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If a a and b b are two successive nodes (thus distant by 1/n 1 / n ), then |f(x) − f(y)| < ϵ | f ( x) − f ( y) | < ϵ for any x, y ∈]a, b[ x, y ∈] a, b [, this is true by construction (specifically, the construction of n n and ϵ ϵ ). We can thus conclude that |f(x) − f(a)| ≤ ϵ | f ( x) − f ( a) | ≤ ϵ and |f(x) − f(b)| ≤ ...
Step 1: Given f (x), find f (a), f (b), f (c), for x= a, b and c, where a < c < b. Where a and b are the points of interest. C is just any convenient point in between them. Step 2: Find the equation of the line that connects the points found for a and b. Step 3: Find that y-coordinate of the line from Step 2 at point C.
- 2 min
- Sal Khan
- Yes. However, it will often be less work (and less prone to error) to use the Calculus method. Anyway here is how to find concavity without calculu...
- Interesting question – I was sure this wasn't true, but it was harder than I expected to come up with a counter example. Try this: `f(x) = sin(eˣ)...
- Typically in mathematics and natural sciences, we don't deal too much with the 3rd derivative. I think the easiest way to understand the 3rd deriva...
- The second derivative is one of the higher order derivatives. One example is the second derivative of the position function, s(t), which is equival...
- First half is correct, but if the slope is negative and f"(x) > 0, then the slope still increases – this means it becomes *less* negative, i.e. bec...
- Yep! That's correct. The second derivative being negative only implies that the first derivative is decreasing (becoming less positive or more nega...
- No, f can be negative, decreasing, both, or neither on that interval. If f'' is positive, it means the graph is concave up (that is, if you fit a c...
Google Classroom. Review your knowledge of concavity of functions and how we use differential calculus to analyze it. What is concavity? 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.
- You have two functions f and g, where f''>0 and g''>0. If you take the second derivative of f+g, you get f''+g'', which is positive. So their sum i...
- In order for 𝑓(𝑥) to be concave up, in some interval, 𝑓 ''(𝑥) has to be greater than or equal to 0 (i.e. non-negative) for all 𝑥 in that inter...
- In general, no. Points where f"(x)= 0 are specifically called inflection points. But in the example provided, -2 is also a critical point (Observe...
- Product rule: d/dx[f(x)*g(x)] = f'(x)g(x) + f(x)g'(x) f(x) = (x-1)³ and g(x) = x
Jul 26, 2016 · Start practicing—and saving your progress—now: https://www.khanacademy.org/math/ap-c... Sal finds the intervals where g (x)=-x_+6x_-2x-3 is concave down/up by finding where its second ...
- 9 min
- 78.1K
- Khan Academy
State the first derivative test for critical points. Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph. Explain the concavity test for a function over an open interval.
Differential equations that can be solved using separation of variables are called separable equations. So how can we tell whether an equation is separable? The most common type are equations where d y d x is equal to a product or a quotient of f ( y ) and g ( x ) .
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How to find concavity without calculus?
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What is concavity in differential calculus?
How do you know if a function is concave?
Dec 21, 2020 · The important \(x\)-values at which concavity might switch are \(x=-1\), \(x=0\) and \(x=1\), which split the number line into four intervals as shown in Figure \(\PageIndex{7}\). We determine the concavity on each.
