We can represent the velocity as the instantaneous rate of change of a particles motion, that is . We solve the problem by taking the derivative of the function that describes the particles motion with respect to time

We then evaluate the velocity equation at the specified time, which in this case was 

The velocity is the instantaneous rate of change of the particles position with respect to time (), so by taking the derivative of the function we can find the particles velocity ant any point in time

Evaluating at the time we specified in the problem statement, we get

To determine the rate of change of the amount of bacteria at a specific time, we must take the derivative of the function:

The derivative was found using the following rules:

, , 

Now, we are asked to find the time corresponding to a certain rate of growth of the bacteria. This is the output of the derivative function, from a certain time input into the function. So, we must solve for the time that gets us 38 as our output:

To find the velocity of the electron, we take the derivative of its position equation. Velocity is the instantaneous rate of change ofposition with respect to time, which we will evaluate at a specified time

Evaluating when 

The cars velocity is the instantaneous rate of change of the position with respect to time, , so by taking the derivative of the function we can get the velocity equation which can be evaluated for any time 

Evaluated at , we get

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

To find the velocity function from a height or position function, we simply need to find the first derivative of the function. Then, to find the velocity at a given value of t, simply plug the value of t into the velocity 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 almost there.

We need v(1.5)

So, we get -264.75. This might seem like a strange velocity for a frisbee, but we don't know the units, so it might make sense within the relevant system.

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

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. Once we have that, we need to plug in -1 for t.

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 acceleration function:

Now, finally, plug in -1 for t and solve.

So, our acceleration is:

-754

We have no units, so although this number may seem a little strange, we can;t really say much about it.