Let P be the point in 3-space with rectangular coordinates

$$( \sqrt { 2 } , - \sqrt { 2 } , 2 \sqrt { 3 } )$$

. (a) Cylindrical coordinates for P are

$$( r , \theta , z )$$

=___, (b) Spherical coordinates for P are

$$( \rho , \theta , \phi )$$

=___

Sketch the graph of each quadratic function and compare it with the graph of

$$y = x^2$$

.

$$g(x) = (x-3/2)^2$$

Exactly 1.0 L of solution of acetic acid gives the same color with methyl red as 1.0 L of a solution of hydrochloric acid. Which solution will neutralize the greater amount of 0.10 M NaOH(aq)? Exp lain.

In the 18th century, the trade winds played an important role in England's merchant fleet crossing the Atlantic Ocean. This is how the trade winds got their name. In the tropics, in which direction do the trade winds drive equatorial ocean currents? A. eastward B. northward C. southward D. westward

The research submarine Alvin, used to locate the wreck of the Titanic, descended at a rate of about 100 feet per minute. Write an integer to represent the distance Alvin traveled in 5 minutes.

Saul solves the equation $(2 x+5)^{2}=6 x+15$ by letting y = 2x + 5. His work is shown below.

$$\begin{aligned}(y)^{2} &=3(y) \\ y^{2}-3 y &=0 \\ y(y-3) &=0 \end{aligned}$$

y = 0 and y = 3, so the solutions to the equation are x = 0 and x = 3. Describe Saul's mistake in his solution. What are the actual solutions?

A 1.50 kg snowball is fired from a cliff 11.5 m high. The snowball's initial velocity is 16.0 m/s, directed

$$41.0 ^ { \circ }$$

above the horizontal. (a) How much work is done on the snowball by the gravitational force during its flight to the flat ground below the cliff? (b) What is the change in the gravitational potential energy of the snowball-Earth system during the flight? (c) If that gravitational potential energy is taken to be zero at the height of the cliff, what is its value when the snowball reaches the ground ?

Suppose that a grandfather clock (a simple pendulum) is running slowly; that is, the time it takes to complete each cycle is greater than it should be. Should you (a) shorten or (b) lengthen the pendulum to make the clock keep correct time?

Use synthetic substitution to find P(c) for the given polynomial P(x) and the given number c.

$$P ( x ) = 6 x ^ { 3 } - x ^ { 2 } + 4 x + 3 ; c = - \frac { 1 } { 3 }$$

Of what does an ac steady-state Thevenin equivalent circuit consist? A Norton equivalent circuit? How are the values of the parameters of these circuits determined?

Find the equation of tangent line and normal line to the curve $y=x \sqrt{x+(x-1)^{2}}$ at the point $(2,2 \sqrt{3})$

Use a CAS to find the numbers $x$ at which (a) $f^\prime(x)=0$ , (b) $f^\prime(x)>0$ (c) $f^\prime(x)<0$.

$$f(x)=\dfrac{x}{\sqrt{x^2+4}}$$

A triangular brace is constructed in the shape of a right triangle. The two legs of the brace are $\sqrt{x}+5$ and $\sqrt{x}$ units long. a. What expression could be used to solve for the length, of the third side of the brace? b. Simplify the equation so it does not contain any radicals. c. Find the value of x for which the length of the third side of the brace is equal to 10.

The direction cosines of

$$\langle 2 , - 1,3 \rangle$$

are

$$\cos \alpha =$$

___,

$$\cos \beta =$$

___, and

$$\cos \gamma =$$

___.