WELCOME to MATH : Calculus I Welcome to the Course. Pe Math Calculus I. Topics: quick review of high school calculus, methods ad applicatios of itegratio, ifiite series ad applicatios, some fuctios of several variables.. College-level level pace ad workload: Moves very fast - twelve sessios to do everythig! Demadig workload, but help is available! YOU ARE ADULTS - how much do you eed to practice each topic? Emphasis o applicatios - what is this stuff good for?. Opportuities to iteract with istructor, TA, ad other studets Outlie for Week Fuctios ad Graphs The idea of a fuctio ad of the graph of a fuctio should be very familiar (a)review of fuctios ad graphs (b)review of its (c)review of derivatives - idea of velocity, taget ad ormal lies to curves (d)review of related rates ad ma/mi problems f ( ) + Questios for discussio.... Describe the graph of the fuctio f() (use calculus vocabulary ary as appropriate).. The graph itersects the y-ais y at oe poit. What is it (how do you fid it)?. How do you kow there are o other poits where the graph itersects the y-ais? y. The graph itersects the -ais at four poits. What are they (how do you fid them)? 5. How do you kow there are o other poits where the graph itersects the -ais? 6. The graph has a low poit aroud, y-. What is it eactly? How do you fid it? 7. Where might this fuctio come from? 5 Liear, quadratic Kids of fuctios that should be familiar: Polyomials, quotiets of polyomials Powers ad roots Epoetial, logarithmic Trigoometric fuctios (sie, cosie, taget, secat, cotaget, cosecat) Hyperbolic fuctios (sih( sih, cosh, tah, sech, coth, csch) 6
Quick Questio The domai of the fuctio 7 Quick Questio Which of the followig has a graph that is symmetric with respect to the y-ais? y 8 f ( ) A. All ecept, B. All < ecept. C. All > ecept. D. All <. E. All >. is... A. B. C. y y y D. E. y y 5 + 9 Quick Questio The period of the fuctio Quick Questio A. B. /5 C. / D. 6/5 E. 5 π f ( ) si 5 is... A. 5 B. 5 C. 5 D. 5 a a If log 5, the a a E. Noe of these ( ) Limits Basic facts about its The cocept of it uderlies all of calculus. Derivatives, itegrals ad series are all differet kids of its. its. Limits are oe way that mathematicias deal with the ifiite. First thigs first... First some otatio ad a few basic facts. Let f be a fuctio, ad let a ad L be fied umbers. The f( ) L a is read "the it of f() as approaches a is L" You probably have a ituitive idea of what this meas. Ad we ca do eamples:
For may fuctios......ad may values of a, it is true that a f ( ) f ( a) Ad it is usually apparet whe this is ot true. "Iterestig" thigs happe whe f(a) is ot well-defied, or there is somethig "sigular" about f at a. Defiitio of Limit So it is pretty clear what we mea by f ( ) L a But what is the formal mathematical defiitio? 5 6 Properties of real umbers Least upper boud property Oe of the reasos that its are so difficult to defie is that a it, if it eists, is a real umber. Ad it is hard to defie precisely what is meat by the system of real umbers. Besides algebraic ad order properties (which also pertai to the system of ratioal umbers), the real umbers have a cotiuity property. If a set of real umbers has a upper boud, the it has a least upper boud. 7 8 Importat eample The set of real umbers such that <. The correspodig set of ratioal umbers has o least upper boud. But the set of reals has the umber I a Advaced Calculus course, you lear how to start from this property ad costruct the system of real umbers, ad how the defiitio of it works from here. Official defiitio f() L meas that for ay ε >, a o matter how small, you ca fid a δ > such that if is withi δ of a, the f()-l < ε i.e., if -a < δ,
For eample. 5 5 because if ε < ad we choose < δ The for all such that 5 < δ we have 5 δ < < 5 +δ ad so ε 9. Top te famous its: + 5 δ + δ < < 5 + δ+ δ which implies ε ε 5 < δ + δ < + < ε.. (A) If < < the (B) If >, the si cos. ad 5. e ad e 6. For ay value of, 7. ad for ay positive value of, si does ot eist! e l 6-8. 9. l( ) + + e Basic properties of its I. Arithmetic of its: If both f ( ) ad g( ) eist, the f ( ) ± g( ) f ( ) ± g( ) a a a a a. If f is differetiable at a, the f ( ) f ( a) a a f '( a) f ( ) g( ) f ( ) g( ) a ad if g( ), the a a a f ( ) f ( ) a a g( ) g( ) a
II. Two-sided ad oe-sided its: f ( ) L if ad oly if a BOTH f ( ) L ad a+ III. Mootoicity: If f() g() for all ear the f( ) g( ) a a a a, f ( ) L 5 IV. Squeeze theorem: If f() g() h() for all ear a, ad if f ( ) h( ), the g( ) eists ad a is equal to the commo value of the other two its. si a si a 6 7 8 Let s s work through a few: Now you try this oe... + 5 + + 5 + t t t + A. B. C. -/ D. E. - F. G. - H. 9 Cotiuity Itermediate value theorem A fuctio f is cotiuous at a if it is true that f() f(a) a (The eistece of both the it ad of f(a) is implicit here). Fuctios that are cotiuous at every poit of a iterval are called "cotiuous o the iterval". The most importat property of cotiuous fuctios is the "commo sese" Itermediate Value Theorem: Suppose f is cotiuous o the iterval [a,b], ad f(a) m, ad f(b) M, with m < M. The for ay umber p betwee m ad M, there is a solutio i [a,b] of the equatio f() p. 5
Applicatio of the itermediate-value theorem f ( ) f ( ) We kow that f() - ad f(), so there is a root i betwee. Choose the halfway poit,. Maple graph Sice f()- ad f()+, there must be a root of f() i betwee ad. A aive way to look for it is the "bisectio method" -- try the umber halfway betwee the two closest places you kow of where f has opposite sigs. Sice f() - <, we ow kow (of course, we already kew from the graph) that there is a root betwee ad. So try halfway betwee agai: f(.5) -.65 So the root is betwee.5 ad. Try.75: f(.75) -.65 f ( ) We had f(.75) < ad f() >. So the root is betwee.75 ad. Try the average,.875 f(.875).8796875 f is positive here, so the root is betwee.75 ad.875. Try their average (.85): f(.85).957 So the root is betwee.75 ad.85. Oe more: f (.785).8986 So ow we kow the root is betwee.75 ad.85. You could write a computer program to cotiue this to ay desired accuracy. Let s s discuss it: Derivatives. What, i a few words, is the derivative of a fuctio?. What are some thigs you lear about the graph of a fuctio from its derivative?. What are some applicatios of the derivative?. What is a differetial? What does dy f '() d mea? Derivatives (cotiued) Derivatives give a compariso betwee the rates of chage of two variables: Whe chages by so much, the y chages by so much. Derivatives are like "echage rates". 6// US Dollar.65 Euro Euro.99 US Dollar (USD) 6// US Dollar.6 Euro Euro.9 US Dollar (USD) Defiitio of derivative: dy d h f ( + h) f ( ) h 5 Commo derivative formulas: p p ( ) p d d ( ) dg df f ( ) g( ) f ( ) + g( ) d d d d d d ( e ) e d ( l ) d d d d d ( si ) cos ( cos ) si d f ( ) g( ) f '( ) f ( ) g'( ) d g( ) ( g( ) ) d d ( f ( g( ) ) f '( g( )) g'( ) Let s s do some eamples.. 6 6
