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# Survey of Calculus MATH 160

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This 17 page Class Notes was uploaded by Breanne Schaden PhD on Saturday October 3, 2015. The Class Notes belongs to MATH 160 at Boise State University taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/218006/math-160-boise-state-university in Mathematics (M) at Boise State University.

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Date Created: 10/03/15

16 RIEMANN SUMS In Chapter 13 we introduced the right and lefthand sums to approximate the definite integral In this chapter we use a program to simplify explorations of these and other types of sums We will add the capability to graphically view the subdivision areas that sum to make the approximation A few words about programs This marks our first use of a stored program so perhaps an introduction is in order A program is a set of commands that are performed in a prescribed order The order is normally the sequential list of commands but there are techniques to alter that order A program is written by pressing PRGM arrowing to NEW and pressing ENTER see Figure 161 You then enter a name and the sequence of commands that are desired Commands are entered from menus by pasting most contain lower case letters which make them easy to spot in the program listing What is not always so easy is where to find the command to be paste remember the 2ndCATALOG option Use 2ndQUIT to exit the program editing mode E EC P EMF LBCreate ew FOURIER FOURIER 3ancos 3LIMIT 4LIMIT 4RSUM 5 MINE 5 THVLOR 6 POLYSDLV 7 RSUM Figure 161 The PRC7P1 submenus A program is activated the more common term is run or executed by pressing PRGM selecting the program name and then pressing ENTER Stepby step instructions can be found in the programming chapter ofthe Guidebook Entering a program from the printed page is quite tedious and you can commonly expect to make a few errors that will only be discovered when executing the program However once a program is correctly entered in a TI calculator it can be transferred to other Tl calculators using the builtin 2ndLNK commands In a classroom it is typical that a program is verified by the instructor and distributed to the class using 2ndLNK You can download additional programs from the Web this is outlined in the Appendix The program pasting error Tip Programs are run from the home screen Be sure the command line is clear before starting to execute a program The most common error when you first use programs is to forget that programs are pasted onto the home screen and then executed For example you might be in the middle of creating a command to sum a sequence when you remember that you have a program to do it Pressing PRGM and selecting the program name RSUM will paste the command PrngSUM at the end ofthe line you were on Some kind of error message results because the PRGM command was not on its own line sunltse 1ltV1 I I 1511gtPP9MRSUHI ERR DHTH TVPE Quit Figure 162 Pasting a command to execute a program can cause an error if the home screen cursor is not on a new line Using the RSUM program to find Riemann sums w I NDow v1xinum Xn1na Xnax25 ng11 Vm1n3915 Vnax15 Vscl1 Xres1 x1zs v5955555 Figure 163 Set V and the window before using the RSUM program A program to automate the Riemann sum process is given at of the chapter It is designed to require a minimum of input to the program Before using it you must define the function in Y1 and set the window to have XMin be the left endpoint and XMax be the right endpoint of the desired interval The program prompts you for the number of subdivisions partitions of the interval and then displays five different kinds of Riemann sums The value labeled TRUE is ntntY1XXMinX1lax and is as such only another numerical approximation SIMP stands for the value using Simpson39s method This is a weighted average of the two previous results specifically the sum of twice the midpoint value and the trapezoid value all divided by three You will find that this value is consistently close to the 39true39 value for small partitions u now TRUE 4305177438 SEEE a gr hf 28 32 39 MID 14168949965 3 MIDPT 0 QUIT N 1039 TRHP 4572397981 4TRHP SIMP 43a3426637 5QUIT GRHPH lteN 1vgt Figure 164 The RSUM program gives six numeric approximations of the de nite integral of V from Xmin to Xnax Mann HSZOSZH ili39 il i EII IWZZBI Figure 165 T hefour graphic representations of the Riemann sum