AP Calculus AB : Derivative as a function

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

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Example Question #1 : Derivative As A Function

A gun sends a bullet straight up with a launch velocity of 220 ft/s. It reaches a height of s=220t-16t^2 after  seconds. What's the acceleration in of the block after it has been ejected?

Possible Answers:

Correct answer:

Explanation:

Since a=\frac{ds^2}{dt^2}, by differentiating the position function twice, we see that acceleration is constant and . Acceleration, in this case, is gravity, which makes sense that it should be a constant value!

Example Question #1 : Conceptualizing Derivatives

The speed of a car traveling on the highway is given by the following function of time:

Consider a second function:

What can we conclude about this second function?

Possible Answers:

It represents the change in distance over a given time .

It has no relation to the previous function.

It represents the total distance the car has traveled at time .

 

It represents another way to write the car's speed.

It represents the rate at which the speed of the car is changing.

Correct answer:

It represents the rate at which the speed of the car is changing.

Explanation:

Notice that the function  is simply the derivative of  with respect to time. To see this, simply use the power rule on each of the two terms. 

Therefore,  is the rate at which the car's speed changes, a quantity called acceleration.

Example Question #1 : Corresponding Characteristics Of Graphs Of ƒ And ƒ'

Find the critical numbers of the function, 

 

 

 

 

 

Possible Answers:

Correct answer:

Explanation:

1) Recall the definition of a critical point: 

The critical points of a function  are defined as points , such that  is in the domain of , and at which the derivative  is either zero or does not exist. The number  is called a critical number of .  

 

2) Differentiate 

 

3) Set to zero and solve for 

The critical numbers are, 

We can also observe that the derivative does not exist at , since the term would be come infinite. However,  is not a critical number because the original function  is not defined at . The original function is infinite at , and therefore  is a vertical asymptote of  as can be seen in its' graph, 

 

  Problem 7 plot

 

 

Further Discussion

In this problem we were asked to obtain the critical numbers. If were were asked to find the critical points, we would simply evaluate the function at the critical numbers to find the corresponding function values and then write them as a set of ordered pairs, 

 

Example Question #1 : Derivative As A Function

The function  is a continuous, twice-differentiable functuon defined for all real numbers. 

If the following are true: 

Which function could be ?

Possible Answers:

Correct answer:

Explanation:

To answer this problem we must first interpret our given conditions: 

  •  Implies the function is strictly increasing.
  •  Implies the function is strictly concave down.

We note the only function given which fufills both of these conditions is .

Example Question #2 : Corresponding Characteristics Of Graphs Of ƒ And ƒ'

A jogger leaves City  at .  His subsequent position, in feet, is given by the function:

,

where  is the time in minutes.

Find the acceleration of the jogger at  minutes.

Possible Answers:

Correct answer:

Explanation:

The accelaration is given by the second derivative of the position function:

For the given position function:

,

,

.

Therefore, the acceleration at  minutes is .  Again, note the units must be in .  

Example Question #1 : Relationship Between The Increasing And Decreasing Behavior Of ƒ And The Sign Of ƒ'

On which interval(s) is the function  increasing and on which interval(s) is it decreasing? 

Possible Answers:

Increasing: 

Decreasing:  

Increasing: 

Decreasing:  

Increasing: 

Decreasing: 

Decreasing: 

Increasing: 

Correct answer:

Increasing: 

Decreasing: 

Explanation:

First we must find out critical points by setting  equal to zero.

Then we reverse foil to get

 

These are our critical points.  We now must test values on each interval defined by our critical points in order to determine the sign of  on each interval. 

Our intervals to check are

We have many choices, but let's choose 

 

Thus on our first interval  is positive meaning  is increasing.

On our second interval  is negative meaning  is decreasing.

And on our third interval  is positive meaning  is increasing.

 

Example Question #1 : Derivative As A Function

What does the sign of the first derivative tell us about whether a function is increasing or decreasing?

Possible Answers:

The sign of the first derivative does not tell us anything about whether a function is increasing or decreasing.

If the first derivative is positive, then the function is decreasing. If the first derivative is negative, then the function is increasing.

If the first derivative is negative, then the function is decreasing, but if the first derivative is positive, then the function is neither increasing nor decreasing.

If the first derivative is negative, then the function is decreasing. If the first derivative is positive, then the function is increasing.

Correct answer:

If the first derivative is negative, then the function is decreasing. If the first derivative is positive, then the function is increasing.

Explanation:

What does the sign of the first derivative tell us about whether a function is increasing or decreasing?

The first derivative test is used to tell whether a function is increasing or decreasing at a certain point or interval. 

To use this test, first find the derivative of your function. Then, plug in the values for the point(s) and see what sign you get on your values. 

If the value of your first derivative is negative, then your function is decreasing. If the value of your first derivative is positive, your original function is increasing. If your first derivative is 0, then you have a point of inflection in your original function.

Example Question #1 : Relationship Between The Increasing And Decreasing Behavior Of ƒ And The Sign Of ƒ'

Use the first derivative test to tell whether f(c) is increasing or decreasing when c=24.

Possible Answers:

Increasing, because our first derivative is negative.

Increasing, because our first derivative is positive.

Decreasing, because our first derivative is positive.

Decreasing, because our first derivative is negative.

Correct answer:

Increasing, because our first derivative is positive.

Explanation:

Use the first derivative test to tell whether f(c) is increasing or decreasing when c=24

Begin by finding the first derivative of f(c)

Next, plug in 24 for c and find the sign of our first derivative.

Now, our first derivative is positive, so our original function must be increasing.

 

Example Question #3 : Relationship Between The Increasing And Decreasing Behavior Of ƒ And The Sign Of ƒ'

Find the derivative of g(t) and tell whether g(t) is increasing or decreasing on the interval [5,6].

Possible Answers:

Decreasing

Increasing

Increasing

Decreasing

Correct answer:

Decreasing

Explanation:

Find the derivative of g(t) and tell whether g(t) is increasing or decreasing on the interval [5,6]

First, find the derivative by decreasing each exponent by 1 and multiplying the coefficient by that number.

Next, plug in our two endpoints of our interval to see what the sign of g'(t) is.

Now, clearly these are both negative, and every point between them will be negative. This means that function g(t) is decreasing on this interval.

Example Question #4 : Relationship Between The Increasing And Decreasing Behavior Of ƒ And The Sign Of ƒ'

Tell whether f is increasing or decreasing when . How do you know?

Possible Answers:

f(x) is decreasing, because 

f(x) is increasing, because 

f(x) is increasing, because 

f(x) is decreasing, because 

Correct answer:

f(x) is decreasing, because 

Explanation:

Tell whether f is increasing or decreasing when . How do you know?

To test for increasing/decreasing, we need to find the first derivative.

In this case, we can use the power rule to do all our differentiation.

Power rule:

 

We will use this on each term in order to find our first and then second derivative.

For each term, we will decrease the exponent by 1, and then multiply by the original exponent.

Now, we need to find the sign of f'(-12). This will tell us if it is increasing or decreasing.

So, we get

So, 

f(x) is decreasing, because 

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