Page 1 39gs and hp 40gs graphing calculators Mastering the hp 39gs & hp 40gs A guide for teachers, students and other users of the hp 39gs & hp 40gs Edition 1.0 HP part number F2224-90010...
Page 2 Notice REGISTER YOUR PRODUCT AT: www.register.hp.com THIS MANUAL AND ANY EXAMPLES CONTAINED HEREIN ARE PROVIDED "AS IS" AND ARE SUBJECT TO CHANGE WITHOUT NOTICE. HEWLETT-PACKARD COMPANY MAKES NO WARRANTY OF ANY KIND WITH REGARD TO THIS MANUAL, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY, NON-INFRINGEMENT AND FITNESS FOR A PARTICULAR PURPOSE.
Page 3: Table Of Contents
What is the HOME view? ...18 Exploring the keyboard ...19 Angle and Numeric settings ...28 Memory Management ...30 Fractions on the hp 39gs and hp 40gs ...33 The HOME History ...37 Storing and Retrieving Memories ...39 Referring to other aplets from the HOME view...40 A brief introduction to the MATH Menu ...41...
Page 4 The Statistics Aplet - Univariate Data...114 The Expert: Simulations & random numbers...120 The Statistics Aplet - Bivariate Data...123 The Expert: Manipulating columns & eqns...133 The Inference Aplet ...141 The Expert: Chi tests & Frequency tables ...147 The Linear Solver Aplet ...150 Example 1 ...150 Example 2 ...150 Example 3 ...151...
Page 5 Deleting downloaded aplets from the calculator ...250 Capturing screens using the Connectivity Kit ...251 Editing Notes using the Connectivity Software...252 Programming the hp 39gs & hp 40gs ...255 The design process ...255 Planning the VIEWS menu ...257 The SETVIEWS command ...259 Example aplet #1 –...
Page 6 Rectilinear Motion...321 Limits...321 Piecewise Defined Functions ...322 Sequences and Series ...322 Transformations of Graphs ...323 Appendix C: The CAS on the hp 40gs ...324 Introduction ...324 Using the CAS ...327 Examples using the CAS ...341 The CAS menus ...358 On-line help ...361 Configuring the CAS...362...
Page 7: Introduction
This book is intended to help you to master your hp 39gs or hp 40gs calculator but will also be useful to users of earlier models such as the hp 39g, hp 40g and hp 39g+. These are very sophisticated calculators, having more capabilities than a mainframe computer of the 1970s, so you should not expect to become an expert in one or two sessions.
Page 8 39g which would ‘convert’ it into an hp 40g and activate the CAS. This is not the case with the hp 39gs & hp 40gs: the internal chips are different and there is no way to ‘convert’...
Page 9: Getting Started
The sketch below shows most of the important keys. As can be seen on the previous pages, the keyboards for the hp 39gs and hp 40gs are exactly the same except for the different color schemes. These keys are the ones which control the operation of the calculator – most others are simply used to do calculations once the important keys have set up the environment to do it in.
Page 10: Some Keyboard Examples
- used to graph the function. PLOT (The PLOT SETUP key is used to choose which aplet is active. There are 12 APLET aplets provided with the calculator and more can be downloaded from the internet. view sets the axes.)
Page 11: Keys & Notation Conventions
& & There are a number of types of keys/buttons that are used on the hp 39gs and hp 40gs. The basic keys are those that you see on any calculator including scientific ones, such as the numeric operators and the trig keys. Most of these keys have two or more functions, with the second function accessed...
Page 12 The Screen keys A special type of key unique to the hp 39gs, hp 40gs and family is the row of blank keys directly under the screen. These keys change their function depending on what you are doing at the time. The easiest way...
Page 13 (see image). In case you symbol comes from the divide key HOME The calculator also comes with an immense number of mathematical functions. They can all be obtained via menus through the from the keyboard. Try pressing the find your screen looks like the screen shot left.
Page 14: Everything Revolves Around Aplets
A built in set of aplets are provided in the and hp 40gs. This effectively mean that it is not just one calculator but a dozen (or more), changing capabilities according to which aplet is chosen. The best way to think of these aplets is as “environments” or “rooms” within which you can work. Although...
Page 15 The Sequence aplet (see page 99) Handles sequences such as recursive and non-recursive sequences. The Solve aplet (see page 105) Solves equations for you. Given an equation such as variable if you tell it the values of the others. The Statistics aplet (see page 114 & 123) Handles descriptive statistics.
Page 16 The standard aplets will cover all of your normal requirements in mathematics. However one of the great strengths of the hp 39gs and hp 40gs is their ability to “download” additional aplets from other calculators and from the Internet. See page 245.