Derivative questio # 7 Derivative questio # 8 Fid f '() if f ( ) + 5 9/5 Fid the equatio of a lie taget to at the poit (,). y 8 + A. /5 B. /5 C. -8/5 D. -/5 E. -/5 F. /5 G. 8/5 H. -/5 A. 6+y6 B. +y C. -y D. 7+8y6 E. 5+y6 F. +5y6 G. +6y H. -y6 Derivative questio # 9 Derivative questio # Calculate A. B. C. D. d f d ( ) e + e ( ) ( ) e + ( ) e + if e f ( ) e ( ) E. F. G. H. ( 5) e + ( ) e + ( ) e + What is the largest iterval o which the fuctio f ( ) is cocave upward? + A. (,) B. (,) C. (, ) D. (, ) E. (, ) F. (, ) G. (, ) H. (/, ) Discussio Here is the graph of a fuctio. Draw a graph of its derivative. The meaig ad uses of derivatives, i particular: (a) The idea of liear approimatio (b) How secod derivatives are related to quadratic fuctios (c) Together, these two ideas help to solve ma/mi problems 7
Basic fuctios --liear ad quadratric. The derivative ad secod derivative provide us with a way of comparig other fuctios with (ad approimatig them by) liear ad quadratic fuctios. Before you ca do that, though, you eed to uderstad liear ad quadratic fuctios. Let s s review Let's review: liear fuctios of oe variable i the plae are determied by oe poit + slope (oe umber): y + (-) 5 6 Liear fuctios Liear fuctios occur i calculus as differetial approimatios to more complicated fuctios (or first-order Taylor polyomials): f() f(a) + f '(a) (-a) (approimately) Quadratic fuctios Quadratic fuctios have parabolas as their graphs: y +, y + 7 8 Quadratic fuctios They also help us tell... Quadratic fuctios occur as secod- order Taylor polyomials: relative maimums from relative miimums -- if f '(a) the quadratic approimatio reduces to f() ) f(a) + f '(a)(-a) + f "(a)(-a) /! (approimately) f() f(a) + f "(a)(-a) /! ad the sig of f "(a) tells us whether a is a relative ma (f "(a)<) or a relative mi (f "(a)>). 8
Positio, velocity, ad acceleratio: You kow that if y f(t) represets the positio of a object movig alog a lie, the v f '(t) is its velocity, ad a f "(t)" is its acceleratio. Eample: For fallig objects, y is the height of the object at time t, where iitial height (at time t), ad v y + v t 6t y is the is its iitial velocity. 9 Review - ma ad mi problems Also, by way of review, recall that to fid the maimum ad miimum values of a fuctio o ay iterval, we should look at three kids of poits:. The critical poits of the fuctio. These are the poits where the derivative of the fuctio is equal to zero.. The places where the derivative of the fuctio fails to eist (sometimes these are called critical poits,too).. The edpoits of the iterval. If the iterval is ubouded, this meas payig attetio to f ( ) ad/or f ( ). ( ) 5 Related Rates 5 More o related rates 5 Recall how related rates work. This is oe of the big ideas that makes calculus importat: If you kow how z chages whe y chages (dz/dy( dz/dy) ) ad how y chages whe chages (dy/d( dy/d), the you kow how z chages whe chages: dz dz d dy dy d Remember the idea of implicit differetiatio: The derivative of f(y) with respect to is f '(y) dy d The idea is that "differetiatig" both sides of a equatio with respect to " " [or ay other variable] is a legal (ad useful!) operatio. This is best doe by usig eamples... Related Rates Greatest Hits A light is at the top of a 6-ft pole. A boy 5 ft tall walks away from the pole at a rate of ft/sec. At what rate is the tip of his shadow s movig whe he is 8 ft from the pole? At what rate is the legth of his shadow icreasig? 5 Greatest Hits... A weather balloo is risig vertically at a rate of ft/sec. A observer is situated yds.. from a poit o the groud directly below the balloo. At what rate is the distace betwee the balloo ad the observer chagig whe the altitude of the balloo is 5 ft? 5 A ma o a dock is pullig i a boat by meas of a rope attached to the bow of the boat ft above the water level ad passig through a simple pulley located o the dock 8 ft above water level. If he pulls i the rope at a rate of ft/sec, how fast is the boat approachig the dock whe the bow of the boat is 5 ft from a poit o the water directly below the pulley? The eds of a water trough 8 ft log are equilateral triagles whose w sides are ft log. If water is beig pumped ito the trough at a rate of 5 cu ft/mi, fid the rate at which the water level is risig whe the depth is 8 i. Gas is escapig from a spherical balloo at a rate of cu ft/hr. At what rate is the radius chagig whe the volume is cu ft? 9
