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Concavity relates to the rate of change of a function's derivative. A function f is concave up (or upwards) where the derivative f ′ is increasing. This is equivalent to the derivative of f ′ , which is f ″ , being positive. Similarly, f is concave down (or downwards) where the derivative f ′ is decreasing (or equivalently, f ″ is ...
- Concavity introduction (video)
Google Classroom. Microsoft Teams. AboutAbout this video....
- Analyzing Concavity
Concavity and convexity are opposite sides of the same coin....
- Concavity introduction (video)
The concavity of the graph of a function refers to the curvature of the graph over an interval; this curvature is described as being concave up or concave down. Generally, a concave up curve has a shape resembling "∪" and a concave down curve has a shape resembling "∩" as shown in the figure below. Concave up.
Dec 21, 2020 · The key to studying f ′ is to consider its derivative, namely f ″, which is the second derivative of f. When f ″ > 0, f ′ is increasing. When f ″ < 0, f ′ is decreasing. f ′ has relative maxima and minima where f ″ = 0 or is undefined. This section explores how knowing information about f ″ gives information about f.
Google Classroom. Microsoft Teams. AboutAbout this video. Transcript. Sal introduces the concept of concavity, what it means for a graph to be "concave up" or "concave down," and how this relates to the second derivative of a function. Created by Sal Khan. Questions. Tips & Thanks.
- 10 min
- Sal Khan
- From that point on, the slope goes from being negative to becoming zero. Hence, it stops decreasing (in other words, it increases till it becomes z...
- Increasing slope can mean one of two things: _more_ positive or _less_ negative. Whichever situation you have, increasing slope always implies conc...
- Short answer: no. Since the function f is not defined by some formula, only by the graph sal draw, you cant say wether or not these are parabolas....
- A point where both f''(x) = 0 and f''(x) changes sign (i.e. f(x) changes concavity) is called a point of inflection of f(x). Visually, the graph of...
- Observe that he was looking at the graph of f'(x). This graph is the derivative of the first graph. So, in the first graph, at that point, the deri...
- When f'(x) is positive, f(x) increases When f'(x) is negative, f(x) decreases When f'(x) is zero, it indicates a *possible* local max or min (use t...
Dec 21, 2020 · 5.4: Concavity and Inflection Points. We know that the sign of the derivative tells us whether a function is increasing or decreasing; for example, when f′(x) > 0 f ′ ( x) > 0, f(x) f ( x) is increasing. The sign of the second derivative f′′(x) f ″ ( x) tells us whether f′ f ′ is increasing or decreasing; we have seen that if f ...
Concavity and convexity are opposite sides of the same coin. So if a segment of a function can be described as concave up, it could also be described as convex down. We find it convenient to pick a standard terminology and run with it - and in this case concave up and concave down were chosen to describe the direction of the concavity/convexity.
- 9 min
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Concavity. Concavity. There are various terms used when the values of a function, its average rates of change, or its second average rates of change (the rates of change of the rates of change) are all positive (or negative), at least on some interval. When the function is differentiable, and especially when it's twice differentiable, there are ...