Page 17 Once an aplet is transferred onto any calculator from the PC, transferring it to another takes only seconds using the built in infra-red link at the top of each calculator on the hp 39gs or using the mini-serial cable on the hp 40gs.
Page 18: The Home View
HOME • Exploring the Keyboard Angle and numeric settings • Memory management • Fractions on the hp 39gs & hp 40gs • History • HOME • Storing and retrieving memories • Referring to other aplets from the •...
Page 19: Exploring The Keyboard
The first step in efficient use of the calculator is to familiarize yourself with the mathematical functions available on the keyboard. If we examine them row by row, you will see that they tend to fall into two categories - those which are specific to the use of aplets, and those which are commonly used in mathematical calculations.
Page 20 Internet written by other programmers. Once these are downloaded into your calculator they can also be accessed via the later in this section, and the chapter entitled “Programming the hp 39gs & hp 40gs” on page 255. The SYMB, PLOT and NUM keys When working mathematically there are three ways that we view functions: •...
Page 21 VIEWS from the Internet. When a programmed aplet is created for the hp 39gs or hp 40gs, a menu is provided by the programmer to let you control and use it. During the programming this menu is tied to the...
Page 22 “The MATH Menus” on page 165. As is usual with all calculators, most of the keys have another function above the key. The hp 39gs and hp 40gs get twice the action from each key by having this second function.
Page 23 ALPHA right of most keys. Pressing SHIFT ALPHA Calculator Tip If you press and hold down the although this doesn’t work for lower case. Many people use this to type in functions by hand rather than going through the views, such as the Notepad, also offer a screen key function that lets you lock either upper or lower case alpha mode.
Page 24 Of course, the Scientific calculator’s idea of scientific notation may not be the same as yours. Since the calculator has no way of displaying powers as superscripts, a result of 3 203 ×10...
Page 25 Moving back to our tour of the keyboard, the next key is key. This is used as an all purpose “I’ve ENTER finished - do your thing!” signal to the calculator. In situations where you woul d normally press the ‘...
Page 26 The negative key Another important key is the hp 39gs and hp 40gs do not treat a negative as being the same as a subtract. If you want to calculate the value of (say) − − − before the 2 and the 9 rather than the subtract key. If you press the subtract key twice, entering ‘subtract, subtract 9’...
Page 27 ‘super delete’ key. For example, if pressing CLEAR would erase one function only in the Calculator Tip Another use for the if you move back into the then pressi in the CLEAR entries.
Page 28: Angle And Numeric Settings
It is critical to your efficient use of the hp 39gs and hp 40gs that you understand how the angle and numeric settings work. For those few who may be upgrading from the original hp 38g released in the mid ‘90s this is particularly important, since the behavior is significantly different.
Page 29 It even resulted in users returning their hp 38g to dealers as being ‘faulty’! Hence the change, which was first made in the hp 39g and hp 40g.
Page 30: Memory Management
One of the major complaints about the original hp 38g was its memory - mainly the lack of it at only 23Kb, but also the inability to easily control or manage it. This problem has been addressed on the hp 39gs and hp 40gs in two ways.
Page 31 Connectivity Kit, or which are supplied to you by your teacher via the infra-red link on an hp 39gs or the cable on an hp 40gs, then you need to bear in mind that most of them have ‘helper’ programs that aid them in performing their tasks.
Page 32 ‘helper’ programs. Calculator Tip Because of the amount of memory available on the hp 39gs & hp 40gs, the Memory View is not one that you will normally need to worry about unless you store a tru y amazing number of Notes. It is probably of more interest to programmers.
Page 33: Fractions On The Hp 39Gs And Hp 40Gs
Most calculators opt for the easy option of switching to a decimal answer in any mixture of fractions and decimals. When making the hp 39gs and hp 40gs HP took a very different approach. Once you select including any decimals.
Page 34 The second point to remember involves the method the hp 39gs and hp 40gs use when converting decimals to fractions, which is basically to generate (internally and unseen by you) a series of continued fractions which are approximations to the decimal entered. The final fractional approximation chosen for display is the first one found which is ‘sufficiently close’...
Page 35 0.666, while adding one more 6 (to take the decimal beyond 4 d.p.) will give the desired result of other words, so long as you understand the approach taken by the hp 39gs and hp 40gs it is capable of producing results which are closer to what was probably intended by the user in entering 0.66666.
Page 36: Fraction Setting
Generally it is not a good idea to go below the default setting of . In fact, a Fraction 4 Fraction 6 A new feature of the hp 39gs and hp 40gs is the setting of in the view. Fraction MODES The results of this new setting can be seen in the image to the right.