left right midpoint and trapezoid In Figure 165 the graphic representations of the Riemann sums are shown in the order of the four menu choices from Figure 164 A set of vertical dots shown in the upper right comer will appear on the active screen as a twinkling This alerts you that the program is in Pause mode Program pauses are needed so that you can inspect the graph before returning to the menu Press ENTER to continue through a pause The Tl82 program RSUM The program listed below takes a considerable amount of time to enter but once entered can be passed to other Tl calculators ofthe same type It is in the public domain It can also be stored on a computer using the TI Graph Link For faster entry you can use a copy technique on repetitive code The Lb1 through Lb5 sections are similar You can enter one section as a new program and use 2ndRCL PRGM EXEC to paste in four copies ofthe section then lightly edit them where they differ FnOff FnOn 1 Lbl 0 Cerome Disp quotFOR Y1 IN WINDOWquot Disp quot u Disp quotSET PARTITIONSquot Input quot0QUIT NquotN If Nlt1 Stop Cerome XMax XMinN gt D Disp quotTRUEquot ntntY 1 X XMin XMax gt U Output1 6 U Disp quotLEFTquot sumseqY XMinID I 0 N 1 1D gtL Output2 6 L Disp quotRTquot sumseqY XMinID I 1 N 1 1D gtR Output3 6 R Disp quotMIDquot sumseqY XMinID I 5 N 1 1D gtM Output4 6 M Disp quotTRAPquot LR2 gt T Output5 6 T Disp quotSIMPquot 2MT3 gt S Output6 6 S Input quotGRAPH 0N lYquot A If A 1 Goto 0 Lbl 6 Cerraw Line0 0 0 I kI nlu SET Pquot quotLEFTquot 1 quotRIGHTquot 2 quotMIDPTquot 3 quotTRAPquot 4 quotQUITquot 5 F0rI 0 N l XMlnID gt X Y1X gt Y Llne 0 X 319 Line Y X Y Line D Y XD 0 End Textl l L Pause Goto 6 Lbl 2 F0rI 0 N l XMmID gt X E199 2039533 Line Y X Y Line D Y XD 0 Tndt l R P ex ause G0t016 F0rI0N l XMmID gt X YXD2 gt Y Y gt Y LI1 2X 0 X Y LineX Y XD Y XD 11 Line Max Y1IX Vlax XMax 0 Tex 8f 1 T Pause Reimann TI83 Program for Finding Reimann Sums CIrDraw FnOff PIotsOff AxesOff ZStandard Text00quotRiemann Sumsquot Lineu1078u678 Text110quotFinds area under a functionquot Text180quoton a given interval usingquot Text250quotRiemann Sums Increasingquot Text320quotthe number of partitionsquot Text390quoteads to better estimatesquot Text4956quotPress Enterquot LineL 11L 8111L I81 Text572quotFrom the Mind of Mike 2003quot Pause AxesOn CIrHome Goto CA Lbl1 CIrHome MenuquotRiemann SumsquotquotSettingsquotSPquotLeft SumquotLquotRight SumquotRquotMidpoint SumquotMquotTrapezoid SumquotTquotDef IntegraIquotDFquotQuitquotN Lbl SP CIrHome MenuquotSettingsquotquotChange aIIquotCAquotFunctionquotCFquotBoundariesquotCBquotPartitionsquotCPquotPrevious menuquot1 Lbl CA Disp quotSettingsquot Input quotfxquotStr1 Str139uY Input quotLower BoundquotStr2 exprStr239uA Input quotUpper BoundquotStr3 exprSt339uB Input quotPartitionsquot0 Goto 8 Lbl CF Input quotfxquotStr1 Str139uY Goto 8 Lbl CB Input quotLower BoundquotStr2 exprStr239uA Input quotUpper BoundquotStr3 exprSt339uB Goto 8 Lbl CP Input quotPaltitionsquot0 Lbl 8 Y fMaxY XAB39uS Y fMinY XAB39uU If uuo en L39I1539L39Ymin 116339quax End If Ult0 and 30 115U39quin u16U quax End If Ult0 and Sgt0 Then SU39L39IQ U1SQ39quin S1SQ39quax End A05BA39uXminB05BAuXmaxYmaxYminl15uYsclXmaxXminl15quc FnOff FnOn 1 DispGraph Text00quotYquotStr1quot quotStr2quot quotStr3quot quot StorePic 2 LineA0AY A LineB0BY B Text5749quotPRESS ENTERquot BAlO39uW sumseqY XAB 5WWW39uC sumseqY XBA5WL IWW39uD Pause CIrDraw RecaIIPic 2 For001 LineAIW0AIWY AIW LineAIWY AIWAIWWY AIW LineAIWWY AIWAIWW0 End Text570quotLeft SumquotroundC5 Goto 1 Lbl R CIrDraw RecaIIPic 2 For001 LineAIW0AIWY AIWW LineAIWY AIWWAIWWY AIWW LineAIWWY AIWWAIWW0 End Text570quotRight SumquotroundD5 Pause Goto 1 Lbl M Cerraw RecallPic 2 sumseqY XA5WBWW39uE For001 LineAW0AWY A5 LineAWY A5WAWWY A5W LineAWWY A5WAWW0 End Text570quotMidpoint SumquotroundE5 Pause RecallPic 2 CDl239uP For001 LineAW0AWY AIW LineAWY AIWAWWY AWW LineAWWY AWWAWW0 nd Text570quotTrapezoid SumquotroundP5 ause Goto 1 End Lbl DF Cerraw RecallPic 2 fnlntY XAB39uV Text570quotDefinite ntegralquotroundV5 LineB0BY B f UEIO Then Shade0Y AB Goto 5 End If S0 Then ShadeY 0AB Goto 5 Shade0Y AB ShadeY 0AB Lbl 5 Pause Goto 1 End Lbl N Cerraw Cerome 15 THE FUNDAMENTAL THEOREM OF CALCULUS The Fundamental Theorem of Calculus will be discussed in two forms as the total change of the antiderivative then as a connection between integration and differentiation Why do we use the Fundamental Theorem The Fundamental Theorem of Calculus states Iffx is a continuous function andft then I ftdt Fb Fa One reason we use this theorem is that it calculates the definite integral in a simple way Unfortunately this use is limited to when we know the antiderivative For example we previously used the Riemann sum to guess that the definite integral fnntsinx x 7139 2 Using the Fundamental Theorem we could do this calculation in our head an antiderivative of fx sinx is Fx cosx so F7r F0 cos7r cos0 1 1 2 A second reason to use the Fundamental Theorem is that it gives us an exact answer which may be required orjust plain useful For example when a growth factor compounds continuously the decimal accuracy is limited to that ofthe calculator This is fine when we are dealing in thousands or millions but sometimes we have amounts that are astronomical and we want an answer that will be exact to whatever number of decimal places are required Think ofthe value of 7t it is roughly 3 or if more accuracy is 22 needed we can use or better yet 31459 In its exact form the symbol 7t represents full accuracy not a decimal or fraction approximation The definite integral as the total change of an antiderivative Let39s look at an example where an exact answer will be found Consider a savings account into which you put a dollar every hour What will it be worth in 20 years if it is compounded continuously at a 10 annual rate This is a thinly disguised definite integral First whatever you deposit needs to be expressed in an annual amount so that all our rates are annual Call this amount P which we will take as 36524 ignoring leap years Deposits are so frequent that we will consider the rate to be continuous The future balance in ten years is then given by the definite integral J39ZO P601207xdx 0 The fnnt function gives a value of over half a million dollars This is probably good enough in this case but the calculator answer does have limited accuracy We now use the Fundamental Theorem to write an exact answer correct to an infinite number of digits This is sometimes also called closed form We first find an antiderivative function Fx Pe2 870 M J 01 and then evaluate it at the upper and lower limits zi 2 i 72 F20 F0 Pe 01 Pe 01 P0I11 e When written in terms of e the expression has full accuracy The two answers are compared in Figure 151 Knowing the closed form solution allows us to accurately find the future value of saving a thousand dollars per minute although we would need a calculator with more internal digits of accuracy to see the difference Figure 15 A de nite integral calculated using the built in numeric integral approximator and the closed form given by the Fundamental Theorem Using fnlnt to check on the Fundamental Theorem 2 Consider the example of finding the value of J xzdx in different ways First let39s graph the function and use CALC 7 Ifxdx X 0 This is shown in Figure 152 where we used a window 47 S x S 47 and 10 S x S10 The last frame is on the home screen 3 where we first use fnlnt and then the Fundamental Theorem with Fx x to evaluate the integral numerically as F2 F0 Luckily we got the same answer all three ways y1x2 v1x2 f nIntXZyX62 quot 2 666666667 Xquot33quotgtVz Don vlt2gt Vzltegt e 2666666667 Lowquot Limit llvny Limit X0 V0 X2 Y39l fxdxz i 7 Figure 152 Di erent aspectx 0f the Fundamental Theorem equation Viewing the Fundamental Theorem graphically In the above examples the lower and upper limits were constants Now consider the upper limit as a variable Specifically let the upper limit be a variable on each side ofthe Fundamental Theorem equation which gives a function in terms of x your Fx Fa We now try an example by setting up the following functions 3 fx Y1 0 1X2 whose antiderivative is Fx Y3 Since a is arbitrary let a 35 This means that our summing starts at zero when x 35 Finally we define the two functions we want to compare and see if they are equal Y2 fnlntYz X 35 X and Y4 Y3X Y3 35 Now using aZDecimal graphing window and the graphing styles shown in the rst frame of Figure 153 you will see thath andY4 have the same graph 71m 31 Hot E 391zEf nIntV1 X 39 35 lt1x393gt3 EV3X V3 3 V Figure 153 An example of J f tdt Fx Fa with ft 0 it f Checking on fx e the function that is its own antiderivative As another example of this kind of comparison we let fx be the famous exponential function whose antiderivative is itself We check that the two sides of the Fundamental Theorem equation are graphically equal by defining and graphing the two functions shown in Figure 154 no NR2 quot9 3 u nu I ITIVI A V3 iic 0 imarnmuewn I zltxgt eelte o l K7 YL0137527 X17 V10137SZ7 Figure 154 An example of fotdf FOC Fa With m5 equot Tip If you are verifying that two functions have the same graph use TRACE and the up arrow to move back and forth between functions Tip39 Graphing with f39nlnt as part of a formula is very slow Patience is required especially for TI82 users TI83 users can speed it up some by increasing the Xres setting Comparing nDerivfnnt and fnlntnDeriv What happens when you find the derivative ofthe integral function It should not be too surprising that you get the original 51 x function back Consider gx whose domain is all nonzero real numbers x In both the graph and table of Figure 155 it looks like Y1 and Y2 match perfectly except at x 0 However there is so mu numerical action taking place in the calculation of Y1 that these values are just close approximations This can be seen in the table ifyou it highlight a Y1 entry and compare it to the corresponding Y2 value they are not exactly the same mu HotZ HokB 2V 5UFr 2 s1n y y 7 xgtlt J 0V2551nXX V3 Vw Vs X V1 V2 ll 1 man s Bsaa 5533 auw auw s 655 555 2 Iasiss J39s as s 2353 2393 3 ouroiu ouroli ixa X V1 V2 X V1 V2 339 E39Ea39lls s ssaas 1 auw l39ll39 1 auw auv 1 555 555 35 555 55 z s ss Iisias 2 HS39IES usws 25 2153 23 2 2393 2393 3 owou mm 2 ovou 039I7039l V1958851025775 Vz98851 77208 Figure 155 Graph and table for nDeriuf nInt 3 with closer inspection of two table entries that should be equal but di er slightly because of approximation errors Let39s reverse the order and integrate the derivative function You might think you will get the original function back again plus some constant which depends on the lower limit you en er rim nmhrmz 1 VzxinXX EV lV TIQ l s1n 1 mm 1 VzEsmXX VE nIntnDer1u smltTgtTyTXX 1X x1 v1ssaa17 x1 VE3339117 Figure 156 Graph showing that f nIntICnDer ivf J di ers from the original mction by a vertical shift of sin I 0841 4 7 Linking and Troubleshooting Internet address information The main internet address for Texas Instruments is httpwwwticom The calculator materials main menu can be obtained at httpwww ti corncalcdocs It is unlikely the above two entry addresses will change Web page designs do change but the following addresses are current as of 1 January 1 1998 and will probably remain stable for finding the following topics Topic Address Calculator Based Laboratory CBL httpwww ti corncalcdocscbl htm Calculator comparison 39 39 Calculator support hl39tn39 www ti 39 39 39 39 htm Classroom activities httpwww ti 39 39 39 39 39 39 htm Frequently Asked QuestionFAQs hl39tn39 www ti 39 39 39 htm G ra phLi nk httpwww ti corncalcdocslink htm G uides to downloading httpwww ti corncalcdocsguides htm Guidebook download WM New calculator news hl39tn39 www ti 39 39 39 39 htm Program Archives httpwww ti corncalcdocsarch htm Ra nger like the CBL httpwww ti corncalcdocscbr htm Resources httpwww ti 39 39 39 htm There are discussion groups available you will find information about these from the main screen of httpwwwticorncalcldocsl Linking calculators The essentials of linking are presented in the TI Guidebook and will not be repeated here But we include the following tips The endjack must be pushed firmly into the socket There is a nal click you can feel as it makes the proper connection If you are experiencing difficulty connecting turn off both calculators check the connection and then turn them on and try again If available try other cables or calculators When selecting items the cursor square is hardly visible when the selection arrow is on the same item As you arrow off of the item doublecheck whether or not it has been selected If you are required to drain your calculator memory before entering an exam use Back UP to keep a copy on some other calculator Even better is to store it on your computer which is the next topic Linking to a computer The TlGraph LinkTM is a cable and software that connects to a PC or Macintosh computer and your T calculator The software is available on the internet so you can order only the cable There are many advantages to using TlGraph LinkTM This is the best way to back up your work It is the preferred way to write and edit programs You can download and transfer programs from the internet archives It allows you to capture the screen in a form for direct printing or use in a word processor Troubleshooting Nothing shows on the screen Check the contrast Check ONIOFF button Pullout the batteries and correctly reinsert the batteries As a last resort remove the backup battery and reinsert all the batteries Warning This will erase all memory including programs Nothing shows up on the graph screen except the axes Press TRACE to see if a function is defined but outside the window There may be no functions selected The function may be graphed along either axis and need the window reset If there is a busy indicator a running line in the upper right comer of the screen then the TI may be still calculating Press ON if you can39t wait If there is a pause indicator twinkling line in the upper right comer of the screen then the TI is paused from a program Press ENTERto continue If there is a full checkerboard cursor then you have a full memory You need to delete something choose some things you no longer need or copy them to your computer Nothing shows up in the table You may have the Ask mode set and need to either enter xvalues or change to Auto in the 2ndTBLSET Check to see that a function is selected get a syntax error screen The most common errors are Parenthesis mismatch Count and match parentheses carefully Subtraction vs negative symbol For example the subtraction sign cannot be used to enter 10 Pasting a command in the wrong place For example a program name must be on a fresh line get an error message This can cover the widest array of problems Read the manual carefully it will tip you off to the kind of error you are looking for If you have no idea what could have caused it consult the appendix of the TI Guidebook for explanation of the error messages I39m getting a result but it is wrong Check for Typing direct letters in place of pasting a command For example SIN X can be entered using the ALPHA keys but it is not the same as sin X using the SIN key Parenthesis mismatch Count and match parentheses carefully Subtraction vs negative symbol For example the subtraction sign cannot be used to enter a 10 Correct default settings My program won39t run Program errors are difficult to diagnose Scroll and check your code with PROGRAM EDIT Add temporary displays and pauses to check the progress and isolate the problem 11 THE RULES OF DIFFERENTIATION Using the definition of the derivative to find a derivative function is cumbersome fortunatelythere are shortcuts to finding derivative functions We will use the calculator to see examples of the main three rules These rules must be proved analytically but a graphical verification gives us added confidence in them Also we can show graphically that some common guesses for the rules are wrong The Product Rule fxgx39 f xgxfxg x We take two common functions and graph their product and the derivative ofthe product as shown in Figure 111 We see from the graph that the product function has a local maximum at x 2 and thus the derivative function should be zero there Y 4512E 8 is essentially zero 71m not rm wINpow vunnmvltvzixm Y X m1nf max Vz VvW Xscl1 VuEnDemUWhX Vmin 1 A Ymax2 Vs Vscl1 Ya Xres1 xz Vquot11ZE E Figure 11 A product function and its derivative Now let39s suppose our guess for the product rule is that the derivative ofthe product is the product ofthe derivatives This is not such a silly guess since the derivative of a sum is the sum of the derivatives but it is wrong We can see this in Figure 112 by graphing the product of the derivatives The product of the derivative functions does not have a zero where it should at x 2 so it is not the derivative ofthe product function 71m Hot rm VzequotltX V35V1V VuEnDerwW X X VSEHDEK IUKV17X gtnDer1UltVzXyX II Z v539i13uz Figure 112 The product of the derivative function does not have a zero at the product function maximum therefore fg 39 f39g P1ol1 er39rm VEnD rivVhXIXV20VL VsEnDemWW X gtltnDer1ultV2yXX vVaEnDerithX A XVzV1nDeriUlt V2XX w V7 xz v5ozua Figure 113 The correct derivative formula function V5 traces over V4 To see that the product rule holds for these two functions in this window enter the true formula with a leading cursor style so that you can see that the Y6 graph traces over the Y4 graph Tl82 users will need to use trace on the two curves to see that they are the same Notice that we interpret y 90214E 8 as zero The Quotient Rule W gltxgt gltxgtfltxgt gx gx Let39s check the quotient rule in the same way This means we can recycle the functions make a minor change to the window setting and get the graph shown in Figure 114 Notice the two zeros of the derivative function are at the local maximum and minimum of the quotient function mu rmz rim UINDOliJ 1 2 X 1n 1 VzequotltX VIEWV VkEnDevazxxy VsnDerivV1X XnDeriuVzXX 0lt X2 V3915115E quotI Figure 114 A quotient mction and its derivative We can again make a feasible but incorrect guess namely that the derivative of the quotient is the quotient ofthe derivatives We see our folly easily since the quotient of the derivatives is not zero asx 2 Now enter the correct formula in Y6 and see that it traces over Y4 mm mm rim 39 quot v1 VsEnDeriuW1 yX XnDerivVzXX oYsEnDer1vV1 X XVzVinDerw V2 V7 yltgtYzgtz v XZ V39I133911039I XZ V 3910le 39l Figure l 15 The incorrect quotient function V 5 does not trace V 4 but the contact formula V 6 does trace the derivative of the quotient function Figure 115 The incorrect quotient function Y5 does not trace Y4 but the correct formula Y6 does trace the derivative of the quotient function The Chain Rule Thinking of a function as a composite function and using the chain rule will often simplify finding the derivative of a function Considery xZ 1100 A straightforward but impractical approach would be to expand the expression write it as a polynomial of degree 200 and then differentiate term by term Instead we apply the chain rule and find the derivative quickly and easily y39 100x2 1 2x 200xx2 1 Figure 116 shows the graphs ofy and y notice the scale 71m rm P1ot3 wINDow V1Xquot1 Xnin393 V2XZ1 Xmax3 V3EV1V Xscl1 VnEnDerwWhX Vnin39180 X Wax 160 V Vscl10 We Xres1 xoazsran vzz7oosu Figure 116 A compasite mction and its derivative Some care should be exercised in entering the function for the chain rule formula The derivative of the outside function is evaluated in terms of the inside function so the outside function is evaluated at Y2 Hence for this calculator we write nDeriVYl X Y2 nDeriVY2 X X We see in Figure 117 that the graphs of the two functions are the same P39Iotl Natl quoton Vz XZH 35 V3 ltgtEnDer1vV3yxy NSEnDerithX nDeriuVzX Figure 117 The correct formula V5 traces the derivative of the composite function The derivative of the tangent function Recall that the window used for graphing trigonometric functions is often crucial In Figure 118 we first use the ZOOM ZStandard window then the ZOOM ZTrig window to view Y1 tanx This should remind you that X values are sampled evenly across the window and the function values at these points are then connected to form the graph It is 7 7 clear from the graph ofthe tangent function that it is an increasing function within an interval such as is x S and that 7 7 it is undefined at multiples of E Thus we expect the derivative to be positive between ilt x lt 7 and undefined at lt39l f7 mui eso 7 p 2 Figure 118 Graph of y tanx with different windows We can fnd the derivative of the tangent function from the quotient definition tanx sinx cosx Set y tanx and use the quotient rule to derive y x f This is always positive and is undefined at multiples of E where the cos x 2 cosine is zero Tip The conventional mathematical way of writing a power of a trigonometric function such as cos2 x gives a syntax error Use cosxZ or better yet cos2 x 1 For Figure 119 first enter Y3 7 2 and graph it Then enter the numeric derivative in Y2 and graph it This creates 00806 7139 a surprise of double vertical lines between the defined intervals What is going on If you trace to a value like i as shown in the third panel then you will see that the numeric derivative has been calculated incorrectly as a large negative number v31ltum ut Ul mun nu Ulll U r Ull K0 V1 l 11 0000003 X 57075 1 V 555555 Figure 119 Graph of the algebraic derivative of tanx and then the numeric derivative The moral of this graph is that we must be constantly vigilant in believing the numeric derivative function at points where the function is undefined A cosmetic remedy to this problem is to make the sampling avoid the bad places In this example we can raise the Xres setting as shown in Figure 1110 and it won t evaluate at multiples of UINDOU Xmin Xnax6 Xres2l X13055565 39v1o173327 Figure 1110 Reset Xres to avoid unde ned values Notes on Xres An Xres setting of 2 as in Figure 1110 will skip every other X value that the calculator normally would use in its sample of xvalues between Xmax and Xmin These 95 samples are formed by adding multiples of Xmax Xmln to Xmin The only ZOOM setting that will change Xres is ZStandard lt resets to Xres 1 Again there is no Xres setting on a Tl82

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