AP Calculus AB : Interpretation of the derivative as a rate of change

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

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Example Question #21 : Interpretation Of The Derivative As A Rate Of Change

Given the position function:

 

 

Find the average velocity across the interval

 

Possible Answers:

Correct answer:

Explanation:

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:

          

Example Question #571 : Derivatives

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.

Possible Answers:

Correct answer:

Explanation:

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

Example Question #41 : Applications Of Derivatives

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

 

Possible Answers:

Correct answer:

Explanation:

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:

 

Example Question #24 : Interpretation Of The Derivative As A Rate Of Change

A fluid cools with time according to the equation

What is the cooling rate at a temperature of 29?

Possible Answers:

Not enough information given

Correct answer:

Explanation:

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:

Example Question #21 : Interpretation Of The Derivative As A Rate Of Change

Banana farmers notice a fungi is destroying bananas. They will use pesticides when the bananas die at a rate of 31 per day. What time will this occur if the amount of dead bananas at any time is given by 

, where t is time in days.

Possible Answers:

None of the other answers

Correct answer:

Explanation:

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:

 

Example Question #22 : Interpretation Of The Derivative As A Rate Of Change

Find the velocity at t=3 given the position function 

Possible Answers:

Correct answer:

Explanation:

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:

Example Question #23 : Interpretation Of The Derivative As A Rate Of Change

Find the time at which the rate of change of the function is equal to 3:

Possible Answers:

Correct answer:

Explanation:

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

Example Question #28 : Interpretation Of The Derivative As A Rate Of Change

Mass accumulates in a bag according to the function:

What is the rate of change of mass in the bag at t=1?

Possible Answers:

None of the other answers

Correct answer:

Explanation:

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!)

Example Question #51 : Applications Of Derivatives

The amount of bacteria in a petri dish at any time, t, is modelled by:

What is the rate of change of the amount of bacteria at t=30?

Possible Answers:

Correct answer:

Explanation:

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

Example Question #51 : Applications Of Derivatives

A family of eight wants to track how much cereal they eat. Determine how fast is the cereal in the box decreasing if the amount of cereal at any time is given by

Possible Answers:

There is not enough information given

Correct answer:

Explanation:

To find the rate of change of the amount of cereal in the box at any time, we must take the derivative of the function:

which was found using the rule

(We can tell, therefore, that the cereal is decreasing at a constant rate.)

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