Page 37: The Home History
SHIFT CLEAR regularly, since the history uses memory that may be needed for other things, even with the immense amount of user memory the hp 39gs & hp 40gs have. You can number of different lines in building your new expression.
Page 38 . This key will display an expression the way you would write it on the page rather than in the somewhat difficult to read style that is forced on the calculator when it must show the whole expression on one line. This works anywhere the label appears, not just in .
Page 39: Storing And Retrieving Memories
X and then for re-evaluation. Clearly, if the expression is complex, this can be very helpful. Calculator Tip 1. The memory to store the current cursor position in the access it in other places. Values stored in next time you use the 2.
Page 40: Referring To Other Aplets From The Home View
The results shown will, of course, depend on your settings in the view. The reason for the QUOTE(X-2) would tell the calculator to use the value currently stored in memory while tells it to use the symbol. The QUOTE(X-2) available through the menu under Symbolic (see page 181).
Page 41: A Brief Introduction To The Math Menu
menu holds all the functions that are not used often enough to be worth a key of their own. There MATH is a very large supply of functions available, many of them extremely powerful, listed in their own chapter beginning on page 165. When you press the key you will see the pop up screen shown MATH...
Page 42: Resetting The Calculator
Thi s method is provided in case the calculator is locked up to the point that the keyboard no longer responds. On the back of the calculator is a small hole. Poke a paper clip or a pin into this hole and press gently on the switch inside.
Page 43 This type of reset will always cause complete loss of data. If you find that the screen fills with garbage, or if the calculator’s in-built diagnostic routine starts to run, then it is just that you have not released them in the right order.
Page 44 If so, clean them carefully, being careful not to get moisture inside the calculator. Have you recently dropped the calculator or spilled liquid on it? If so, this is not good. It is probably • permanently dead. Did you need a very expensive paperweight? Check the USB and Serial ports at the top of the calculator.
Page 45: Summary
• simply putting the letter in the expression in place of the number. You can easily reboot the calculator if it locks up, generally without loss of memory. Make sure you • know how to do this in case it happens during a test or an examination.
Page 46: The Function Aplet
The Function aplet is probably the one that you will use most of all. It allows you to: graph equations • find intercepts • • find turning points (maxima/minima) • find areas under curves find areas between curves • find gradients •...
Page 47 This is the Notice that the screen title is supplied so that you will know where you are (if you didn’t already). Calculator Tip Pressing ENTER Whenever there is an obvious choice pressing produce the desired effect.
Page 48 One of the easiest ways to set up the axes properly for a function whose shape is not known in advance is to let the calculator suggest a suitable scale using the Auto Scale option in the menu covered on the next VIEWS page.
Page 49: Auto Scale
See “The Expert” chapter beginning on page 62 for more information on how to find good choices for axes. The Auto Scale function is also covered on page 89. key. Use the arrow keys to scroll . The calculator will adjust the y view. If you look PLOT SETUP keys you will see the word setting that removes the ‘thick’...
Page 50: The Plot Setup View
SHIFT PLOT the right. The highlight should be on the first value of XRng. Enter the value -4. Calculator Tip Don’t use the subtract key to enter a negative. You MUST use the negative key labeled subtract. For example, Type in 4 for the other...
Page 51 Simultaneous The first option controls whether each graph is drawn separately (one after the other) or whether they Simult are all drawn at the same time, sweeping from left to right on the screen. My preference is to turn this off. I find that if there are more than two functions defined then drawing them all at the same time can be confusing.
Page 52: The Default Axis Settings
In any of these modes the up/down arrows move the cursor from function to function, while the left/right arrows move along the currently selected function. Calculator Tip Pressing SHIFT directly to the right or left side of the screen.
Page 53: The Menu Bar Functions
In the examples and explanations which follow, the functions and settings used are: Trace is quite a useful tool. The dot next to the word means that it is currently switched on. If yours shows underneath to turn it on. Leave it on for now. Press the left arrow 5 or 6 times to see a similar display to that shown right.
Page 54 Calculator Tip example, jump to a value such as e intersection, then jumping to a value of to that point.
Page 55 The Zoom Sub-menu The next menu key we’ll examine is pops up a new menu, shown right. The list which follows covers the purpose of the first nine options shown right, down to “Set Factors”. The four final options which follow these are also on the and are covered on page 85 as part of the detailed examination of the menu.