Itegrals Start with d -- this meas "a little bit of " or "a a little chage i " If we add up a whole buch of little chages i, we get the "total" chage of " -- A tautology questio: If you add up all the chages i as chages from to 7, what do you get? A. B. C. 5 D. 7 E. caot be determied 55 We write this i itegral otatio as: If y f(), the we write dy f '() d. To add up all the "little" chages i y" as chages from to 7, we should write or 7 df f '( ) d d 7 d... ad the aswer should be the total chage i y 7 as chages from to 7, i other words df d f (7) f () d This is the cotet of the fudametal theorem of calculus! 7 56 d 5 The fudametal theorem of calculus gives the coectio betwee derivatives ad itegrals. It says you ca calculate b g ( ) d a precisely if you ca fid a fuctio whose derivative is g(). Ad the result is the differece betwee the value of the "ati-derivative" fuctio evaluated at b mius the same fuctio evaluated at a. 57 Basic atiderivative formulas: + d + C + ecept for - cos( ) d si( ) + C si( ) d cos( ) + C d arcsi( + C ) d l( ) + C e d e + C d arcta( ) + C + 58 A quick eample 59 Fudametal Theorem Workout 6 Fid the value of A. 7/ B. C. D. 5/ E. F. / G. / H. / ( + ) d Let Fid the value of f '() -- the derivative of f at. A. B. 8 C. D. f ( ) t dt E. 5 F. G. 6 H.
6 6 A problem that was aroud log before the ivetio of calculus is to fid the area of a geeral plae regio (with curved sides). Ad a method of solutio that goes all the way back to Archimedes is to divide the regio up ito lots of little regios, so that you ca fid the area of almost all of the little regios, ad so that the total area of the oes you ca't measure is very small. By Newto's time, people realized that it would be sufficiet to hadle regios that had three straight sides ad oe curved side (or two or oe straight side - - the importat thig is that all but oe side is straight). Essetially all regios ca be divided up ito such regios. Ameba These all-but but-oe-side-straight straight regios look like areas uder the graphs of fuctios. Ad there is a stadard strategy for calculatig (at least approimately) such areas. For istace, to calculate the area betwee the graph of y - ad the ais, we draw it ad subdivide it as follows: 6 Sice the gree pieces are all rectagles, their areas are easy to calculate. The blue parts uder the curve are relatively small, so if we add up the areas of the rectagles, we wo't be far from the area uder the curve. For the record, the total area of all the gree rectagles is: 6 5 6 whereas the actual area uder the curve is: d Also for the record, 6/5 9.8 while / is about.6667. We ca improve the approimatio by dividig ito more rectagles: 65 Limits of Riema sums 66 Area 6 boes Now there are 6 boes istead of, ad their total area is: 78 which is about.97. Gettig better. We ca i fact 675 take the it as the umber of rectagles goes to ifiity, which will give the same value as the itegral. This was Newto's ad Leibiz's great discovery -- derivatives ad itegrals are related ad they are related to the area problem. A kid of it that comes up occasioally is a itegral described as the it of a Riema sum. Oe way to recogize these is that they are geerally epressed as somethig, where i the somethig depeds o as well as o i.
