This problem, although not technically complex, requires some understanding of the derivative and how it relates to elementary physics. 

For part a)

Given a function that returns an objects position at time t, we can take the derivative of the position function to find the velocity function

We are given the position function:

We can apply the general power rule for derivatives to find the derivative of this function:

And from our understanding that the derivative of the position function is the velocity function, we can make the substitution

Thus, our part a answer is:

For part b)

Furthermore, we can take the derivative of the velocity function to find our acceleration function.

We found the velocity function to be:

Thus, the derivative of the velocity function (the acceleration function) is simply:

The full answer should be:

To answer this question, we must first understand how to find average velocity:

We can figure out the position values by plugging in our final and initial times to our position equation as follows:

We can repeat this process at the first endpoint for the initial position:

Now that we have both the final position and the initial position, we simply plug in to find our average velocity:

Thus, the correct answer should be:

Given that h(t) represents the height of a frisbee as a function of time, find the equation which models the frisbee's velocity as a function of time.

To find the velocity function from a height or position function, we simply need to find the first derivative of the function.

Recall the power rule:

This states that we can find the derivative of any polynomial by taking each term, multiplying by its exponent, and then subtracting one from the exponent.

Doing so yields:

Notice that our 14 dropped out. Any constant terms will drop our while differentiating.

Change it to v(t) and we are all set

Given that h(t) represents the height of a frisbee as a function of time, find the  frisbee's acceleration function.

To find the acceleration function, we need to recall that velocity is the first derivative of position, and that acceleration is the first derivative of velocity. In other words, we need to find our second derivative.

Recall the power rule:

This states that we can find the derivative of any polynomial by taking each term, multiplying by its exponent, and then subtracting one from the exponent.

Doing so yields:

Notice that our 14 dropped out. Any constant terms will drop our while differentiating.

Change it to v(t) and we are almost there.

Next, find the derivative of v(t), again with the power rule.

Making our answer:

The equation given is the rate of change of temperature itself, so to find the cooling rate at a temperature (T) of 29, we simply plug this into the rate function:

In order to find the time at which the banana death rate is 31, we must take the derivative of the function for dead bananas, whose rate of change - the death rate - is given by:

The derivative was found using the following rules:

, 

To find the time at which the banana death rate is equal to 31, we set the derivative function equal to 31 and solve for t:

The velocity is determined by taking the derivative of the position function, as velocity is the rate of change of position:

The derivative was found using the following rules:

, 

Now, we simply plug in the given t value into the velocity function:

To find the rate of change of the function at any instant, we must take its first derivative:

The derivative was found using the following rule:

Now, to find the value of t at which the rate of change is equal to 3, we use algebra:

The problem statement tells us that t cannot be negative, so our only answer is

The rate of change of mass in the bag at any instant is given by the first derivative of the function:

which was found using the following rule:

Notice that the rate of change is a constant value, not dependent on t. (You can compare the original equation to a line which has a slope of 2; the rate of change of the line is always 2, regardless of the t value you are at!)

The rate of change of the amount of bacteria in the dish at any time, t, is given by the first derivative:

and was found using the following rules:

, 

Evaluating the first derivative at t=30, we get