Increasing Intervals - CLEP Calculus

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:

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:

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.

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.

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.

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.

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

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

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 .

Is f(x) increasing, decreasing, or flat at ?

Recall that to find if a function is increasing or decreasing, we can use its first derivative. If f'(x) is positive, f(x) is increasing. If f'(x) is negative, f(x) is decreasing.

So, given:

We get

Then:

Therefore, f(x) is decreasing at the point, because f'(x) is negative.

Tell whether g(t) is increasing or decreasing on the interval \[4,7\]

To find increasing and decreasing, find where the first derivative is positive and negative. If g'(t) is positive, then g(t) is increasing and vice-versa.

Then,plug in the endpoints of \[4,7\] and see what you get for a sign.

So, since g'(t) is positive on the interval, g(t) is increasing.

To find the interval(s) on which the function is increasing, we must find the intervals on which the first derivative of the function is positive.

The first derivative of the function is:

and was found using the rule

Now we must find the critical value, at which the first derivative is equal to zero:

Now, we make the intervals on which we look at the sign of the first derivative:

On the first interval the first derivative is positive, while on the second it is negative. Thus, the first interval is our answer, because over this range of x values, the first derivative is positive and the function is increasing.

Take the first derivative of :

by quotient rule

is increasing whenever is positive, that is, whenever both the numerator and denominator are of the same sign. The function is certainly positive for all values of greater than because and since

is positive for all positive , it is increasing on the interval, too. It will never be negative. For the same reason, the numerator is always positive. With the numerator and denominator always positive everywhere on the given interval, the derivative is always positive and the function is always increasing. So for any interval of nonzero length within , is increasing.

NOTE: Interestingly the opposite of the choice > is also true. on the entire interval because at , we have

. So the numerator is larger to begin with, and since:

for all (or any for that matter), the derivative of the numerator is greater, too. This means the numerator will always be larger, so this condition coincides with the condition of being positive.

If is continuous everywhere and always increasing (i.e. for all ), then it must be true that after has attained its root, it can never do so again because it can't "return" to the -axis. NOTE: this is not automatically true of functions that aren't continuous. As for the other choices, the possibility of at least having one more root is automatically false and a simple counterexample to the notion thay has to have an inflection point is a simple increasing lines. It has a constant positive derivative, but possesses no upward or downward concavity and has no inflection points.

To find the intervals on which the function is increasing, we must find the intervals where the first derivative is positive. To do this, we must find the first derivative, and find its critical values (at which the first derivative is equal to zero):

The derivative was found using the following rule:

Now, write the intervals of the function for which c is the upper and lower bound:

Note that at the critical value, the derivative is neither positive nor negative.

Now, we analyze the sign of the derivative within each interval; on the first interval, the derivative is always negative, but on the second interval, the first derivative is always positive. In other words, for this set of values - - the function is increasing.

To determine the intervals on which the function is increasing, we must determine the intervals on which the first derivative of the function is positive. To start, we must find the first derivative:

The derivative was found using the following rule:

The first derivative is a positive constant, therefore the function is increasing on the entire domain, .

To find where the function is increasing, you must first find the derivative of the function so you can test critical points. The derivative of the function is . Then, set that equal to to find the critical points. When you set that equal to , you get . Then set up a number line so you can test values to determine when the function is increasing and decreasing. We know it's changing direction at our critical point, . So let's pick a point to the left of and plug it in to the derivative. I'll pick : . Since the answer is negative, we know that the function is decreasing. Pick a point to the right of 2. I'll pick 3: . Since the answer is positive, the function is increasing. Therefore, the function is increasing from .

To find the intervals on which the function is increasing, we must find the intervals on which the first derivative is positive.

To start, we must first find where the first derivative of the function is zero:

The first derivative was found using the following rule:

Now, set this function equal to zero to get the critical values (values at which the first derivative is equal to zero):

Next, we create the intervals using the critical value as the upper and lower bound of the limits, respectively:

On the first interval, the first derivative is negative, but on the second interval, the first derivative is positive, meaning that on this interval the function is increasing. (Simply plug in any point on the interval into the first derivative function and check the sign.)

The answer is

To solve, you must first differentiate the function once and then find where the derivative is positive. To differentiate, use the power rule:

Thus,

Now you must find where this is greater than 0, and therefore increasing.

Therefore, our answer is when x is greater than 2. Thus, .