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# Calculus for Physical Scientists I (GT MATH 160

CSU

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This 14 page Class Notes was uploaded by Melvina Keeling on Monday September 21, 2015. The Class Notes belongs to MATH 160 at Colorado State University taught by Hilary Smallwood in Fall. Since its upload, it has received 64 views. For similar materials see /class/210086/math-160-colorado-state-university in Mathematics (M) at Colorado State University.

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Date Created: 09/21/15

MATH 160 Sections 41 44 Practice Problems 1 Items 1 6 give information about the rst and second derivative of three continuous and dif ferentiable functions 917 and Match the given pieces of information 1 6 With the mathematical statements a k they must indicate about the behavior of their respective functions f17910r MI 7 1 WP gt 0 2 hz gt 0 on 721 73 9 0 0 74h 1lt 0 7 5 y 0 0 76 lt0forzgt1 a the function has a global maximum herei b the function is increasing herei c the function could change concavity herei d the function must change concavity herei e the function is concave up here f the function is decreasing herei g the function has a critical point here h the function has a local minimum here i the function has a local maximum herei the function is concave down here k the function has a global minimum herei 2 b The hne y z c on me yayh you drew abuve CIRCLE all Lhelucal schema ufz is one Dime asymytma Di Lhe yayh a on me yayh you drew abuve BOX 1th5 m ech n yum Name MATH 160 Calculus for Physical Scientists I Fall 2011 Section Smallwoog 001 Calculator Investigation Date due Mondax November 7 Calculator Calculating Riemann Sums Evaluating a Riemann integral with pencil andpaper from the de nition is usually impractical However it is possible and even practical to approximate most Riemann integrals very accurately from the definition by using a calculator The investigations in this lab require a calculator that can produce tables of function values and traceable graphs While many makes and models of calculators have these capabilities the author used Texas Instrument calculators as he wrote this lab The lab does not include instructions for using a calculator Use the manual for your calculator to learn how to perform the tasks in this investigation e iciently and accurately Manuals for Texas Instrument calculators can be read from the Texas Instrument web site Go to httpeducationticomusglobalguideshtrnl You can find instructions for many d erent makes and models of calculators on the web The following factors will be considered in scoring your lab report 0 Completeness Each investigation must be completed entirely recorded fully and explained or interpreted clearly Mathematical and computational accuracy Clarity and readability Explanations must be written in complete sentences with correct spelling capitalization and punctuation and with reasonable margins and spacing Handwriting must be legible Tables and graphs must accurate and presented in a clear readable format We recommend thatyou keep a copy ofyour completed lab report You may want to refer to the work you did on this lab before it is graded and returned Also lab reports have been known to be lost We encourage you to work with IVIATH 160 classmates on homework and calculator investigations However you must write up your own answers in your own words Submitting material copied from a lab report or homework assignment that someone else wrote either this semester or in a previous semester is an example of plagiarism and a violation of the policy onAcademic Integrity When there is compelling evidence that plagiarism has occurred on a lab report or homework assignment in keeping with university policy an academic penalty will be imposed You have the right to examine the evidence which is the basis for an allegation of plagiarism to provide additional information and to appeal any finding of academic dishonesty andor penalty imposed You are invited but not required to sign thefollowing a irrnation I a irrn that although I may have worked with classmates on this investigation this final lab report is written in my own words and accurately indicate my 1 139 ofthe 1 1 39 and ideas in this investigation signature I worked on this investigation with thefollowing people listfirst and last names 2006 Kenneth F Klopfenstein Fort Collins CO 2 n Recall that J x3 1 dx means the limit of the Riemann sums Sn Z 0 1 Ax In these sums n is 1 k the number of subintervals into which the interval from 1 to 2 is divided by the equally spaced partition points 1 x0 ltx1 lt ltxn1 ltxn 2 The length of each subinterval is Ax The subintervals don t have to be all the same length but calculations are easier when they are Each ck is a number from the k3911 2 subinterval This limit statement tells us that J x3 1 dx is the number that can be approximated as closely as 1 VI anyone could want though not necessarily exactly by Riemann sums 0 1 Ax computed using a k1 partitioning of the interval 1 S x S 2 into a large enough number of equal subintervals Thepartition points x0 ltx1 ltx2 lt ltxn1 ltxn that divide the interval of integration 1 Sx S 2 into subintervals are at the heart ofthe idea ofa Riemann sum But the evaluation points 01 02 c3 c4 cm on are more prominent in calculating the value of aRiemann sum The key to learning to use your calculator to evaluate Riemann sums is to learn to calculate with LISTS of evaluation poinm 2 11 We begin by calculating several Riemann sums S6 for the integral J x3 1 dx based on dividing the 1 interval of integration into 6 subintervals of equal length On the graph of fx x3 1 next page divide the interval 1 2 on the xaxis into 6 equal subintervals Mark and