CALCULUS. Taimur Khalid

Transcription

1 CALCULUS Taimur Khalid 5/20/2015

2 Chapter 1 What is Calculus? What is it? I m sure you ve all already heard about what calculus is from the other studets i my calculus class, but just i case you eed a refresher, here you go. I my opiio, to create a cocise, sigle setece defiitio of calculus that thoroughly coveys the true idea of what calculus is about would be close to impossible. I thik istead, that to properly teach the idea behid the essece of calculus (what calculus really is) it would require much more tha just a setece. But seeig as I did ot choose to preset to you all solely about what calculus is, I ll give it my best shot to sum it up i oe setece. As simple as I ca make it, Calculus is the brach of mathematics that deals with rate of chage ad accumulatio. If you do t believe that Calculus is importat or useful i the slightest, hopefully I ca chage your mid a little bit before I begi the actual lesso. Why is it Useful? Rather tha give you the exact details, I ll simply tell you where Calculus is used i the real world. For starters, math related occupatios use calculus such as actuaries, mathematicias, ad statisticias. Calculus is also used i early every sigle type of egieerig you ca thik of, Biological Sciece, Medical Sciece, Physical Sciece, Ecoomists, Social Scietists, Medical ad Health Service Maagers, Roofers, ad eve Fueral Directors. Oe of my persoal favorite thigs about Calculus is that it s the first time you ll realize you do t eed aywhere ear the amout of iformatio ormally give i problems to solve them! To Ifiity ad Beyod! Oe of the recurrig themes i calculus is the cocept of ifiity. Studets usually have a hard time (at least i my opiio :P) wrappig their head aroud it, so I m goig 2

3 to discuss that as part of this presetatio. Ifiity is t actually a umber, it is, as I previously metioed, a cocept. There s obviously o way to have a ifiite amout of somethig. I additio, dealig with ifiity as a umber ca lead to some wacky false results i math. Lets see what would happe if we treated ifiity as a umber. If we were to keep addig the coutig umbers we would get ifiity. So = Now we re assumig is a umber, so let s subtract 1 from both sides. This results i But we also kow that is ifiite! So we have = = 1. Subtractig from both sides, we get 0 = 1, which is obviously ot true! However mathematicias, beig the clever people they are, have foud ways to tame ifiity. For example, a geometric sequece is a list of umbers i which the previous terms is multiplied by a fixed umber to get the ext term. Let s cosider the geometric sequece 1, 1 2, 1 4, 1 8,... What would happe if we were to add all of these terms up? I this case, we actually get a umber! Each terms is gettig smaller ad smaller, so we might assume that it ca oly add so much before it reaches some sort of limit. So let s say that the value of the sum is X. The we have If we multiply both sides by 2, we get = X = 2X. 8 But we kow the value of a huge chuk of this! Let s take a closer look at the secod equatio I wrote out = 2X. 8 From our first equatio we kow the boxed part is just X! So substitutig we get 2 + X = 2X. So what is X? You should be able to aswer that for yourself! :) I this sese, we ca actually use the cocept of ifiity to produce extremely accurate results. Say you wated to pay a cashier the exact amout that you were charged. If you oly had a 100 dollar bill, your goal probably would t be accomplished. But if istead you 3

4 had te 10 dollar bills, you would have a higher chace of beig able to pay him. Eve better say you had oe hudred 1 dollar bills. What would be eve better tha that? If you had te thousad peies, the you would be able to pay the cashier exactly what you were charged. Basically, oe of the big ideas of calculus is lookig at extremely large amouts of thigs to better the approximatio dealig with that thig. The more ad more you have, the closer your approximatio gets. Or the less you have of somethig, the closer it gets. Limits I my previous example with the geometric series, the sum gets closer ad closer to the umber 2, but it does t actually ever reach that umber. Techically speakig, the sum will always be a very very small amout away from the umber 2. But what we ca do, is we ca say that as the amout of terms approaches ifiity, the sum gets ifiitesimally close to the umber 2. Mathematically, this is called the limit. If we wat to write this formally, we would say lim #of terms = 2. 8 Limits are essetially how we tur approximatios ito exact values. So What s the Area of a Circle? We all kow the area of a circle is πr 2 Pretty easy huh? But does ayoe kow why? How do you actually come up with that formula that seems so well kow? Well let s try to figure it out together. I math we always try to break dow complex thigs ito simpler thigs that we kow how to deal with. Oe of my favorite quotes: The essece of mathematics is ot to make simple thigs complicated, but to make complicated thigs simple. S. Gudder Breakig it Dow How do you fid the area of a triagle? Most probably everyoe oly remembers the classic base height 2. However, we ca also express the area slightly differetly, which I m goig to require for derivig the area of a circle. 4

