Preface Here are my online notes for my course tat I teac ere at Lamar University. Despite te fact tat tese are my class notes, tey sould be accessible to anyone wanting to learn or needing a refreser in some of te early topics in calculus. I ve tried to make tese notes as self contained as possible and so all te information needed to read troug tem is eiter from an Algebra or Trig class or contained in oter sections of te notes. Here are a couple of warnings to my students wo may be ere to get a copy of wat appened on a day tat you missed. 1. Because I wanted to make tis a fairly complete set of notes for anyone wanting to learn calculus I ave included some material tat I do not usually ave time to cover in class and because tis canges from semester to semester it is not noted ere. You will need to find one of your fellow class mates to see if tere is someting in tese notes tat wasn t covered in class. 2. Because I want tese notes to provide some more examples for you to read troug, I don t always work te same problems in class as tose given in te notes. Likewise, even if I do work some of te problems in ere I may work fewer problems in class tan are presented ere. 3. Sometimes questions in class will lead down pats tat are not covered ere. I try to anticipate as many of te questions as possible wen writing tese up, but te reality is tat I can t anticipate all te questions. Sometimes a very good question gets asked in class tat leads to insigts tat I ve not included ere. You sould always talk to someone wo was in class on te day you missed and compare tese notes to teir notes and see wat te differences are. 4. Tis is somewat related to te previous tree items, but is important enoug to merit its own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Using tese notes as a substitute for class is liable to get you in trouble. As already noted not everyting in tese notes is covered in class and often material or insigts not in tese notes is covered in class.
2007 Paul Dawkins 2 ttp://tutorial.mat.lamar.edu/terms.aspx
Te Definition of te Derivative In te first section of te last capter we saw tat te computation of te slope of a tangent line, te instantaneous rate of cange of a function, and te instantaneous velocity of an object at x a all required us to compute te following limit. f x f a lim x a x a We also saw tat wit a small cange of notation tis limit could also be written as, f ( a+ ) f ( a) lim 0 (1) Tis is suc an important limit and it arises in so many places tat we give it a name. We call it a derivative. Here is te official definition of te derivative. Definition Te derivative of f ( x ) wit respect to x is te function f ( x) ( + ) f ( x) lim f x f x 0 and is defined as, (2) Note tat we replaced all te a s in (1) wit x s to acknowledge te fact tat te derivative is f x as f prime of x. really a function as well. We often read Let s compute a couple of derivatives using te definition. Example 1 Find te derivative of te following function using te definition of te derivative. 2 f ( x) 2x 16x+ 35 Solution So, all we really need to do is to plug tis function into te definition of te derivative, (1), and do some algebra. Wile, admittedly, te algebra will get somewat unpleasant at times, but it s just algebra so don t get excited about te fact tat we re now computing derivatives. First plug te function into te definition of te derivative. f ( x+ ) f ( x) f ( x) 0 2 16 35 2 16 35 0 2 2 ( x+ ) ( x+ ) + ( x x+ ) Be careful and make sure tat you properly deal wit parentesis wen doing te subtracting. Now, we know from te previous capter tat we can t just plug in 0 since tis will give us a 2007 Paul Dawkins 3 ttp://tutorial.mat.lamar.edu/terms.aspx
division by zero error. So we are going to ave to do some work. In tis case tat means multiplying everyting out and distributing te minus sign troug on te second term. Doing tis gives, 2 2 2 2x + 4x + 2 16x 16 + 35 2x + 16x 35 f ( x) 0 2 4x + 2 16 0 Notice tat every term in te numerator tat didn t ave an in it canceled out and we can now factor an out of te numerator wic will cancel against te in te denominator. After tat we can compute te limit. ( 4x+ 2 16) f ( x) 0 4x+ 2 16 0 4x 16 So, te derivative is, f ( x) 4x 16 Example 2 Find te derivative of te following function using te definition of te derivative. t g( t) t + 1 Solution Tis one is going to be a little messier as far as te algebra goes. However, outside of tat it will work in exactly te same manner as te previous examples. First, we plug te function into te definition of te derivative, g( t+ ) g( t) g ( t) 0 1 t+ t 0 t+ + 1 t+ 1 Note tat we canged all te letters in te definition to matc up wit te given function. Also note tat we wrote te fraction a muc more compact manner to elp us wit te work. As wit te first problem we can t just plug in 0. So we will need to simplify tings a little. In tis case we will need to combine te two terms in te numerator into a single rational expression as follows. 2007 Paul Dawkins 4 ttp://tutorial.mat.lamar.edu/terms.aspx