67 68 Gree graph Agai, recall that oe way to look at itegrals is as areas uder graphs, ad we approimate these areas as sums of areas of rectagles. This is a picture of the right edpoit approimatio to the itegral of a fuctio. has width a + ( f a + thus i( b a) i( b a) b a ) b a i ( b a ) ( b a)f( a+ ) i approimatig of f( ) If we are approimatig the itegral iterval from a to b usig rectagles, the each rectagle. The right side of the ith rectagle is at, ad so the area of the ith rectagle is. The sum of the areas of the rectagles is approaches ifiity is the itegral, ad the it of this sum as b f ( ) d. A eample will help a over the 69 7 i Eample... What is? i First, we eed a / for our (b-a)/. solutio So we ca rewrite the epressio uder i the summatio sig as ( )( ). Now we eed to figure out a ad b. For (b-a)/ /, we eed b-a. Ad i/ appears i the other factor, we should choose a. If f(), i ( b a ) ( b a)f ( a+ ) i the we'll have ( ). Thus the it of the sum is equal to d. Positio, velocity, ad acceleratio: 7 Eample... 7 Sice velocity is the derivative of positio ad acceleratio is the derivative of velocity, Velocity is the itegral of acceleratio,, ad positio is the itegral of velocity. (Of course, you must kow startig values of positio ad/or velocity to determie the costat of itegratio.) A object moves i a force field so that its acceleratio at time t is a(t) t -t+ (meters per secod squared). Assumig the object is movig at a speed of 5 meters per secod at time t, determie how far it travels i the first secods.
Solutio... First we determie the velocity, by itegratig the acceleratio. Because v() 5, we ca write the velocity v(t) as 5 + a defiite itegral, as follows: t t t v( t) 5 + a( τ ) dτ 5 + τ τ + dτ 5 + + t The distace the object moves i the first secods is the total chage i positio. I other words, it is the itegral of d as t goes from to. But d v(t) dt. So we ca write: (distace traveled betwee t ad t) t t 5 + + t dt t v ( t) dt 95/ 6.666... meters. 7 Methods of itegratio Before we get too ivolved with applicatios of the itegral, we have to make sure we're good at calculatig atiderivatives. There are four basic tricks that you have to lear (ad hudreds of ad hoc oes that oly work i special situatios):. Itegratio by substitutio (chai rule i reverse). Trigoometric substitutios (usig trig idetities to your advatage). Partial fractios (a algebraic trick that is good for more tha doig itegrals). Itegratio by parts (the product rule i reverse) We'll do # this week, ad the others later. LOTS of practice is eeded to master these! 7 Substitutio I some ways, substitutio is the most importat techique, because every itegral ca be worked this way (at least i theory). The idea is to remember the chai rule: If G is a fuctio of u ad u is a fuctio of, the the derivative of G with respect to is: 75 e could be cosidered as e u where u. To differetiate d e d For istace... e the, we use that the derivative of e u is e u : d du u d u ( e ) ( ) e ( ) d e 76 dg d G'(u) u'() Now we ll tur this aroud... To do a itegral problem... 77 I geeral... I substitutio, you 78 For a problem like we suspect that the should be cosidered as u ad the d is equal to du/. Ad so: e e d u du u u d e e du e + C e + C. Separate the itegrad ito factors. Figure out which factor is the most complicated. Ask whether the other factors are the derivative of some (compositioal) part of the complicated oe. This provides the clue as to what to set u equal to.
( + ) d 5 Here s s aother oe: -- the complicated factor is clearly the deomiator (partly by virtue of beig i the deomiator!) ad the rest (( d) ) is a costat times the differetial of -- but it's a good idea to try ad make u substitute for as much of the complicated factor as possible. Ad if you thik about it, d is a costat times the differetial of +5!! So we let u +5,, the du d, i other words d du /. So we ca substitute: d ( ) du + C + u 5 8( + 5) u + C 79 Now you try a couple... A) B) / C) π / cos d D) π/ Ε) π 8 π/ 8 Fid sec si(ta ) d A) π/ E) π/ - si B) -π/ C) si D) - cos F) π/ + cos G) + π/ H) + ta