Page 56 As you move the cursor to a position at the diagonally opposite corner of a rectangle, the selection box will appear on the screen. Pressing expands the box to fill the screen. You’ll notice that the scale has been disrupted so that the labels are no longer very helpful.
Page 57: The Fcn Menu
In this case that means moving it past the turning point. Calculator Tip If you are working with a function which has asymptotes then make sure the cursor is positioned on the same side of the asymptote as the root.
Page 58 X axis, or the other function are shown right. F2(X) Calculator Tip When you find an intersection or a root the value of the x coordinate is stored in the memory . If you immediately change to the and type See “The Expert”...
Page 59 If you now press again to accept the end point, the hp 39gs or ENTER hp 40gs will calculate the signed area and display the result at the bottom of the screen. Calculator Tip It should be c early understand that although the label at the...
Page 60 As you do the area will be shaded by the calculator. The current position is shown at the bottom of the screen. When you reach the end point you are looking for, press the and the area will be calculated as before.
Page 61 Extremum from the menu. You should find that the cursor will jump to the position of the maximum. Calculator Tip If your graph has asymptotes then make sure that the cursor is positioned on the side of the asymptote containing the extremum before initiating the process.
Page 62: The Expert: Working With Functions Effectively
Part of the answer is to know your function – this is why we still expect you to learn mathematics instead of expecting the calculator to do it all! If you know, for example, that your function is hyperbolic then that immediately gives information about what to expect.
Page 63 Change into the view and scroll through the window from zero to 100. As you do so, take note of the values that the function takes. From the display it seems that the function peaks around y=30 and then declines steadily. Change into the view and enter an x axis of 0 to 100 PLOT SETUP...
Page 64 If the highlight is now positioned on each of these in turn, and the performed. The result is shown in the right hand snapshot. Notice that the calculator is smart enough to realize in not, unfortunately, smart enough to keep track of the implications for the domain, which are that be defined only for non-negative x.
Page 65 These functions can all be graphed but the speed of graphing is slowed if you don’t press internally re-evaluated for each point graphed. The hp 39gs and hp 40gs are fast enough that the result is still satisfactory but if you have an old 39g or 40g they are slowed to the point of being unusable.
Page 66 Differentiating There are different approaches that can be taken to differentiating, most of which are best done in the SYMB The syntax of the differentiation function is: ∂X ( function ) where function is defined in terms of X. The function can be already defined in the in the screen shot above.
Page 67 Algebraic differentiation is most easily handled in the your function as and its derivative as F1(X) Calculator Tip Doing your differentiation in the Function aplet is much easier and offers the additional advantage of being able to graph the two functions. Circular functions There are two issues that influence the graphing of circular functions, both related to the scale chosen.
Page 68 This is shown in the second snapshot above. The reason for this is that when the calculator draws the graph it does so by ‘joining the dots’. For the default scale of -6.5 to 6.5 this is not a problem since the edges of the two half circles at -3 and 3 fall on a pixel. This means that the last segment of the graph plotted extends is from 2.9 to 3 and the circle reaches right down to...
Page 69 0, 0.0923077, 0.1846154... In particular, near x=3 the pixel values are 2.953846 and 3.046154. This means that the calculator can't draw anything past 2.953846 because the next value doesn't exist, being outside the circle. This is what causes the gap in the circle. There's nothing to join to past that last point.
Page 70 Retaining calculated values When you find an extremum or an intersection, the point is remembered until you move the cursor even if it is not actually on a value that would normally be accessible for the scale you have chosen. For example, if you find an intersection and then immediately return to menu and choose Slope, the slope calculated will be for the intersection just found rather than for the nearest pixel point.
Page 71 Automatic vs. Build Your Own Looking at the view you will see an entry called NUM SETUP The alternative to Automatic is the setting of Build Your Own. Under this setting the waiting for you to enter your own values for Typing in the values of (for example): 3 ENTER (-) 2 ENTER 5 ENTER In this situation the function values are being calculated as you input the...
Page 72 Integration: The definite integral using the The situation for integration is very similar to that of differentiation. As with differentiation, the results for algebraic integration are better in the Function aplet. The The syntax of the integration function is: ∫ ( , , a b function name ) where: a and b are the limits of integration...
Page 73 - 1, X ) This is shown above, together with the results of highlighting the answer and pressing result may seem odd but is caused by calculator assuming that other variable and integrating accordingly as a ‘partial integration’. While mathematically correct, this is not what most of us want.
Page 74 ‘+c’! Calculator Tip There are strict limits to what the hp 39gs can integrate. For example, on the hp 39gs if you try to evaluate able to do it.
