### All Calculus 1 Resources

## Example Questions

### Example Question #1 : Decreasing Intervals

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

**Possible Answers:**

Never

Always

**Correct answer:**

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 #2 : Decreasing Intervals

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

**Possible Answers:**

Always

Never

**Correct answer:**

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 #3 : Decreasing Intervals

Is increasing or decreasing on the interval ?

**Possible Answers:**

Increasing because the second derivative is positive on the interval .

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

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

The function is neither increasing nor decreasing on the interval .

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

**Correct answer:**

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

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 #4 : Decreasing Intervals

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

**Possible Answers:**

Decreasing, because is negative on the interval .

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

Decreasing, because is negative on the interval .

Increasing, because is positive on the interval .

Increasing, because is positive on the interval .

**Correct answer:**

Increasing, because is positive on the interval .

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 #5 : Decreasing Intervals

Find the intervals where the following function is decreasing.

**Possible Answers:**

**Correct answer:**

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 #6 : Decreasing Intervals

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

**Possible Answers:**

Interval B

Interval E

Interval C

Interval D

Interval A

**Correct answer:**

Interval E

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 #7 : Decreasing Intervals

Let .

On which open interval(s) is **decreasing?**

**Possible Answers:**

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

**Correct answer:**

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 #8 : Decreasing Intervals

On what interval(s) is the function decreasing?

**Possible Answers:**

**Correct answer:**

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 #9 : Decreasing Intervals

For which values of is the function decreasing?

**Possible Answers:**

and

and

and

and

This function is never decreasing.

**Correct answer:**

and

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 #10 : Decreasing Intervals

For which values of is the function decreasing?

**Possible Answers:**

,

and

and

This function is never decreasing.

and

**Correct answer:**

,

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

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