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Review your knowledge of concavity of functions and how we use differential calculus to analyze it.
- 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
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.
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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. Explain the relationship between a function and its first and second derivatives.
Dec 21, 2020 · If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points.
Review 5.6 Determining Concavity for your test on Unit 5 – Analytical Applications of Differentiation. For students taking AP Calculus AB/BC.
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Concavity. Definition. Let f be a differentiable function on the open interval I. Let Ta(x) = f′(a)(x − a) + f(a), ∀ a ∈ I. This is the tangent line to. at x = a. If ∀a ∈ I, f(x) > Ta(x) ∀x ∈ I − {a}, then we say f is concave up. If ∀a ∈ I, f(x) < Ta(x) ∀x ∈ I − {a}, then we say f is concave down. Theorem.
