Evaluate f(3). $f ( x ) = \int _ { 0 } ^ { \infty } x ^ { - t } d t$

Prove each statement by mathematical induction.

$$\sqrt { n } < \frac { 1 } { \sqrt { 1 } } + \frac { 1 } { \sqrt { 2 } } + \cdots + \frac { 1 } { \sqrt { n } }$$

, for all integers

$$n \geq 2$$

.

A sewing machine needle moves with a rapid vibratory motion, rather like SHM, as it sews a seam. Suppose the needle moves 8.4 mm from its highest to its lowest position and it makes 24 stitches in 9.0 s. What is the maximum needle speed?

For what values of x is the product (x + 4)(x – 6) positive? Explain

A positive number

$$\epsilon$$

and the limit L of a function f at

$$+ \infty$$

are given. Find a positive number N such that |f(x)-L|<

$$\epsilon$$

if x>N.

$$\lim _ { x \rightarrow + \infty } \frac { 4 x - 1 } { 2 x + 5 } = 2 ; \epsilon = 0.1$$

Use a calculator to find the value of the expression. $( 4.1 ) ^ { 2 }$

The designations 1A through 8A used for certain families of the periodic table are helpful for predicting the charges on ions in binary ionic compounds. In these compounds, the metals generally take on a positive charge equal to the family number, and the nonmetals take on a negative charge equal to the family number minus 8. Thus the compound formed from sodium and chlorine contains $\mathrm { Na } ^ { + }$ and $\mathrm { Cl } ^ { - }$ ions and has the formula $\mathrm { NaCl } .$ Predict the formula and the name of the binary compound formed from the following pairs of elements. Rb and F

What is the type of the equation

$$u_{xx} − 4u_{xy} + 4u_{yy} = 0?$$

Show by direct substitution that u(x, y) = f (y + 2x) + xg(y + 2x) is a solution for arbitrary functions f and g.

Find the indicated term of each arithmetic sequence.

$$a _ { 1 } = - 1 , d = - 5 , n = 18$$

Exhibit all Sylow 3-subgroups of $SL_2(\mathbb{F}_3)$.

Simplify the expression. Write your answer using only positive exponents.

$$\left( \frac { 2 } { 3 } \right) ^ { 4 }$$

Tell which number is greater. 15. 80%, 7/9

$$\begin{array}{l} \text { Find } d y / d x \text { by implicit differentiation. } \end{array}$$

$$\sqrt { x + y } = x$$

sketch the curve represented by the parametric equations (indicate the orientation of the curve) x = 4 cos θ , y = 2 sin θ