5 A B α D I the diagram above, we have ABC with ABC = α ad height AD. Now usig the simple area of a triagle, we ca say that the area of ABC is BC AD 2 However, if we ca recall some of our trigoometry kowledge, we have or si α = AD AB AB si α = AD Pluggig this i we get that the area of the triagle is simply. BC AB si α 2 C Gettig Back To The Circle To approximate the area of a circle, I m goig to use regular polygos. Regular polygos are just shapes that have equal sides ad equal agles. As I previously metioed, the more ad more of somethig we have, the more accurate it gets (usually). Let s look at the diagrams below. 5

6 = 3 I this diagram, a regular triagle is iscribed withi the circle (a regular triagle is just a equilateral triagle). As you ca see, the triagle is a pretty terrible approximatio of the area. The area of the triagle is i blue, ad the remaiig area is i red. I the followig diagrams, the sides of the polygos will gradually icrease. Be sure to ote what happes to the red area (the differece betwee the area of the shape ad the circle). = 4 = 5 6

7 = 9 = 12 Aha! As we icrease the amout of sides of the polygo, the closer ad closer it gets to the area of the circle. But there s still a problem. We do t kow how to deal with icreasig the sides of the polygo ad it s relatio with area :( So istead, lets try to make it ito somethig we do kow. We ca split up a polygo ito smaller triagles with oe vertex at the ceter of the circle ad the others at the vertices of the polygo. = As we figured out before, for us to make the shape as close to possible as a circle, we wat it to have as may sides as possible. So we wat to take the umber of sides, ad make it approach ifiity. 7

8 There are triagles ad the cetral agle formed by each triagle will be 360 because there are sides, ad each of the blue segmets has legth r, where r is the radius of the circle. So if we re goig to use our triagle method, makig approach ifiity would make the cetral agle approach 0. The reaso for this is because the larger ad larger gets, the smaller ad smaller 360 gets (dividig by a larger umber results i a smaller fractio). Now keepig i mid all these facts, we ca go ahead ad try to express the area of this polygo usig idividual triagles, ad at the same try to maximize that area. The area of the polygo will be equal to 1 2 r2 si 360 Now here is where it gets a bit complicated. We wat to take the limit of this expressio as i order to make it as close to a circle as possible (icreasig the # of sides, remember?). But evaluatig limits is somethig you ll lear i calculus. Ayways, we proceed. 1 lim 2 r2 si 360 The above expressio will essetially evaluate the area of a polygo with ifiite sides, or a circle. But to evaluate we must use a clever trick. We let. The as So the expressio becomes 1 = x x 0 r 2 si 360x r 2 si 360x lim = 180 lim x 0 2x x 0 360x Now we ca let 360x = y, ad it further simplifies to r 2 si y lim y 0 y Oe of the special limits you will evetually lear i Calculus goes as follows: si lim 0 = 1 Usig this we have the area of the circle is equal to 180 lim y 0 r 2 si y y 8 = 180r 2