Page 75 Integration: The definite integral using PLOT variables As was discussed earlier, when you find roots, intersections, extrema or signed areas in the view, the results are stored into variables for PLOT later use. For example, if we use Root to find the x intercept of −...
Page 76 Suppose we want to find the area between f x − 2 and ( ) = x ( ) 0.5x −1 from x = -2 to the first positive intersection of the two g x = graphs. From the hand shaded screenshot shown above right it can be seen that to find the area we need to split it into two sections, with the boundaries being -2 and the two intersections.
Page 77 Piecewise defined functions It is possible to graph piecewise defined functions using the Function aplet, although it involves literally splitting the function into pieces. ⎧ x + 3 ⎪ For example: f x ( ) = ⎨ x ⎪ 3 − x ⎩...
Page 78 ‘Nice’ scales As discussed earlier, the reason for the seemingly strange default scale of -6.5 to 6.5 is to ensure that each dot on the screen is exactly 0.1 apart. There are other scales, basically multiples of these numbers, that also give nice values if you want to along the graph.
Page 79 Use of brackets in functions One problem commonly encountered by new users is misinterpretation of brackets. The hp calculator will correctly interpret F1(X) = X (X+1) but will not understand .
Page 80 ) instead of the true situation of the bottom being roughly twice size of the top. This error is most likely to happen with limits involving power functions as they will overflow for smaller values of x. The hp 40gs can instead evaluate limits algebraically using the CAS (see page 324). An example is shown right to illustrate the results.
Page 81 There are naturally a whole range of numbers which will all round off to the same value of 1.00000000003, so that (for that range of numbers) the expression (1+ 1 / X ) X is equivalent mathematically (on the HP) to 1.00000000003 X . This produces a short section of an exponential graph, which only looks linear because you don't see enough of it.
Page 82 39gs to numerically evaluate limits. Because of the CAS on the hp 40gs this situation is less likely to be a problem for that model. The solution to all problems of this type is to simply be aware of their existence and to allow for them rather than simply accepting the results shown in view.
Page 83 Gradient at a point as the limit of the slope of a chord The true gradient at a point is available in a number of ways. For example, via the view or via the δ differentiation operator. For students first being introduced to calculus a common task PLOT is to investigate the slope of the chord joining two points as the length of the chord tends towards zero.
Page 84 This method works equally well for complex roots. See page 309 for details on finding roots of real and complex polynomials using the CAS on the hp 40gs. Calculator Tip This trick is particularly helpful if you are working with complex roots.
Page 85: The Views Menu
It may seem odd to devote an entire chapter to what might appear to be an inconsequential key. In fact, however, this button is very useful to the effective use of the calculator, and crucial if you intend to use aplets downloaded from the internet.
Page 86 Plot-Detail Choosing Plot-Detail from the menu splits the screen into two halves and re-plots the graph in each half. The right hand side can now be used to without affecting the left screen. The idea is that you the left screen and the result appears on the right screen. For example a Box zoom shows the result on the right allowing easy comparison of ‘before’...
Page 87 Plot-Table The next item on the menu is Plot-Table. This option plots the VIEWS graph on the right, with the Numeric view on the right half screen. Using the left/right arrow keys moves the cursor in both the graph and the numeric windows.
Page 88 Nice table values What makes this view even more useful is that the table keeps its ‘nice’ scale even while the usual tools are being used. As you can see in the screenshot left, the table is automatically repositioned to show the closest pixel value to that of the extremum found.
Page 89 Auto Scale Auto Scale is an good way to ensure that you get a reasonable picture of the graph if you are not sure in advance of the scale. After using Auto Scale you can then use the It is important to understand two points about how Auto Scale works. 1. Auto Scale uses the X-axis range that is currently chosen in range to include as much of the graph as possible.
Page 90 The example right uses zoom factors of 2x2 with Recenter: Calculator Tip In the graphs above the cursor is at x = . The coordinates at the bottom of the screen should show F1(X)=0 but doesn’t due to the fact that the value of π...
Page 91: Downloaded Aplets From The Internet
The most powerful feature of the hp 39gs & hp 40gs is that you can download aplets and programs from the internet to help you to learn and to do mathematics. Two quick examples of aplets that are available are shown here.
Page 92: The Parametric Aplet
T range to that of the X and Y ranges. Calculator Tips The default setting for TStep is 0.1. In my experience this is too large and can result in graphs that are not sufficiently smooth. It is worth developing the habit of changing it to 0.05...
Page 93 The effect of TRng The X and Y ranges control the lengths of the axes. They determine how much of the function, when drawn, will be visible. See the examples below. Notice that in both cases, followed by an ordered pair giving (X,Y). Unlike &...
Page 94 Calculator Tip Decreasing ƒ graphing process without smoothing the graph any further. 0.05 is generally enough. Since trig functions are often used in parametric equations, one ƒ should always be careful that the angle measure chosen in MODES As usual, the view gives a tabular view of the function.
Page 95: The Expert: Vector Functions
Apart from the normal mathematical and engineering applications of parametric equations, some interesting graphs are available through this aplet. Three quick examples are given below. Example 1 Try exploring variants of the graph of: = 3sin 3t x t = 2sin 4t Example 2 Try varying the values of A and B in the equations: x1(t) = ( A B)cos(t...
Page 96: Vectors
The Parametric aplet can be used to visually display vector motion in one and two dimensions. Example 1 A particle P is moving in a straight line. Its velocity v (in ms ( ) = 2t − 5t + 2t − 3 Enter the motion equation from (v) as X(T) and enter Y(T)=T.
Page 97 Example 2 Two ships are traveling according to the vector motions given below, where time is in hours and distance in kilometers. Illustrate their motion during the first ten hours. ⎛ x = ⎜ Ship A : ⎝ ⎛ ⎜...
Page 98: The Polar Aplet
This aplet is used to graph functions of the type where the radius r is a function of the angle the Parametric aplet, it is very similar to the Function aplet and so the space devoted to it here is limited mainly to the way it differs.
Page 99: The Sequence Aplet
This aplet is used to deal with sequences, and indirectly series, in both non-recursive form (where T function of n) and implicit/recursive/iterative form (where T Recursive or non-recursive Examples of these types of sequences are: (explicit/non-recursive) T = 3n − 1 ... 2, 5,8,11,14,...
Page 100 U1(1) The value of will be ignored in the U1(2) calculator automatically in the Convenient screen keys provided There are a number of very convenient extra buttons provided at the bottom of the screen when entering sequences. Two of these - moves onto the line (see right).
Page 101 view offers more useful features. Change to that view NUM SETUP now and change the value to 10. If you then swap back to NumStep view you will see (as right) that the sequence jumps in steps of 10. In case you don’t realize… ⋅...
Page 102: The Expert: Sequences & Series
Defining a generalized GP and the sum to n terms for it. If we define our GP using memory variables then it becomes far more flexible. The advantage of this method is that you now need only change the values of in the change the sequence.
Page 103 Population type problems are also easily dealt with in this way. For example, “A population of mice numbers 5600 and is growing at a rate of 12.5% per month. How long will it be until it numbers more than one million?” Pressing (above ) clears out the existing expressions, and...
Page 104 Modeling loans Suppose that I need to see the progress of a loan of $10,000 at a compound interest of 5.5% p.a. calculated each quarter, starting Jan. 1 1995, with a quarterly repayment rate of $175. This problem can be modeled by a sequence. To do this, set up as shown above.
Page 105: The Solve Aplet
This aplet will probably rival the Function aplet as your ‘most used’ tool. It solves equations, finds zeros of expressions involving multiple variables, and even involving derivatives and integrals. Equations vs. expressions To ensure that we are using the same terminology, let's define our terms first. An equation includes an = sign, and can usually be solved: ⎫...
Page 106 Suppose you had the problem: “What acceleration is needed to increase the speed of a car from 16 67 m/s (60kph or ~38mph) to 27 78 m/s (100kph or 60mph) in a distance of 100m (~110 yd)?” ⋅ We’ll assume that you have already entered the equation into Solving for a missing value If you press to change to the...
Page 107 Multiple solutions and the initial guess Our first example was fairly simple because there was only one solution so it did not much matter where we began looking for it. When there is more than one possible answer you are required to supply an initial estimate or guess.
Page 108 (eventually) of 2.4495. The delay is caused by the repeated integrations as the calculator searches for better solutions. It is important to remember that the calculator does not use algebra in Solve – it uses an algorithm which is essentially a more sophisticated version of “guess, check & improve”.
Page 109 Example 4 “Let X be a random variable, representing the heights of basketball players. If X is normally distributed, with the tallest 5% of players.” function which allows you to work with the normal distribution is MATH gives the upper-tailed probability. The syntax is In the Solve aplet, set P=UTPN(M,V,X) Enter the...
Page 110 The result is a quadratic intersecting a line and the reason for this lies in how Solve interprets your equation. When you select by highlighting it, the calculator substitutes the supplied values in the other variables except and graphs the left and right sides of the equation as two separate graphs. This may not always be obvious because the substitution may produce graphs which aren’t visible on the default scale.
