This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Math 006 (Lecture 25) Second Derivative
Deﬁnition 1. For y = f (x), the second derivative of f , is d2 f d f (x). (x) = f (x) = 2 dx dx Example 1. Find the second derivatives of (a) f (x) = √ 3 x2 + 3x3 + 1 x7 (b) y = x3 + √ x (c) y = x+1 x2 + 1 1 Concavity
Example 2. Find the ﬁrst and the second derivatives of f (x). √ (a) f (x) = x2 (b) f (x) = x Deﬁnition 2. Concavity of f (x) 1. The graph of a function is concave upward on the interval a < x < b if f (x) is increasing on a < x < b. 2. The graph of a function is concave downward on the interval a < x < b if f (x) is decreasing on a < x < b. Theorem 1. For the interval a < x < b, 1. f (x) is an concave upward on a < x < b if and only if f (x) > 0 on a < x < b. 2. f (x) is an concave downward on a < x < b if and only if f (x) < 0 on a < x < b. Example 3. Determine the interval(s) where the graph of f is concave upward and the interval(s) where the graph of f is concave downward. (a) f (x) = x3 (b) g (x) = x4/3 (c) h(x) = √ x+1 (d) k (x) = ln x 2 Inﬂection Points
Deﬁnition 3. An inﬂection point is a point on the graph of the function where the concavity changes. Theorem 2. If y = f (x) has an inﬂection point point at x = c, then 1. f (c) = 0, or 2. f (c) does not exist. Example 4. Find the inﬂection point(s) of (a) y = x3 − 9x2 + 24x − 10. (b) f (x) = ln(x2 − 4x + 5) 3 ...
View Full Document