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We've looked at increasing/decreasing behavior of a function, as determined by the _____ of the derivative. Another important property is concavity, which refers to the way the graph ______. Informally, a curve is concave up if it _____ ____ and concave down if it _____ _____
- sign
- bends
- bends up
- bends down
- **figure 1 (pg 190)
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To analyze concavity in a precise fashion, let's examine how concavity is related to tangent lines and derivatives. Observe in Figure 2 (pg 191) that when f is concave up, f' is ______ (the slopes of the tangent lines _______ as we move to the right). Similarly, when f is concave down, f' is _______. What is the general idea of concave up and concave down as it pertains to the slope of the tangent line?
- increasing
- increase
- decreasing
- Concave up: Slopes of tangent lines are increasing
- Concave down: Slopes of tangent lines are decreasing
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Formally Define Concavity
- Let f be a differentiable function on an open interval (a,b). Then:
- f is concave up on (a,b) if f' is increasing on (a,b)
- f is concave down on (a,b) if f' is decreasing on (a,b)
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The concavity is determined by the sign of its ______ ______. Indeed, if ______ then f' is increasing and hence f is concave up. Similarly, if _______ then f' is decreasing and f is concave down.
- second derivative
- if f''(x) > 0
- if f''(x) < 0
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State: The Test for Concavity Theorem
- Assume that f''(x) exists for all x ∈ (a,b):
- If f''(x) > 0 for all x ∈ (a,b), then f is concave up on (a,b)
- If f''(x) < 0 for all x ∈ (a,b), then f is concave down on (a,b)
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Of special interest are the points on the graph where the concavity changes. We say that P = (c,f(c)) is a _____ ____ ______ of f if the concavity changes from up to down or from down to up at x = c
point of inflection
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Figure 5 (pg 192) shows a curve made up of two arcs-one is concave down and one is concave up (the word "arc" refers to a piece of a curve). The point P where the arcs are joined is a ______ _____ _______.
points of inflection (denoted in graphs by a solid green square)
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Whats the difference between a critical point and a point of inflection?
A critical point c is just a single number, whereas a point of inflection (c,f(c)) is a point in the xy-plane
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According to Theorem 1 (Test for concavity), the concavity of f is determined by the sign of ______. Therefore, a point of inflection is a point where f''(x) ______ _____
- f''(x)
- f''(x) changes sign
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State: The Test for Inflection Points Theorem
If f''(c) = 0 or f''(c) does not exist and f''(x) changes sign at x = c, then f has a point of inflection at x = c
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There is a simple test for critical points based on concavity. suppose that f'(c) = 0. As we see in figure 10 (pg 193), f(c) is a local max if f is concave _____, and it is a local min if f is concave _____. Concavity is determined by the sign of _____
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State: The Second Derivative Test
- Let c be a critical point of f(x). If f''(c) exists, then:
- f''(c) > 0 ⇒ f(c) is a local minimum
- f''(c) < 0 ⇒ f(c) is a local maximum
- f''(c) = 0 ⇒ inconclusive, f(c) may be a local min, a local max, or neither
**See mnemonic device in fig 11 (pg 193)
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