To find the rate of change of crystal growth at a given time, we must take the derivative of the growth rate function and evaluate it at a specific time.

The derivative of the function is

and was found using the following rules:

, 

Now, to find the growth rate at t=5, simply plug in this value for t into the derivative function:

If p(t) gives the position of a planet as a function of time, find the function which models the planet's acceleration.

Velocity is the first derivative of position. Acceleration is the first derivative of velocity.

Therefore, all we need to do to solve this problem is to find the second derivative of p(t).

We can do this via the power rule and the rule for differentiating sine and cosine.

1) 

2) 

So, we these rules in mind, we get:

So our velocity function is:

Next, differentiate v(t) to get a(t).

So, our acceleration function is

If p(t) gives the position of a planet as a function of time, find the planet's acceleration when t=0.

Velocity is the first derivative of position. Acceleration is the first derivative of velocity.

Therefore, all we need to do to solve this problem is to find the second derivative of p(t) and then plug in 0 for t and solve.

We can do this via the power rule and the rule for differentiating sine and cosine.

1) 

2) 

So, we these rules in mind, we get:

So our velocity function is:

Next, differentiate v(t) to get a(t).

So, our acceleration function is

Next, plug in 0 and simplify.

Our answer is 12. Since we are not given any units, we can leave it as 12

The car's acceleration is the instantaneous rate of change in its velocity with respect to time (), so we can find the value of the cars acceleration at any time by taking the derivative of the velocity equation

Evaluating at , we get

We are given the function describing the position of the body given a time "t":

We are also given the interval that the function can be applied over: 

First, we are tasked to find the speed of the body at each of the endpoints (0 and 2 seconds, respectively). 

To figure this, we must understand that speed is the absolute value of velocity. 

To find the velocity function, we must take the derivative of the position function with respect to time:

Now that we have the function of velocity given a time "t", we can find the speed of the body given a time "t" by simply taking the absolute value of the velocity function. 

Now, to find the speed of the endpoints:

We can repeat the same process to find the speed at 2 seconds.

For part a, the speeds at 0 and 2 seconds are 3 and 1 [m/s], respectively. 

Part b asks us to find the acceleration at the endpoint times (0 and 2 seconds). To do this, we must first understand that to find acceleration at a time "t", we must take the derivative, with respect to time, of the velocity function. 

From part a, we found the velocity function:

Thus, to find acceleration, we derive this function with respect to time.

The result of our derivation tells us that no matter what time is plugged into the function, the acceleration shall always return 2[m/s^2].

So, for part b, the acceleration at 0 and 2 seconds are 2 and 2 [m/s^2], respectively. 

We are given the position equation:

To find the acceleration when the velocity is equal to zero, we must first find the function to describe the velocity given a time "t".

We know that the velocity function is found by deriving the position function with respect to time. 

Thus, the velocity function is given by the equation: 

To find the acceleration when the velocity is equal to zero, we must set the velocity function equal to zero and solve for the times. 

 (multiply c factor by the a coefficient)

 (divide out the original a coefficient)

Thus, the velocity is zero at both 3 and 1 seconds. The question asks for the acceleration at these times. 

To do the next part, we must understand that the acceleration function is the derivative of the velocity function, with respect to time. We already know the velocity function:

So, if we derive this function with respect to time and plug in our times, we will know the acceleration when the velocity is equal to zero. 

Our derivation tells us that given any time "t", the acceleration of the body can be found by the function:

So, to find the acceleration at times 3 and 1, we simply plug in the values to our acceleration function above. 

So, the acceleration at time 1 is -6[m/s^2] and the acceleration at time 3 is 6[m/s^2].

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: