha + \beta )\ d. \sin 2 \alpha\ e. \cos \frac { \beta } { 2 }.$$

$$\tan \alpha = \frac { 4 } { 3 } , \pi < \alpha < \frac { 3 \pi } { 2 } , \text { and } \tan \beta = \frac { 5 } { 12 } , 0 < \beta < \frac { \pi } { 2 }$$

Determine the amplitude of each function. Then graph the function and y = sin x in the same rectangular coordinate system for

$$0 \leq x \leq 2 \pi.$$

y = 5 sin x

$$\begin{array} { l } { \text { Show that the correspondence } x \rightarrow 5 x \text { from } Z _ { 5 } \text { to } Z _ { 10 } \text { does not pre- } } \\ { \text { serve addition. } } \end{array}$$

The coordination number for $Mg^{2+}$ ion is usually six. Assuming this assumption holds, determine the anion coordination number in the following compounds: MgO

Solve each compound inequality.

$$2 t \leq - 4 \text { or } 7 t \geq 49$$

Select the representations that do not change the location of the given point.

$$\left( 7,140 ^ { \circ } \right)$$

$$a. \ \left( - 7,320 ^ { \circ } \right) \\ b. \ \left( - 7 , - 40 ^ { \circ } \right) \\ c. \ \left( - 7,220 ^ { \circ } \right) \\ d. \ \left( 7 , - 220 ^ { \circ } \right)$$