# Stanford MATH 19 - Lecture Outline (2 pages)

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## Lecture Outline

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Math 19 Calculus Winter 2008 Instructor Jennifer Kloke Lecture Outline More Implicit Differentiation Friday February 15 Announcements 1 No Class or OH on Monday 2 18 08 holiday 2 Quiz 5 due on Wednesday 2 20 08 3 Homework 5 due on Wednesday 2 20 08 Recap Last time we began learning about implicit differentiation Recall how we do this 1 start with a complicated expression involving x0 s and y 0 s this makes y an implicit function of x 2 compute the derivative of both sides of this expression don t forget those pop up 3 solve for dy 0 s dx which dy dx dy 4 celebrate because dx gives you the slope of the tangent line to the curve by your original complicated expression Examples Let s do some more examples 1 Find y 0 where x2 xy y 3 3 Now find the equation of the tangent line that passes through 1 1 2 Find the equation of the tangent line to 2 x2 y 2 2 25 x2 y 2 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 1 Motivation Over the last few weeks we ve gotten really good at taking derivatives of functions and equations We ve used this skill to be able to find the equation of the tangent line to just about any curve which tells us the instantaneous rate of change Derivatives have many more applications that just this The application that we are going to start talking about can be described as the shape of the curve If we have a function f x with some physical meaning maybe f x is a population 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 about this in a theoretical setting we will apply this to optimization problems where we try to maximize or minimize some quantity revenue for example Definitions Let 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 the interval f a f b Less formally a function f x is increasing on an interval if bigger values 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 the interval f a f b Less formally a function f x is decreasing on an interval if bigger values of x give smaller values of f x How does this relate to f 0 x Notice that if f x is increasing on an interval the slope of the tangent line at any point in that interval is positive We can check this with the limit definition of the derivative Thus if f x is increasing f 0 x 0 Similarly if f x is decreasing on an interval f 0 x 0 2

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