Review how we use differential calculus to find the intervals where a function increases or decreases.
Log in akuppili45 8 years agoPosted 8 years ago. Direct link to akuppili45's post “Is this also called the 1...” Is this also called the 1st derivative test? • (19 votes) Bruh 8 years agoPosted 8 years ago. Direct link to Bruh's post “In summation, it's the 1s...” 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 First derivative test: If f' changes from (+) to (-) at a critical number, then f has a local max at that critical number (78 votes) bhunter3 5 years agoPosted 5 years ago. Direct link to bhunter3's post “I'm finding it confusing ...” 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: 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. • (29 votes) bhunter3 5 years agoPosted 5 years ago. Direct link to bhunter3's post “I found the answer to my ...” 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. (35 votes) Aztec Binaynay 8 years agoPosted 8 years ago. Direct link to Aztec Binaynay's post “for the notation of findi...” 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? • (4 votes) Cesar Sandoval 7 years agoPosted 7 years ago. Direct link to Cesar Sandoval's post “Yes. For example, the fun...” 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. (8 votes) Maria 6 years agoPosted 6 years ago. Direct link to Maria's post “What does it mean to say ...” 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. • (3 votes) bhunter3 5 years agoPosted 5 years ago. Direct link to bhunter3's post “I think that if the probl...” 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. (7 votes) Daniel Leles 6 years agoPosted 6 years ago. Direct link to Daniel Leles's post “Is x^3 increasing on (-∞,...” Is x^3 increasing on (-∞,∞) or is it increasing on two open intervals and is increasing on (-∞,0)U(0,∞)? • (4 votes) Mark Geary 6 years agoPosted 6 years ago. Direct link to Mark Geary's post “f(x) = x³ is increasing o...” 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. x³ is not strictly increasing, but it does meet the criteria for an increasing function throughout it's domain = ℝ (4 votes) mitchellqmj 6 years agoPosted 6 years ago. Direct link to mitchellqmj's post “Using only the values giv...” 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) • (2 votes) Jerry Nilsson 6 years agoPosted 6 years ago. Direct link to Jerry Nilsson's post “𝑓(−4) < 𝑓(−1), so 𝑓 ca...” 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) (2 votes) James a year agoPosted a year ago. Direct link to James's post “how would a question on t...” how would a question on the test look like for these types of questions? • (1 vote) Venkata a year agoPosted a year ago. Direct link to Venkata's post “They'll probably give you...” 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. (4 votes) SIRI MARAVANTHE 5 years agoPosted 5 years ago. Direct link to SIRI MARAVANTHE's post “How do we decide if y=cos...” How do we decide if y=cos3x increasing or decreasing in the interval [0,3.14/2] • (2 votes) Jasper a year agoPosted a year ago. Direct link to Jasper's post “We can tackle the trigono...” We can tackle the trigonometric functions in the same way we do polynomials or rational functions! We take the derivative of y, giving us dy/dx = -3sin3x. Then, we find where this derivative is equal to zero or is undefined - this tells us all the possible x-values where the derivative might change from positive to negative, or negative to positive. Then we figure out where dy/dx is positive or negative. Since we know functions are increasing where their derivatives are positive, and decreasing where their derivatives are negative, we can then use this knowledge to figure out if the function is increasing or decreasing. (1 vote) anisnasuha1305 6 years agoPosted 6 years ago. Direct link to anisnasuha1305's post “for the number line we mu...” for the number line we must do for all the x or the value of crtitical number that is in the domain? • (2 votes) Jasper a year agoPosted a year ago. Direct link to Jasper's post “We only need to look at t...” We only need to look at the critical values of x; that is, whether or not the function's derivative changes signs at those points, so that we can figure out if the derivative is positive or negative on its domain. (1 vote) emmiesullivan96 5 years agoPosted 5 years ago. Direct link to emmiesullivan96's post “If a graph has positive a...” 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? • (1 vote) Alex 5 years agoPosted 5 years ago. Direct link to Alex's post “Given that you said "has ...” Given that you said "has negative slope", no. It is increasing perhaps on part of the interval. 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If f'(x) < 0 on an interval, then f is decreasing on that interval
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.f'(x) = ((x-1)^2) / (x-4)
If f(b) > f(a) for all b>a, the function is said to be strictly increasing.
(–4, –1)
(–1,1)
(–1, 2)
