CALCULUS IN A NUTSHELL INTRODUCTION:

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Mervin Moody
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1 CALCULUS IN A NUTSHELL INTRODUCTION: Studets are usually itroduced to basic calculus i twelfth grade i high school or the first year of college. The course is typically stretched out over oe year ad ivolves over stuffed books reachig legths of five hudred pages or more ad ofte taught by teachers poorly prepared for hadlig of the topic. It is our purpose here to show that calculus is a very simple ad straight forward form of mathematical aalysis dealig predomiatly with cotiuous fuctios. A stadard calculus course cosists of two parts both of which deal with cotiuous fuctios f() defied over a iterval a<<b. The first is termed Differetial Calculus. It is strictly cocered with the slope (derivative) of a fuctio. The secod part deals with the area udereath a fuctio f() ad the ais ad forms the subject of Itegral Calculus. Both Isaac Newto(64-77) ad Gottfried Leibitz(646-76) are credited with the ivetio of calculus. DIFFERENTIAL CALCULUS: Here we deal with a cotiuous fuctio f() etedig from =a to =b whose slope we wish to measure at a poit. we have the picture- If we take a eighborig poit at + ad let approach zero we get the slope of f() to be-

2 f ( ) f ( ) ta()= I calculus we call this ratio the derivative of f() at. A commo way to epress this derivative is as f() or df()/d. We also ca costruct a secod derivative as- d f ( ) f ( ) f ( ). f() = d ( ) f ( ), agai requirig to approach zero. Eve higher derivatives ca be geerated by a rapeated applicatio of the last procedure. Let us work out a few derivatives of differet fuctios. Cosider first the power fuctio f()=. Here f() equals- d d ( ) We used the first two terms of a biomial epasio to get this result. The secod derivative reads (-) - ad the third derivative is (-)(-) -. The vaishig of the first derivative at =c implies the fuctio f(c) has zero slope at that poit. Thus the cubic equatio f()= - + has zero slope at =/sqrt(). Its secod derivative vaishes at =. This is kow as a iflectio poit. At such a poit the curvature of f() chages sig. Here is a graph of this cubic-

3 Aother fuctio whose derivative we wat to take a look at is f()=a, where a is a costat. There we have- d( a ) d a a This is a iterestig result sice it shows that there is just oe value of a for which the term i the curly bracket equals eactly oe so that the fuctio equals its ow derivative. This occurs oly for the irratioal umber- e ! which is oe of the best kow mathematical costats appearig i the literature. More geerally oe has that the term i the above curly bracket as goes to zero is l(a), where l refers to the atural logarithm which ca be looked up o oe s had calculator. So if a= we fid that d( )/d= l(). To get the derivative of the sie fuctio we write si( )] si( ) si( ) d cos( ) The quotiet ivolvig ds is a idicatio that the limitig form of has already bee take. Note the sig chage for cos()]/d=-si(). The derivative of a product is gotte as follows- d [ f ( ) )] )] f ( )] f ( ) ) d d d while that for the quotiet goes as- f ( ) / )] d f ( ) )' ) f ( )' [ )] I terms of derivatives oe ca epad ay cotiuous fuctios with fiite derivatives as a Taylor series which reads- f ( ) f ( ) f ( )' f ( )"( ) /!... This allows us to say that

4 si( ) ( ) ( )!, cos( ) ( ) ()! ad ep( )! INTEGRAL CALCULUS: This secod part of ay calculus course ivolves fidig the area udereath a curve f(). It is referred to as itegral calculus. To fid the area betwee the curve f() ad the ais i the rage a,<b we break the area up ito N=(b-a)/ small sub-areas of width ad height f( ). This allows us to write- N b Area f ( ) f ( ) d a as goes to zero. Now if we let f()=)]/d we get that the area just equals b)- a). Relaig the ed coditios we arrive at what is kow as the Fudametal Theorem of Calculus, amely,- g ( ) f ( ) d where d ) / d f ( ) Usig this last result allows us to state that the area udereath the parabola f()= etedig from = to = is /. This is epressed mathematically as Area= d [ ] The geeric form of the itegral epressio is- )] ) f ( ) d with f ( ) d Here is a short itegral table- ) f() + /(+) ep(a) a ep(a) si(a)/a cos(a) cos(a)/a -si(a) ta(a) (+ta(a)^)a (/a)arcta(/a) /(a + ) l() /

5 Sih() ep(-a)si(b) ep(-a ) -aep(-a ) arccos() -/sqrt(- ) cosh() ep(-a){-a si(b)+b cos(b)} arcsi() /sqrt(- ) sqrt(- ) (-)/sqrt(-) You ca use this table to calculate, for istace, the area betwee a origi cetered circle of radius R= give by +y = ad a chord y=a with a<. The chord cuts the circle at =sqrt(-a ). We have- a Area= [ a] d a a arcsi[ sqrt( a )] This result makes sese sice the area goes to half the circle area / as a vaishes. Also the area is zero whe a=. Oe ca also use a similar type of itegral to predict the remaiig volume i a partially filled horizotal cylidrical tak usig a dip-stick as is ofte doe at service statios. Itegrals may also be used to geerate ifiite series for certai fuctios ).Take the case of )=l(-). Here we have- l( ) d ( ) d ( ) [...] For =- this reads- ) 4 l( - This is a etremely slowly coverget series which evetually will approach.6947 A faster coverget series follows from- l d ( ) d m Here if =/ we get- l( ) {...} CONCLUDING REMARKS:

6 The above has codesed a years worth of calculus ito a few pages givig you calculus i a utshell. For those studets about to eter your first calculus course the above should be a coveiet aide as a supplemet to the usual tets which ca be hudreds of pages log. Remember whe teachig ew cocepts oe should always keep i mid to keep thigs as simple as possible. The more esoteric aspects of a topic ca be taught ad leared later. This approach has served me very well durig my forty year career of teachig applied mathematics first at Resselaer Polytechic Istitute ad later at the Uiversity of Florida. It has led to a total of eight teachig awards over that time spa icludig the Teacher of the Year award at the Uiversity of Florida i 99. U.H.Kurzweg March, 8 Gaiesville, Florida

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0, Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative