5.1 Estimating with Finite Sums Calculus


 Clarence Conley
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1 5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during this time? Emple: Sketch grph modeling the sitution in the ove emple. Geometricll, how cn we indicte the totl distnce trveled? Emple: Wht if the velocit ws NOT constnt. S, for instnce the velocit in miles per hour is given the function vt () = 10t t, where t is in hours, nd we wnted to know the totl distnce trveled during the first 10 hours. Sketch this grph elow. Geometricll speking, do ou think we cn find the totl distnce trveled in the sme ws s efore? Wh or wh not? Word of cution to those rve few who re ctull reding this The following prgrphs re etremel importnt to the conceptul understnding of wht we re out to do in Clculus. However, since ou rell hven't done nthing et, it might mke ou little dizz t first, so come ck nd red it gin lter. If ou're reding this for the first time, ou might eperience tht sme feeling ou get when ou've een on the Tilt A Whirl one too mn times t the fir. (Never een on the Tilt A Whirl? well, tke m word for it, it's not something ou wnt to ride 10 minutes fter eting corn dog nd funnel cke!) Well, I wrned ou, ut ou've kept on reding nw, so here it goes The ke to finding the totl distnce trveled in the lst emple in method similr to the first emple is to rek the time intervls into such short segments, tht the velocit over those time segments is lmost constnt (this will require quite few intervls). If the velocit is lmost constnt for ech time intervl, then we cn find the distnce trveled for ech time intervl (which is just the re of n etremel thin rectngle) nd dd ll the res of ll the rectngles together. Sounds simple enough, right? Cn ou guess wht etremel importnt clculus concept is involved? We will spend MUCH more time with this lter, ut it turns out tht if we re given the grph of rte of chnge (like velocit in miles per hour) we will e le to find the totl ccumulted chnge over n intervl (like totl distnce trveled, in mile) finding the re under the curve. OK, tht lst prgrph or two m not hve mde perfect sense to ou YET! For now, THE GOAL is to figure out w to find the re under the curve. This chpter ctull discusses 5 ws to pproimte this re, ut we're onl going to del with 4 of them. 8
2 The Are Prolem nd the Rectngulr Approimtion Method (RAM) The limit process cn e used to find the re under curve, nd we will get into this in more detil in the net section. Suppose we wnted to know the re of the region ounded curve, the is, nd the lines = nd =, s shown elow. The first step is to divide the intervl from to into suintervls. The emples elow show 4 nd 8 suintervls, respectivel. After dividing the given intervl into suintervls, we cn then drw rectngles using the width of ech suintervl. The height of ech rectngle is determined the function vlue t point in the specific suintervl, nd cn e determined using 3 different methods. We could use the left endpoint of ech suintervl (clled LRAM), the right endpoint of ech suintervl (RRAM), or the midpoint of ech suintervl (MRAM). Which method is shown elow? Emple: The totl re under the curve then is pproimtel equl to the totl re of ll the rectngles. Which of the grphs ove gives etter pproimtion of the re under the curve? Wh? How could it e further improved? Summr of the Process: A sketch is lmost mndtor! Step 1: Divide (or Prtition) the intervl into n suintervls. Step : Crete n rectngles whose se equls the width of ech suintervl nd whose height is determined the function vlue t the left endpoint, the right endpoint, or the midpoint of the suintervl. Step 3: Find the re of ll n rectngles nd dd them together. 83
3 5 Emple: The grph of = is shown twice elow. On the left picture pproimte the re under the curve from = 1 to = 5 using LRAM with 4 rectngles. On the right picture, pproimte the re under the curve from = 1 to = 5 using RRAM with 4 rectngles. Sketch the rectngles on ech curve. Emple: Approimte the re under the curve from = 1 to = 5 using MRAM with 4 rectngles. Sketch the rectngles on the curve. Emple: It is not necessr to hve grph to estimte the re. Suppose the tle elow shows the velocit of model trin engine moving long trck for 10 seconds. Estimte the distnce trveled the engine, using 10 suintervls of length 1 with () left endpoint vlues (LRAM) nd () right endpoint vlues (RRAM) Time (sec) Velocit (in./sec) Time (sec) Velocit (in./sec) : All the emples on this pge hd suintervl length equl to 1. This is not lws the cse, ut ws done to mke the initil emples strightforwrd. Tr doing the first emple gin using 5 rectngles insted of 4. 84
4 The Trpezoidl Rule (Rell 5.5) While rectngles mke firl good pproimtion, it's es to see tht we're going to need lot of them to provide good estimte. We cn find etter estimte in less time if we use trpezoids. If we were to prtition the intervl into suintervls like we did efore, we cn use ech suintervl to crete trpezoid if we just connect the function vlues of the left nd right endpoints. Before we egin, let's mke sure ou understnd the re formul for trpezoid. 1 Are of Trpezoid: A = h ( 1+ ) While not ll trpezoids must look like this, the one's we're going to e using will, so we'll stick with this picture. Lel ll the prts of the re formul on the picture elow. The iggest difference will e the orienttion of the trpezoid. The ones we re going to e drwing will look like Drw set of es on the picture ove nd function tht goes through the top left nd top right points of the trpezoid. The "height" of the trpezoid is just the width of suintervl, nd the "ses" re going to e the function vlues of the left nd right endpoints. Emple: Let's go ck to the sme function we used efore. Use 4 trpezoids to pproimte the re under the 5 curve = from = 1 to = 5. Sketch the trpezoids on the curve. 85
5 While ll we're rell doing is finding the re of unch of trpezoids, there is lws formul ville. The Trpezoid Rule To pproimte the re under curve on the intervl [, ] use h T = ( n 1+ n) ( ) where [, ] is prtitioned into n suintervls of equl length h =. n Proof: For ll ou formul memorizers the ke words in the ove formul is EQUAL LENGTH. It doesn't work so well in the following emple: Emple: [1998 AP Clculus AB #85 with clcultor] The function f is continuous on the closed intervl [, 8] nd hs vlues tht re given in the tle elow f () Using the suintervls [, 5], [5, 7], nd [7, 8], wht is the trpezoidl pproimtion of the re under the curve? A) 110 B) 130 C) 160 D) 190 E) 10 B the w The trpezoid rule connects the left nd right hnd endpoints with segment. This method of pproimtion turns out to e prett good, ut if ou were to connect the endpoints with curve (nmel prol) the pproimtion would e even etter. Connecting the endpoints with prol nd finding the re of the resulting shpe is the sis ehind the fifth method of pproimtion clled Simpson's Rule. You cn red out it on pges if ou find ourself just ding of curiousit, ut it's not on the AP em. 86
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