Q: for the notation of finding the increasing/decreasing intervals of a function, can you use the notation Union (U) to express more than one interval?

Yes. For example, the function -x^3+3x^2+9 is decreasing for x<0 and x>2. Another way we can express this: domain = (-∞,0) U (2, +∞). This is known as interval notation.

Question 1

I'm finding it confusing when a point is undefined in both the original function and the derivative. While not mentioned in the video on critical points, it's mentioned in the comments and practice problems that a point is not a critical point if it's undefined in both the derivative and in the original function.

On the other hand, in the practice problems, we're given something like:

`f'(x) = ((x-1)^2) / (x-4)`

and asked to find the intervals over which the original function is increasing. The question states that the original function is undefined at x = 4. According to the definition, x = 4 should not be a critical point because it's undefined in both the derivative and the original function. However, it is a point of interest as f'(x) > 0 only when x > 4. If we don't consider x = 4 we won't find the right answer.

Is this an issue with the definition of critical points, the practice problem itself, or this method of finding increasing or decreasing intervals?

If it's the practice problem, I could imagine that maybe it's impossible for a function with that derivative to be undefined at 4 (though it seems unlikely.)

If it's this method, it seems like we need to consider points that aren't strictly critical points as per the definition.

I think a little more clarity around this particular case in this section and the one before would be helpful.

Accepted Answer

I found the answer to my question in the next section. Under "Finding relative extrema (first derivative test)" it says:

_When we analyze increasing and decreasing intervals, we must look for all points where the derivative is equal to zero and all points where *the function or its derivative are undefined*. If you miss any of these points, you will probably end up with a wrong sign chart._

I'll leave my question here because I think it's confusing for this section to only discuss critical points and not to mention this.

Question 2

Is this also called the 1st derivative test?

Accepted Answer

In summation, it's the 1st derivative test. Specifically, it's the 'Increasing/Decreasing test':

_Increasing/Decreasing test:_

If *f'(x) > 0* on an interval, then f is *increasing* on that interval
If *f'(x) < 0* on an interval, then f is *decreasing* on that interval

_First derivative test:_

If *f' changes from (+) to (-) at a critical number, then f has a local max* at that critical number
If *f' changes from (-) to (+) at a critical number, then f has a local min* at that critical number
If f' has no sign changes at that critical number, then f' has no local min nor max at the critical number.

Question 3

for the notation of finding the increasing/decreasing intervals of a function, can you use the notation Union (U) to express more than one interval?

Accepted Answer

Yes. For example, the function -x^3+3x^2+9 is decreasing for x<0 and x>2. Another way we can express this: domain = (-∞,0) U (2, +∞). This is known as interval notation.
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Question 4

Is x^3 increasing on (-∞,∞) or is it increasing on two open intervals and is increasing on (-∞,0)U(0,∞)?

Accepted Answer

f(x) = x³ is increasing on (-∞,∞).

A function f(x) increases on an interval I if f(b) ≥ f(a) for all b > a, where a,b in I.
If f(b) > f(a) for all b>a, the function is said to be *strictly* increasing.

x³ is not strictly increasing, but it does meet the criteria for an increasing function throughout it's domain = ℝ

Question 5

What does it mean to say that the _slope_ of a function is increasing or decreasing? Not when the function is increasing or decreasing, but the _slope_. Is it the same thing? I'm having some trouble with calculus homework that is treating it as if they aren't the same thing.

Accepted Answer

I think that if the problem is asking you specifically whether the slope of the tangent line to the function is increasing or decreasing, then it is asking whether the *second derivative* of the function is positive or negative.

When we want to know if the function is increasing or decreasing, we take the derivative of the function and check if the derivative (slope of the tangent) is positive or negative. But if we want to know whether that derivative is increasing or decreasing (whether the slope is increasing or decreasing), we'd take its derivative. The derivative of the "slope" would be the second derivative of the original function.

I'm betting we get to this a bit later when we start talking about using second derivatives to analyze functions.

Question 6

Using only the values given in the table for the function, f(x) = x3 – 3x – 2, what is the interval of x-values over which the function is decreasing?

(–4, 1)
(–4, –1)
(–1,1)
(–1, 2)

Accepted Answer

𝑓(−4) < 𝑓(−1), so 𝑓 can not be decreasing over (−4, −1) and thereby not over (−4, 1) either.

𝑓(−1) = 𝑓(2), so 𝑓 can not be decreasing over (−1, 2)

𝑓(−1) > 𝑓(1), so it is _possible_ that 𝑓 is decreasing over (−1, 1)

Question 7

how would a question on the test look like for these types of questions?

Accepted Answer

They'll probably give you a function and ask you to write out the intervals on which it increases and decreases. You'll have to then show the whole process of taking derivatives, finding critical points, using either the first or second derivative test, and figuring out the behaviour of the function on each of the intervals.

Question 8

If a graph has positive and negative slopes on an interval, but the y value at the end of the interval is higher than y value at the beginning, is it increasing on the interval?

Accepted Answer

Given that you said "has negative slope", no. It is increasing perhaps on part of the interval.

Interval	$x$ ‍-value	$f^{'} (x)$ ‍	Verdict
$x < - 3$ ‍	$x = - 4$ ‍	$f^{'} (- 4) = 15 > 0$ ‍	$f$ ‍ is increasing. $↗$ ‍
$- 3 < x < 1$ ‍	$x = 0$ ‍	$f^{'} (0) = - 9 < 0$ ‍	$f$ ‍ is decreasing. $↘$ ‍
$x > 1$ ‍	$x = 2$ ‍	$f^{'} (2) = 15 > 0$ ‍	$f$ ‍ is increasing. $↗$ ‍

Interval	$x$ ‍-value	$f^{'} (x)$ ‍	Verdict
$x < 0$ ‍	$x = - 1$ ‍	$f^{'} (- 1) = - 21 < 0$ ‍	$f$ ‍ is decreasing. $↘$ ‍
$0 < x < \frac{5}{2}$ ‍	$x = 1$ ‍	$f^{'} (1) = - 9 < 0$ ‍	$f$ ‍ is decreasing. $↘$ ‍
$\frac{5}{2} < x$ ‍	$x = 3$ ‍	$f^{'} (3) = 243 > 0$ ‍	$f$ ‍ is increasing. $↗$ ‍

AP®︎/College Calculus AB

Course: AP®︎/College Calculus AB > Unit 5

Increasing & decreasing intervals review

How do I find increasing & decreasing intervals with differential calculus?

Example 1

Example 2

Check your understanding

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