9 . But that does t look quite right. 180 is i degrees, so we must covert this to radias to get our fial aswer. If you remember, we have the followig relatio 180 = π Thus the area of the circle is πr 2 There you have it! We successfully derived the area of a circle! The power of Calculus... :) Aother thig that Calculus is useful for is fidig istataeous rates. If you remember, i precalculus you are taught how to graphically represet the average rate of chage by drawig a secat lie through two poits o the graph, the fidig the slope of that lie. The istataeous rate of chage however, is the slope of the taget lie at the poit you are fidig the rate. Now agai, the same cocept appears! If you were to take the secod poit o the secat lie ad keep o movig it closer ad closer to your first poit, evetually you would get the taget lie. The Derivative Let s start of this sectio with a little otatio. x is defied as the chage i x ad f(x) = y. Now let s say we selected two poits (c, f(c)) ad (c + x, f(c + x)). The slope of the lie betwee these two poits would be f(c + x) f(c) (c + x) c = f(c + x) f(c) x Now because we wat to make this the istataeous rate of chage, we wat the two poits to be as close together as possible. I other words, we wat to make the chage betwee the x values, x as small as possible. So we set up the followig expressio for the istataeous rate of chage f(c + x) f(c) lim x 0 x This expressio is kow as the derivative. The most commo otatios used for the derivative are dy dx f (x) ad y (x). Fortuately eough, people have come up with ways to simplify this extremely tedious/tiresome computatio of the derivative. Istead of usig the limit defiitio, there ow exist shortcuts (although you should ever just memorize, always kow why they 9

10 are what they are). For example, if I were to ask you to evaluate the derivative of x 4, the there would be two ways to do it. Oe would iclude usig the limit defiitio of a derivative, ad the other would be usig the shortcut. The shortcut says if you have some fuctio of the form f(x) = ax the dy f(x) = ax 1 dx. Ayways, I m ot goig to sped too much time goig i depth o these thigs, because you ll lear them oce you get to Calculus. 10

11 Chapter 2 Itroductio to Itegrals That Looks Like a Stretched Out S The itegral is basically the opposite of the derivative. Just as divisio is to multiplicatio, the itegral is to the derivative. The itegral is a accumulatio fuctio, ad is deoted by what looks like a stretched out S. The symbol for the itegral is f(x)dx It must always be eclosed as such. Usig Itegrals Oe of the most basic fuctios of itegrals is fidig area udereath curves. Agai, to build a idea of what itegrals do we must first start off with our favorite cocept! Let s take a look at a radom curve f(x) = x 2, ad say we wat to fid the area udereath this curve from whe x = 0 to whe x = 8. Now we eed someway to approximate this area, because of course we do t kow of ay other way to deal with it! Geerally i math we try to make thigs as simple as possible, so to approximate the area, we just choose rectagles. Whe there are 2 rectagles, each with width 4, we see that our guess for the area is pretty off (just as we saw with the circle). So what should we do to make our approximatio more accurate? We icrease the umber of rectagles. So let s take a look at 4 rectagles, each with width 2. Better! But still, its missig a lot of area. I m pretty sure you get the idea ow. Here s a summary of a few of the basic itegratios formulas you ll lear. 11

12 kf(u)du = k f(u)du [f(u) ± g(u)]du = du = u + C f(u)du ± g(u)du u du = u C O aother ote, whe itegrals have bouds (umbers below them ad above them) that tells you from where to where you re fidig the area, ad they are called defiite itegrals. Whe they do ot have bouds, they are called idefiite itegrals. To evaluate a itegral with bouds, you first evaluate it as it would be without bouds, the plug i each of your bouds ad subtract the lower oe from the upper oe. For example: Evaluate 3 0 x 2 dx The solutio to this would be to first evaluate x 2 dx which we fid to be (usig the fourth item) The we have [ x 3. 3 ] 3 0 x 3 3. = = 9 Where Algebra 2 ad Precalc Come Ito Play Do you remember your trig. formulas or partial fractio decompositio? Well you re goig to have to for AP Calc. Here s a example of oe of the questios you could ecouter dealig with trigoometric idetities. Evaluate For this questio you must kow that π 0 cos 2 (θ) = cos 2 (θ)dθ 1 + cos (2θ) 2 12

13 Aother possible example of a itegratio questio would be: Fid the idefiite itegral 1 4x 2 1 dx For this oe it is ecessary to kow partial fractio decompositio. This would be the first step: 1 4x 2 1 = A 2x 1 + B 2x + 1 From here we would rearrage it to get the equatio ad the we would solve for A ad B. A(2x + 1) + B(2x 1) = 1 So that s a itroductio to calculus, thigs you ll ecouter, ad the uses of the subject. I hope you ejoyed it! 13