### All Calculus 1 Resources

## Example Questions

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

On what intervals does f(x) = (1/3)x^{3} + 2.5x^{2 }– 14x + 25 increase?

**Possible Answers:**

(2, ∞)

(–7, 2), and (2, ∞)

(–∞, –7), (–7, 2), and (2, ∞)

(–∞, –7) and (2, ∞)

(–∞, –7)

**Correct answer:**

(–∞, –7) and (2, ∞)

We will use the tangent line slope to ascertain the increasing / decreasing of f(x). To this end, let us begin by taking the first derivative of f(x):

f'(x) = x^{2} + 5x – 14

Solve for the potential relative maxima and minima by setting f'(x) to 0 and solving:

x^{2} + 5x – 14 = 0; (x – 2)(x + 7) = 0

Potential relative maxima / minima: x = 2, x = –7

We must test the following intervals: (–∞, –7), (–7, 2), (2, ∞)

f'(–10) = 100 – 50 – 14 = 36

f'(0) = –14

f'(10) = 100 + 50 – 14 = 136

Therefore, the equation increases on (–∞, –7) and (2, ∞)

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

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

**Possible Answers:**

Always

Never

**Correct answer:**

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

First, take the derivative:

Set equal to 0 and solve:

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

Since the values that are positive is when x=-6 and 2, the interval is increasing on the intervals that include these values. Therefore, our answer is:

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

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

**Possible Answers:**

Always

Never

**Correct answer:**

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

First, take the derivative:

Set equal to 0 and solve:

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

Since the value that is positive is when x=0 and 10, the interval is increasing in both of those intervals. Therefore, our answer is:

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

Is increasing or decreasing on the interval ?

**Possible Answers:**

Decreasing. on the interval .

Cannot be determined from the information provided

Increasing. on the interval .

Decreasing. on the interval .

Increasing. on the interval .

**Correct answer:**

Increasing. on the interval .

To find increasing and decreasing intervals, we need to find where our first derivative is greater than or less than zero. If our first derivative is positive, our original function is increasing and if g'(x) is negative, g(x) is decreasing.

Begin with:

If we plug in any number from 3 to 6, we get a positve number for g'(x), So, this function must be increasing on the interval {3,6}, because g'(x) is positive.

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

Is increasing or decreasing on the interval ?

**Possible Answers:**

Decreasing, because is negative.

Increasing, because is negative.

is neither increasing nor decreasing on the given interval.

Decreasing, because is positive.

Increasing, because is positive.

**Correct answer:**

Increasing, because is positive.

To find out if a function is increasing or decreasing, we need to find if the first derivative is positive or negative on the given interval.

So starting with:

We get:

using the Power Rule .

Find the function on each end of the interval.

So the first derivative is positive on the whole interval, thus g(t) is increasing on the interval.

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

Is the following function increasing or decreasing on the interval ?

**Possible Answers:**

Increasing, because is positive on the given interval.

Decreasing, because is negative on the given interval.

Decreasing, because is positive on the given interval.

Increasing, because is negative on the given interval.

The function is neither increasing nor decreasing on the interval.

**Correct answer:**

Increasing, because is positive on the given interval.

A function is increasing on an interval if for every point on that interval the first derivative is positive.

So we need to find the first derivative and then plug in the endpoints of our interval.

Find the first derivative by using the Power Rule

Plug in the endpoints and evaluate the function.

Both are positive, so our function is increasing on the given interval.

### Example Question #7 : How To Find Increasing Intervals By Graphing Functions

On which intervals is the following function increasing?

**Possible Answers:**

**Correct answer:**

The first step is to find the first derivative.

Remember that the derivative of

Next, find the critical points, which are the points where or undefined. To find the points, set the numerator to , to find the undefined points, set the denomintor to . The critical points are and

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

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

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

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

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

So the second and the last ranges are the ones where is increasing.

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

Below is the complete graph of . On what interval(s) is increasing?

**Possible Answers:**

**Correct answer:**

is increasing when is positive (above the -axis). This occurs on the intervals .

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

**Function A**

**Function B**

**Function C**

**Function D**

**Function E**

5 graphs of different functions are shown above. Which graph shows an **increasing/non-decreasing** function?

**Possible Answers:**

Function E

Function D

Function B

Function A

Function C

**Correct answer:**

Function E

A function is increasing if, for any , (i.e the slope is always greater than or equal to zero)

Function E is the only function that has this property. Note that function E is increasing, but not *strictly increasing*

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

Find the increasing intervals of the following function on the interval :

**Possible Answers:**

**Correct answer:**

To find the increasing intervals of a given function, one must determine the intervals where the function has a positive *first *derivative. To find these intervals, first find the critical values, or the points at which the first derivative of the function is equal to zero.

For the given function, .

This derivative was found by using the power rule

.

When set equal to zero, . Because we are only considering the open interval (0,5) for this function, we can ignore . Next, we look the intervals around the critical value , which are and . On the first interval, the first derivative of the function is negative (plugging in values gives us a negative number), which means that the function is decreasing on this interval. However for the second interval, the first derivative is positive, which indicates that the function is increasing on this interval .

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