AP Calculus AB : Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ'

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

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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:

Decreasing: 

Increasing: 

Increasing: 

Decreasing:  

Increasing: 

Decreasing:  

Increasing: 

Decreasing: 

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 #2 : Relationship Between The Increasing And Decreasing Behavior Of ƒ And The Sign Of ƒ'

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

Possible Answers:

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

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

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.

Decreasing, because our first derivative is positive.

Increasing, 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 #4 : 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:

Increasing

Increasing

Decreasing

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 #2 : 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 increasing, because 

f(x) is decreasing, 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 

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

Determine the intervals on which f is decreasing, for its entire domain:

Possible Answers:

The function is never decreasing

Correct answer:

Explanation:

To determine the intervals on which the function is decreasing, we must determine the intervals on which the function's first derivative is negative.

The first derivative of the function is equal to

and was found using the following rule:

Next, we must find the critical values, at which the first derivative is equal to zero:

Using the critical values, we now create intervals over which to evaluate the sign of the first derivative:

Notice how at the bounds of the intervals, the first derivative is neither positive nor negative.

On the first interval, the first derivative is negative, while on the remaining intervals, the first derivative is positive. Thus, the function is decreasing on the first interval, 

Example Question #7 : Relationship Between The Increasing And Decreasing Behavior Of ƒ And The Sign Of ƒ'

Determine the intervals on which f is decreasing:

Possible Answers:

The function is never decreasing

Correct answer:

The function is never decreasing

Explanation:

To determine the intervals on which the function is decreasing, we must determine the intervals on which the function's first derivative is negative.

The first derivative of the function is equal to

and was found using the following rule:

Next, we must find the critical values, at which the first derivative is equal to zero:

The square root of a negative number is not valid for a critical value, so we only have one critical value for this function.

Using the critical value, we now create intervals over which to evaluate the sign of the first derivative:

Notice how at the bounds of the intervals, the first derivative is neither positive nor negative.

On the first interval, the first derivative is positive, and on the second interval, the first derivative is positive as well. The function is therefore never decreasing because the first derivative is never negative. 

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

Determine the intervals on which f is increasing:

Possible Answers:

The function is never increasing

Correct answer:

Explanation:

To determine the intervals on which the function is increasing, we must determine the intervals on which the function's first derivative is positive.

The first derivative of the function is equal to

and was found using the following rules:

, ,  

Next, we must find the critical values, at which the first derivative is equal to zero:

Note that we stopped writing critical values based on the given interval in the problem statement.

Using the critical values, we now create intervals over which to evaluate the sign of the first derivative:

Notice how at the bounds of the intervals, the first derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the first derivative function, we find that on the first interval, the first derivative is negative, while on the second interval, the first derivative is positive. Thus, the function is increasing on the second interval, .

Example Question #5 : Relationship Between The Increasing And Decreasing Behavior Of ƒ And The Sign Of ƒ'

Determine the local minima of the following function:

Possible Answers:

The function has no local minima 

Correct answer:

The function has no local minima 

Explanation:

To determine the local minima of the function, we must determine the points at which the function's first derivative changes from negative to positive.

The first derivative of the function is

and was found using the following rules:

Next, we must find the critical values, at which the first derivative is equal to zero:

Using the critical value as an endpoint, we now create intervals over which to evaluate the sign of the first derivative:

Notice how at the bounds of the intervals, the first derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the first derivative function, we find that on the first interval, the first derivative is negative, and on the second interval, the first derivative is also negative. The first derivative never changed in sign from negative to positive, thus the function has no local minima.

 

 

Example Question #10 : Relationship Between The Increasing And Decreasing Behavior Of ƒ And The Sign Of ƒ'

Find the local maxima for the following function:

Possible Answers:

No local maxima exist

Correct answer:

No local maxima exist

Explanation:

To determine the local maxima of the function, we must determine the points at which the function's first derivative changes from positive to negative.

The first derivative of the function is

and was found using the following rules:

Next, we must find the critical values, at which the first derivative is equal to zero:

Using the critical value as an endpoint, we now create intervals over which to evaluate the sign of the first derivative:

Notice how at the bounds of the intervals, the first derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the first derivative function, we find that on the first interval, the first derivative is positive, and on the second interval, the first derivative is positive as well. There is no sign change of the first derivative, thus there are no local maxima (the function is always increasing).

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