Page 111 Now press and you will see the calculator find the nearest solution to your guess. Finish by pressing is valid. See page 106 for more information regarding this. Obviously the next step is to change back to the cursor near to the second intersection and...
Page 112 16 67 When you press there are a number of possible positive responses. They are: Zero - The calculator tried to find a value of • shown above, it is reporting that it succeeded. Sign reversal - This also indicates a correct solution, since normally one expects to find an •...
Page 113: The Expert: Examples For Solve
Easy problems Have you ever thought “There has to be an easier way!” when confronted in a test with something like: x −1 3 − x − = − If you’re sure there is only one answer to a problem, as there is in this case, then solving it is simply a matter of entering the equation into the Harder problems When you know or suspect that there is going to be more than one...
Page 114: The Statistics Aplet - Univariate Data
One of the major strengths of the hp 39gs & hp40gs is the tools they provide for dealing with statistical data. The Statistics aplet and its companion the Inference aplet provide very powerful yet easy to use tools with which to analyze statistical data.
Page 115 As you can see in the screens above right, the calculator gives not only the standard statistics that any scientific calculator would give, but also the minimum and maximum values, the median and the upper and lower quartile cutoffs.
Page 116 Registering columns as ‘in use’ Change into the SYMB view and edit yours so that it looks like the one on the right. You must make sure that checked columns will show in the Note that a screen key is provided to give you the letter having to use the key.
Page 117 If you use the left/right arrows and look at the bottom of the screen you’ll see that the frequencies and ranges are listed. It is probably worth tidying up this graph up a little by going into second page) setting the value to be 5 instead of 1.
Page 118 50 - 59 As with most calculators, the hp 39gs & hp 40gs provide only limited methods to deal with data of this form. Summary statistics can be obtained by entering the mid-points of the intervals as the data values but these will only be approximations, as nature of the data itself does not allow calculation of exact values.
Page 119 variable controls the width of the HWidth columns, with the initial starting value and end value set by . In the frequency HRng table on the previous page the interval width was 10. By setting to 10 in the HWidth view as shown right we can PLOT SETUP produce the graph shown below.
Page 120: The Expert: Simulations & Random Numbers
New columns as functions of old You have already seen the use of one trick when we created a new column view. This can be used to create new columns as functions of any number of others. For example, HOME a set of data that you suspect is exponential could be ‘straightened’...
Page 121 Simulation of a normal die Similarly the expression INT(RANDOM*6+1) die. This means that MAKELIST(INT(RANDOM*6+1),X,1,500,1) simulate 500 rolls of a normal die. We therefore need only store the resulting list into a Statistics aplet column to analyze and graph it. This is shown in the series of screen shots to the right.
Page 122 Its mean turned out to be 2.067 (3 decimal places.). Yours will be different of course - after all, that’s the point of using random numbers! Calculator Tip RANDOM computer. If you use the calculators just out of the box then you wil see the same set of numbers...
Page 123: The Statistics Aplet - Bivariate Data
As mentioned in the Univariate section, one of the major strengths of the hp 39gs & hp 40gs is the tools they provide for dealing with statistical data. Unlike the others, the Statistics aplet begins in the offers easy input and editing of values, while the data and which ones frequencies, as well as for indicating pairing of columns for bivariate data.
Page 124 The which to choose, or you can use the name. Entering data as ordered pairs Calculator Tip You can enter the x enter it as ordered pairs in brackets. i.e. as Returning to the data from the previous page, having entered it into the and Auto Scale to produce a plot (this generally produces very satisfactory results), but let's have a look at the PLOT SETUP screen instead.
Page 125 The cursor If you now press you will see the result shown right. If you look at PLOT the screen you will see a small cross and, at the bottom of the screen, a listing of . This is telling you that the cross is currently on the S1[1]: 1,5 first point in data set whose value is...
Page 126 User Defined - discussed on the following page. Calculator Tip If you want the value of L calculated automatically for the Logistic model then store a value of zero into known, you can store a positive real value into memory L prior to the curve fit and this will be used.
Page 127 When you set the model to user defined it means that you are expected to supply the complete equation, including the values of any coefficients. The calculator will not calculate the values of any variables you include. For example, if you were to supply an equation of use the values of currently in memory.
Page 128 Calculator Tip If you have trouble seeing the small dots that the calcu ator uses in its scatter-graphs by defau t then you will be interested in the settings circled on the ri set you are using and press below from which you can choose a different mark. The contrast is illustrated below.
Page 129 PLOT SETUP From the view, press the Calculator Tip Make sure that your data set is defined and view before you try to obtain these results. Results are only SETUP given for data sets that are defined and...
