Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeLecture Outline (More ImplicitDifferentiation)Friday, February 15Announcements1. No Class or OH on Monday, 2/18/08 - holiday2. Quiz 5 due on Wednesday, 2/20/08.3. Homework 5 due on Wednesday, 2/20/08.RecapLast time we began learning about implicit differentiation.Recall how we do this:1. start with a complicated expression involving x0s and y0s; this makes y an implicitfunction of x.2. compute the derivative of both sides of this expression (don’t forget thosedydx0s whichpop up!)3. solve fordydx4. celebrate, becausedydxgives you the slope of the tangent line to the curve by youroriginal complicated expressionExamplesLet’s do some more examples.1. Find y0where x2+ xy + y3= 3. Now find the equation of the tangent line that passesthrough (1, 1).2. Find the equation of the tangent line to 2(x2+ y2)2= 25(x2− y2) at the point (3,1).3. Find the derivative of ln(x). You can memorize the formula you get.4. Find the derivative of y = arctan(x). You can memorize the formula you get.1MotivationOver the last few weeks we’ve gotten really good at taking derivatives of functions and equa-tions. We’ve used this skill to be able to find the equation of the tangent line to j ust aboutany curve (which tells us the instantaneous rate of change). Derivatives have many moreapplications that just this.The application that we are going to start talking about can be described as the ”shapeof the curve”. If we have a function f(x) with some physical meaning (maybe f(x) is a pop-ulation at time x), we want to be able to compute where f (x) is increasing and decreasing,where the maximum and minimum points are, and other information. After talking aboutthis in a theoretical setting, we will apply this to optimization problems, where we try tomaximize or minimize some quantity (revenue, for example.)DefinitionsLet’s talk about some basic ways to describe the graph of a function f(x).A function y = f (x) is increasing on an interval if, whenever a < b are numbers in theinterval, f (a) < f (b). Less formally, a function f (x) is increasing on an interval if biggervalues of x give bigger values of f (x).A function y = f(x) is decreasing on an interval if, whenever a < b are numbers in theinterval, f(a) > f (b). Less formally, a function f (x) is decreasing on an interval if biggervalues of x give smaller values of f (x).How does this relate to f0(x)? Notice that if f(x) is increasing on an interval, the slopeof the tangent line at any point in that interval is positive. We can check this with the limitdefinition of the derivative. Thus if f (x) is increasing, f0(x) > 0.Similarly, if f (x) is decreasing on an interval, f0(x) <
View Full Document
Unlocking...