label the endpoints of these subintervals x0 x1 x2 x3 x4 x5 and x6 The seven endpoints of these six subintervals are the partition points We ll first calculate a Riemann sum using the le endpoinm of the subintervals as the evaluation points With this choice the evaluation poinm 01 02 c3 c4 05 06 coincide with the first six partition points x0 x1 x2 x3 x4 x5 Step 1 Read your calculator manual to find out how to create a list Enter the list 1 05 0 05 1 15 of left endpoinm of the subintervals from above into your calculator Store this list under a name that suggests Left Endpoints 1 suggest LFTND or Iftnd Step 2 To compute a Riemann sum S6 we need to evaluate the integrand fx x3 1 at each evaluation point in the list LLFTND T1 calculators will create a new list by evaluating a function at each entry of a specified list Enter the integrand fx x3 1 as Y1 on the Y screen of your Tl83 or Tl84 or as y on the GRAPH screen of your T189 Return to the home screen and enter Y1 LLFTND On a Tl83 or Tl84 to enter LLFTND press 2m LIST scroll to LLFTND and press ENTER On a T189 to enter Iftnd press 2nd VARLINK scroll to Iftnd and press ENTER Press ENTER a second time A list of the values of the integrand at the six evaluation poinm in the list LLFTND will be displayed You may want to store this list under a name that suggests Function Values 1 suggest FNVLS or fnvls The numbers in this list are the successive heighm of the 6 rectangles that approximate the area under the graph and hence the integral Draw these approximating rectangles on the graph on the next page 2006 Kenneth F Klopfenstein Fort Collins CO cummme Page af7 szp 3 Thelenghaf 6 15mm number Ta nmhe areaafeauh um appmxxmahngrenangesmLheRxemann sum mulhply 5 u TL A r l m Step t mmesxnthe 1m uf areas yvujust created The sum cammm adds 511 m enmesm shit mrm axT184 puss2mUSU scmllm MATH andchaase sum rm Lhamenu on 5T1789 pxess2mWAYHscmlltaUSTpuss39he ngmtmaw m c as sum rm Lhamenu Appmh sum cammmdtathzhstafarus ufxenangeswse swamANS andvenfy um 55mmapmmwumms 3 25n969n74 rm mum 2 E s 2 g a i h 2 mm 2mm Kennsz Klapfenstem n Cnllms co Calculator Lab Calculating Riemann Sums Page 2 of 7 2 12 The approximation for J x3 1 dx calculated inll is not very accurate 71 a Use the graph of x 1x3 1 above to explain why S 6 calculated with left endpoint evaluations is 2 smaller than the actual value of J x3 1 dx 71 b Without evaluating the integral or fin ding the actual area estimate the amount not the percentage by which S6 calculated with left endpoint evaluations differs from the actual value of Z J x3 1 dx Explain how you arrived at this estimate Your method for estimating error must be 71 plausible Your explanation must be consistent with method you used to estimate the error 2 c Riemann sum approximations for J x3 1 dx calculated using left endpoints as evaluation points will 71 always be smaller than the actual value of the integral What property of the function accounts for this Use a graph and words to explain why it is that if a function y fx has the property you just stated then 2 every Riemann sum approximation for I f x dx calculated using left endpoints as evaluation points 71 will be smaller than the actual value of the integral Page total 2006 Kenneth F Klopfenstein Ft Collins CO cummme Page W7 2 m NeXLcalcu ateLheRxemannsum 55 furthemlegal I x3l 11x mmgthznghtendpmnlsuflhe 4 m mmmmxnsjmmmmnapmms Tamuxmmmmnapmms 717n5nn51 1 Sfmm mm mypress 2quot USU mu m m mm Wu gave the m a evaluauanpmnts andpxess ENYER Adang munbm alxstadds thenumbexta eachenkyafthz 15L salmtadd 5 mums 1st Emu 5 2 ms mm SimeOannwhstundex nnamethatsuggzs39s RxgmtEndpamls I suggest RTND ax mm explmn a mpsuf mscalcu auanan39he gaphaf x x3l bdaw h 2 mm 2mm Kennsz Klapfenstem n Cnllms co Calculator Lab Calculating Riemann Sums Page 4 of 7 2 112 The approximation for J x3 1 dx using right endpoint evaluations is not very accurate 1 a Use the graph of x 1x3 1 above to explain why S 6 calculated with right endpoint evaluations is 2 larger thanthe actual value of J x3 1 dx 1 b Without evaluating the integral or fin ding the actual area estimate the amount not the 2 percentage by which S6 calculated with right endpoint evaluations differs from J x3 1 dx 71 Explain how you arrived at this estimate 2 c Riemann sum approximations for J x3 1 dx calculated using right endpoints as evaluation poinm 1 will always be larger than the actual value of the integral What property of the function accounm for this Use a graph and words to explain why it is that if a function y fx has the property you just stated then 2 every Riemann sum approximation for I f x dx calculated using right endpoints as evaluation poinm 1 will be larger than the actual value of the integral Page total 2006 Kenneth F Klopfenstein Ft Collins CO cummme PageS af7 nu Calcu a39z 55 uangmxipmnlewluauans Yau dm39thave m enter the midpmms asamw 1st Yunnan S mlhegaphaf x x3 belww wuthmk 5a m 2 mm 2mm Kennsz Klapfenstem n Cnllms co Calculator Lab Calculating Riemann Sums Page 6 of 7 lVl a Most calculators will evaluate integrals numerically Use the manual for your calculator to learn how to Z evaluate integrals numerically Evaluate J x3 1 dx and record the result 71 b Which of the approximations to the integral calculated above is closest to the value your calculator gives for the integral Explain why one might expect that c If possible draw the graph of a continuous function y fx on the interval fl 2 and a partition of the interval fl 2 so that the Riemann sum calculated using left endpoint as evaluation points gives a 2 better approximation for I f xdx than the Riemann sum calculated using midpoints as evaluation 71 points If it is not possible to graph such a function and partition explain how you know 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