Calculus 1 : Decreasing Intervals

Example Questions

Example Question #1 : Intervals

Find the interval(s) where the following function is decreasing. Graph to double check your answer.

Never

Always

Explanation:

To find when a function is decreasing, you must first take the derivative, then set it equal to 0, and then find between which zero values the function is negative.

First, take the derivative:

Set equal to 0 and solve:

Now test values on all sides of these to find when the function is negative, and therefore decreasing. I will test the values of -6, 0, and 2.

Since the only value that is negative is when x=0, the interval is only decreasing on the interval that includes 0. Therefore, our answer is:

Example Question #1 : How To Find Decreasing Intervals By Graphing Functions

Find the interval(s) where the following function is decreasing. Graph to double check your answer.

Never

Always

Explanation:

To find when a function is decreasing, you must first take the derivative, then set it equal to 0, and then find between which zero values the function is negative.

First, take the derivative:

Set equal to 0 and solve:

Now test values on all sides of these to find when the function is negative, and therefore decreasing. I will test the values of 0, 2, and 10.

Since the only value that is negative is when x=0, the interval is only decreasing on the interval that includes 2. Therefore, our answer is:

Example Question #1 : How To Find Decreasing Intervals By Graphing Functions

Is  increasing or decreasing on the interval ?

The function is neither increasing nor decreasing on the interval .

Increasing because the second derivative is positive on the interval .

Decreasing, because the first derivative of  is negative on the function .

Increasing, because the first derivative is positive on the interval .

Decreasing, because the first derivative is positive on the interval .

Decreasing, because the first derivative of  is negative on the function .

Explanation:

To find the an increasing or decreasing interval, we need to find out if the first derivative is positive or negative on the given interval. So, find  by decreasing each exponent by one and multiplying by the original number.

Next, we can find  and  and see if they are positive or negative.

Both are negative, so the slope of the line tangent to  is negative, so  is decreasing.

Example Question #1 : Intervals

Is  increasing or decreasing on the given interval? How do you know?

Decreasing, because  is negative on the interval .

There is not enough information to tell whether  is increasing or decreasing on the interval .

Increasing, because  is positive on the interval .

Decreasing, because  is negative on the interval .

Increasing, because  is positive on the interval .

Increasing, because  is positive on the interval .

Explanation:

Recall that a function is increasing at a point if its first derivative is positive, and a function is decreasing if its first derivative is negative at that point. Therfore, we should start by finding f'(x). However, I will start by combining like terms and putting f(x) in standard form:

Next, plug in each of our endpoints to see what the sign of f'(x) is.

So f'(x) is positive on the given interval, so we know that f(x) is increasing on the given interval.

Example Question #2 : How To Find Decreasing Intervals By Graphing Functions

Find the intervals where the following function is decreasing.

Explanation:

The first step is to find the first derivative.

We can factor a  out to get

.

Now we need to solve for when  to get the critical points. Notice how factoring the 2 made the expression a little easier to simplify.

The final step is to try points in all the regions  to see which range gives a negative value for .

If we plugin in a number from the first range into , i.e , we get a positive number.

From the second range, , we get a negative number.

From the third range, , we get a positive number.

So the second range gives us values where the function is decreasing because  is negative during that range, so  is the answer.

Example Question #1 : Intervals

On which interval is the function shown in the above graph strictly decreasing?

Interval D

Interval A

Interval E

Interval C

Interval B

Interval E

Explanation:

A function  is strictly decreasing on an inverval if, for any  in the interval,  (i.e the slope is always less than zero)

Interval E is the only interval on which the function shows this property.

Example Question #2 : How To Find Decreasing Intervals By Graphing Functions

Let .

On which open interval(s) is  decreasing?

There are no open intervals on which the function is decreasing.

Explanation:

is decreasing on intervals where .

First, differentiate .

Then, find the values for x for which the derivative is negative by solving

.

Next, test the intervals.

Test them by substituting values for x:

Substitute -2,

.

The function is increasing on this interval since the derivative is positive on this interval.

Substitute 0,

.

The function is deecreasing on this interval since the derivative is negative on this interval.

Substitute 2,

.

The function is increasing on this interval since the derivative is positive on this interval.

Thus,  is the only interval on which the function is decreasing.

Example Question #4 : How To Find Decreasing Intervals By Graphing Functions

On what interval(s) is the function  decreasing?

Explanation:

The function is decreasing when the first derivative is negative. We first find when the derivative is zero. To find the derivative, we apply the quotient rule,

.

Therefore the derivative is zero at . To find when it is negative plug in test points on each of the three intervals created by these zeros.

For instance,

.

Hence the function is decreasing on

.

Example Question #1 : How To Find Decreasing Intervals By Graphing Functions

For which values of  is the function decreasing?

and

and

and

and

This function is never decreasing.

and

Explanation:

To determine where the function is decreasing, differentiate it:

What we are interested in are the points where . To determine these points, factor the equation:

this has solutions at

This splits the graph into 4 regions, and we can test points in each to determine if is greater than or less than 0. If it is less than zero, the function is decreasing.

negative/decreasing

positive/increasing

negative/decreasing

positive/increasing

Example Question #1 : How To Find Decreasing Intervals By Graphing Functions

For which values of  is the function decreasing?

and

This function is never decreasing.

and

,

and

,

Explanation:

The function is decreasing where . To determine where this is happening, differentiate the function and find where . This will split the function into intervals where it is either increasing or decreasing.

To determine where this equals zero, factor:

this has solutions for .

Test a point in each region to determine if it is increasing or decreasing within these bounds:

positive/increasing

negative/decreasing

negative/decreasing

positive/increasing