Determine whether the number is divisible by each of the following numbers: 2, 3, 4, 5, 6, 8, 9, 10, and 12. If you are using a calculator, explain the divisibility shown by your calculator using one of the rules of divisibility.

$$238,632$$

What is the sum of two complementary angles in degrees? in radians?

Graph each equation in a rectangular coordinate system. If two functions are indicated, graph both in the same system. Then use your graphs to identify each relation's domain and range.

$$x ^ { 2 } + y ^ { 2 } + 4 x - 6 y - 3 = 0$$

Suppose you have two triangles. Triangle A is similar to triangle B, and the measures of the sides of triangle A are less than the measures of the sides of triangle B. The scale factor is 0.25. Which is the original triangle? Explain.

Find the exact value of each expression. $\sin \frac{\pi}{18} \cos \frac{5 \pi}{18}+\cos \frac{\pi}{18} \sin \frac{5 \pi}{18}$

Compare the graph of $y=\log _{3}(x+1)$ to the graph of its inverse $y=3^{x}-1 .$ How are the graphs alike? How are they different?

A circle has a radius of 15 feet. What is the length of the arc generated by a

$$45^\circ$$

angle?

Use the graph of the function to determine whether the function is even, odd, or neither. $f(x)=\tan \left(x-\frac{\pi}{2}\right)$

Which expression shows the prime factorization of 100?

$$A. 2 ^ { 2 } \times 5 ^ { 2 }$$

$$B. 10 \times 10$$

$$C. 10 ^ { 10 }$$

$$D. 2 \times 5 \times 10$$

Suppose an automobile has 2000 J of kinetic energy. When it moves at twice the speed, what will be its kinetic energy? What’s its kinetic energy at three times the speed?

Find the determinant of the elementary matrix. (Assume k ≠ 0.)

Tell whether the function represents exponential growth or exponential decay. Then graph the function.

$$y = 0.5 e ^ { x }$$

The integral

$$\vec{a}=\int_Sd\vec{a}$$

is sometimes called the vector area of the surface S.If S happens to be flat, then $|\vec{a}|$ is the ordinary (scalar) area, obviously. Find the vector area of a hemispherical bowl of radius R. Show that $\vec{a}=0$ for any closed surface. Show that $\vec{a}$ is the same for all surfaces sharing the same boundary. Show that

$$\vec{a}=\dfrac{1}{2}\oint \vec{r}\times d\vec{l}$$

where the integral is around the boundary line. [Hint: One way to do it is to draw the cone subtended by the loop at the origin. Divide the conical surface up into infitesimal triangular wedges, each with vertex at the origin and opposite side $d\vec{l}$, and exploit the geometrical interpretation of the cross product.] Show that

$$\oint(\vec{c}\cdot\vec{r})d\vec{l}=\vec{a}\times\vec{c}$$

for any constant vector $\vec{c}$.

Prove that if g is a direct variation over the real numbers, then for every pair of real numbers a and c, g(a+c)=g(a)+g(c).