When asked to evaluate the function, you must realize that you are going to be plugging in a value for t. Therefore, we need to figure out what the output value is telling us when t=1. Since we are simply given the position function, adn now asked to integrate it or differentiate it, our output will simply tell us the position of our object when t=1. Thus,

When evaluating the limit through direct substitution, one ends with the indeterimate form 0/0, therefore another approach must be followed. Two approaches would work here:

1.

One could use L'Hospital's Rule, which states that if a function is indeterminate when evaluating, one should then take the derivative of the top function over the derivative of the bottom. 

In this case one would find 

2.

Alternatively, one could simply factor both the numerator and the denomenator, removing common factors.

To answer this problem one must know the Mean Value Theorem, 

First f(b),f(a), and f'(x).

Plugging into the theorem:

Now solving for x we may find our mean value

The units of the derivative of the function always follow the order of the function.  .  Therefore, the units have to be   Since we are plugging a value for time, we know that this must be the instantaneous rate of change at a specific time.  Therefore, the derivative has to be the rate of change after  of the 

To find the rate of flow, you need to differentiate the function.  This is  substituting  in for  gives the answser of .

Given the length of the cube is 10 inches, the length and the width is also 10 inches.  However, the height of the water is unknown.  Let's assume this height is .

Write the volume of the water in terms of .

Differentiate the volume equation with respect to time.

Substitute the rate of the water flow into .

The water is rising at a rate of  inches per second.

This question tells you that there is a leak of  mL/hour, and then asks you to identify how quickly the leak is causing butter to be lost.

The key is to identify the units, mL/hour, see that there are  mL/hour being moved, and recognize that the units are negative, as  mL are being removed every hour.

Thus the rate of change in the volume of the butter is .

The first step we may take is to write out the formula for the volume of a cylinder:

Where r represents the radius and h the height.

With this known we can find the rate change of volume with respect to height, by deriving these functions with respect to height:

Since we're interested in the rate change of height, the dh term, let's isolate that on one side of the equation:

dV, the rate change of volume, is given to us as 1.5 liters per second, or 1500 cm³/second.  Plugging in our known values, we can thus solve for dh:

We can determine the rate of flow by taking the first derivative of the volume equation for the time provided.

Given

, we can apply the power rule,

. 

Then . 

Therefore, at , 

 liters per minute. 

We can determine the rate of flow by taking the first derivative of the volume equation for the time provided.

Given, 

, we can apply the power rule,

 to find 

.  

Therefore, at , 

liters per minute.