1. Find all critical points (

-values) for the function

and then list them (separated by commas) in the box below.

List of critical numbers: ???

Critical numbers are numbers where there is a change of sign of the derivative. Take the derivative of your function and set it equal to zero and solve for x. The x's you get will be critical values that you will set up on your number line to do your sign chart. Critical values are potential minimum and maximum values for your original function. BUT, they are NOT necessarily minimum or maximum values!
what is this mean??

2. The top of a 28 foot ladder, leaning against a vertical wall, is slipping down the wall at the rate of 4 feet per second. How fast is the bottom of the ladder sliding along the ground when the bottom of the ladder is 7 feet away from the base of the wall?

LADDER PROBLEM.doc See this attachment to see a picture of what this scenario looks like.

You will use the pathagorean theorum for your equation to solve for this.

The wall (y) is one of the legs of your triangle, the floor (x) is the other and the ladder is your hypotenuse.

So,
$\displaystyle x^2 + y^2 = 28^2$

First let's take the derivative because this is how we solve rate problems:
$\displaystyle 2x dx + 2y dy = 0$

*Now, let's look back at the equation and see which of these values we are already given:

$\displaystyle x = 7 ft$

$\displaystyle ladder = 28 ft$

$\displaystyle dy = 4 ft/sec$

*this is the rate of how fast the ladder is moving down the wall which is what we are calling "y" Now we need:
$\displaystyle y = ?$

$\displaystyle dx = ?$

We are solving for dx (the rate at which the ladder base is moving when it is 7 feet from the base of the wall.)

We can find y from the pathagorean theorum above:
$\displaystyle (7)^2 + y^2 = 28^2$

$\displaystyle y = \sqrt{735}$

So now we have everything we need to solve for dx in the derivative of the equation. Can you take it from here?
3. Gravel is being dumped from a conveyor belt at a rate of 50 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 14 feet high?

Recall that the volume of a right circular cone with height h and radius of the base r is given by