Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeLecture Outline (What f0says about f.)Wednesday, February 20Announcements1. Homework 6 due on Wednesday, 2/27/08.2. Midterm 2 is on Thursday, 2/28/08, 7:00-9:00 PM. A review session will be heldWednesday, 2/27/08, at 7:00 in 380W.3. Ziyu has to move his office hours next week from 7:00-10:00 on Tuesday to 7:00-10:00PM on Monday. Plan accordingly.RecapRecall from Friday the following facts:f(x) is increasing on an interval if and only if f0(x) > 0.f(x) is decreasing on an interval if and only if f0(x) < 0.DefinitionsAnother way to describe the graph of a function is to talk about concavity. Informally,f(x) is concave down on an interval if it “bends down” on that interval, whereas f(x) isconcave up on an interval if it “bends up” on that interval.It is important to know that on an interval, any combination of increasing/decreasingand concavity can occur.Let’s do an example. Label the intervals where the graph is increasing, decreasing,concave up and concave down.Definition: A function has an inflection point at x = a if the concavity f(x) changes at a.1Theory: Information from the First and Second deriva-tiveAs we said above:1. f0(x) > 0 on an interval if and only if f(x) is increasing on that interval.2. f0(x) < 0 on an interval if and only if f(x) is decreasing on that interval.3. f0(x) = 0 on an interval if and only if f(x) is flat on that interval.Now, suppose that f(x) is concave up on an interval, so that it is “bending up.” Wecan see that as x increases, the slope of the tangent line to the curve at x increases. Thissays that f0(x) is increasing. But according to the above statement, if f0(x) is increasing,then f00(x) > 0. Similarly if f (x) is concave down on an interval, f0(x) is decreasing andf00(x) < 0. So we have the following statements:1. If f00(x) > 0 on an interval, then f(x) is concave up on that interval.2. If f00(x) < 0 on an interval, then f(x) is concave down on that interval.Examples1. For the function f(x) = x3+3/2x2− 18x+1, find all the intervals on which the functionis increasing, decreasing, concave up, and concave
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