( t+ )( t+ 1) t( t+ + 1) ( + + 1)( + 1) 1 g ( t) 0 t t 1 0 t t 2 2 t + t + t + t + t + t ( + + 1)( + 1) 1 0 ( t+ + 1)( t+ 1) Before finising tis let s note a couple of tings. First, we didn t multiply out te denominator. Multiplying out te denominator will just overly complicate tings so let s keep it simple. Next, as wit te first example, after te simplification we only ave terms wit s in tem left in te numerator and so we can now cancel an out. So, upon canceling te we can evaluate te limit and get te derivative. 1 g ( t) 0 ( t+ + 1)( t+ 1) 1 ( t+ 1)( t+ 1) 1 2 t + 1 Te derivative is ten, ( t) g 1 ( t + 1) 2 Example 3 Find te derivative of te following function using te definition of te derivative. R( z) 5z 8 Solution First plug into te definition of te derivative as we ve done wit te previous two examples. R( z+ ) R( z) R ( z) 0 5( z+ ) 8 5z 8 0 In tis problem we re going to ave to rationalize te numerator. You do remember rationalization from an Algebra class rigt? In an Algebra class you probably only rationalized te denominator, but you can also rationalize numerators. Remember tat in rationalizing te numerator (in tis case) we multiply bot te numerator and denominator by te numerator except we cange te sign between te two terms. Here s te rationalizing work for tis problem, 2007 Paul Dawkins 5 ttp://tutorial.mat.lamar.edu/terms.aspx
( z) R 0 ( ) 5z+ 5 8 ( 5z 8) ( 5( + ) 8+ 5 8) ( 5( + ) 8+ 5 8) ( 5( + ) 8+ 5 8) 5 z+ 8 5z 8 5 z+ 8+ 5z 8 0 z z 5 0 z z z z Again, after te simplification we ave only s left in te numerator. So, cancel te and evaluate te limit. 5 R ( z) 0 5 z+ 8 + 5 z 8 5 5z 8+ 5z 8 5 2 5z 8 And so we get a derivative of, 5 R ( z) 2 5z 8 Let s work one more example. Tis one will be a little different, but it s got a point tat needs to be made. Example 4 Determine f ( 0) for f ( x) x Solution Since tis problem is asking for te derivative at a specific point we ll go aead and use tat in our work. It will make our life easier and tat s always a good ting. So, plug into te definition and simplify. f 0 0 0 0 ( 0+ ) ( 0) f f 0+ 0 2007 Paul Dawkins 6 ttp://tutorial.mat.lamar.edu/terms.aspx
We saw a situation like tis back wen we were looking at limits at infinity. As in tat section we can t just cancel te s. We will ave to look at te two one sided limits and recall tat if 0 if < 0 lim because < 0 in a left-and limit. 1 0 0 0 + + 0 0 + 0 1 lim because > 0 in a rigt-and limit. 1 1 Te two one-sided limits are different and so lim 0 doesn t exist. However, tis is te limit tat gives us te derivative tat we re after. If te limit doesn t exist ten te derivative doesn t exist eiter. In tis example we ave finally seen a function for wic te derivative doesn t exist at a point. Tis is a fact of life tat we ve got to be aware of. Derivatives will not always exist. Note as well tat tis doesn t say anyting about weter or not te derivative exists anywere else. In fact, te derivative of te absolute value function exists at every point except te one we just looked at, x 0. Te preceding discussion leads to te following definition. Definition A function f ( x ) is called differentiable at x a if f ( a) exists and f x is called differentiable on an interval if te derivative exists for eac point in tat interval. Te next teorem sows us a very nice relationsip between functions tat are continuous and tose tat are differentiable. Teorem If f ( x ) is differentiable at x a ten f ( x ) is continuous at x a. 2007 Paul Dawkins 7 ttp://tutorial.mat.lamar.edu/terms.aspx
See te Proof of Various Derivative Formulas section of te Extras capter to see te proof of tis teorem. Note tat tis teorem does not work in reverse. Consider f ( x) x and take a look at, So, f ( x) x lim f x lim x 0 f 0 x 0 x 0 is continuous at x 0 but we ve just sown above in Example 4 tat f ( x) x is not differentiable at x 0. Alternate Notation Next we need to discuss some alternate notation for te derivative. Te typical derivative notation is te prime notation. However, tere is anoter notation tat is used on occasion so let s cover tat. Given a function y f ( x) all of te following are equivalent and represent te derivative of f ( x ) wit respect to x. df dy d d f x y f x y dx dx dx dx ( ) Because we also need to evaluate derivatives on occasion we also need a notation for evaluating derivatives wen using te fractional notation. So if we want to evaluate te derivative at xa all of te following are equivalent. df dy f ( a) y x a dx dx x a x a Note as well tat on occasion we will drop te (x) part on te function to simplify te notation somewat. In tese cases te following are equivalent. f x f As a final note in tis section we ll acknowledge tat computing most derivatives directly from te definition is a fairly complex (and sometimes painful) process filled wit opportunities to make mistakes. In a couple of sections we ll start developing formulas and/or properties tat will elp us to take te derivative of many of te common functions so we won t need to resort to te definition of te derivative too often. Tis does not mean owever tat it isn t important to know te definition of te derivative! It is an important definition tat we sould always know and keep in te back of our minds. It is just someting tat we re not going to be working wit all tat muc. 2007 Paul Dawkins 8 ttp://tutorial.mat.lamar.edu/terms.aspx