Page 130 As long as you press from whatever you had previously chosen to User Defined instead.
Page 131 There are two methods of dealing with this. The first is to use another measure of goodness of fit. The second is to ‘linearize’ the data (discussed on the next page). The calculator provides an alternative measure of goodness of fit via the...
Page 132 The curve which results in the PLOT the equation comes out as Y = ⋅ EXP (0.693147 X ) This “ “ is the calculator’s notation for Y = ⋅ e EXP( which then changes to Y = 2 Checking the key shows that the correlation is unchanged at 0.9058 even when the new equation clearly fits the data perfectly.
Page 133: The Expert: Manipulating Columns & Eqns
‘multiply the mean by 3.5’ is not hard. The values shown on the screen can also be retrieved for use on the calculator relatively easily. For example, the set of data below contains a suspected outlier (erroneous value). In this case one might suspect a missing comma between the last two values.
Page 134 People often find it easier to simply type HOME them. You can obtain the summation sign in Calculator Tip The values of the mean and standard deviation retrieved are those of last set calculated If you have more than one set of data in the...
Page 135 Obtaining coefficients from the fit model The function from PREDY MATH This means that you must use the make sure is set to the correct fit model, and also use the SYMB SETUP ensure that your set of data was the last one graphed and that it has had its curve of best fit displayed. Until the curve has been displayed, the coefficients are not available or, worse, might belong to another data set.
Page 136 (unlike the correlation) changes as the independent and dependent variables swap roles and can’t be simply algebraically reversed in this way. It should not be thought that the hp 39gs & hp40gs are unusual in this odd interpretation. Most calculators’ equivalent of the function behave in the same manner.
Page 137 While the value of S will not change if the roles of independent and dependent columns are reversed, the value of on the bottom means that this formula will give a different value if you change which column is regarded as x (independent) and which as y (dependent). This different value for b will also mean a different value for a and these will not be the values which would result from the simple inverse function.
Page 138 Now position the highlight on column screen (shown right) enter SORT SETUP column. This will have the effect of pairing columns then sorting column into ascending order, re-arranging column retain the existing data pairings. When ready, press of this sort are shown right. The final column has not been re-arranged.
Page 139 eg. 2 A population of bacteria is known to follow a growth pattern governed by the equation N = N e colonies of bacteria and also that at t = 10 hours there are 10 000 colonies. i. Find the values of ii.
Page 140 (iii) Find t so that N = 2N The value of is the y intercept of the line of best fit. These values from the curve of best fit are not directly accessible but can be retrieved using the function (see page 135). This is shown in the screen PREDY shown right.
Page 141: The Inference Aplet
This aplet is a very flexible tool for users investigating inference problems. It provides critical values for hypothesis testing and confidence intervals, and does this not only quickly but in a visually helpful format. It will be assumed in the explanations that follow that the reader is familiar with the concepts of hypothesis testing.
Page 142 Change now to the NUM SETUP Rather than entering them by hand, press the more than one copy of the Statistics aplet (under other names) then you will be presented with a list of aplets from which to choose. Once you have chosen the aplet, you need to nominate the column from which to import the data.
Page 143 µ Confidence interval: T-Int 1- In the previous example we found that the evidence of our sample indicated that the mean number of matches in the boxes was not 50. Suppose we now want to know, at the 95% confidence level, within what range of values the true population mean lies.
Page 144 µ µ Hypothesis test: T-Test A farmer compared the 15-day mean weight of two sets of chicks, one group receiving feed supplement A and the other supplement C. Twenty two chicks only one day old were assigned randomly to the two groups. To distinguish between the two groups of eleven, which were caged together to minimize other influences, the heads of the chicks were stained red and purple respectively with a harmless vegetable dye.
Page 145 view shows the critical values. We can see that the probability of obtaining a test student-t value of 3.38 is 0.0015 and this is well below the permitted test level of 1%. view also shows that the vertical line representing the value of PLOT −...
Page 146 The hypotheses are: The sample is drawn from a population whose mean is the same as the standardized population The sample is drawn from a population whose mean is larger than that of the standardized population Change to the view, you can use the NUM SETUP import facility to import the summary statistics from the Statistics aplet.
Page 147: The Expert: Chi 2 Tests & Frequency Tables
We will start with a small digression to look at a simple inferential problem which can be solved using only the Statistics and Solve aplets. Using the Chi test on a frequency table “Four coins are tossed 400 times and the number of heads noted for each toss. The results are shown below.
Page 148 In the menu, Probability section (see page 208), there is a MATH function called (Upper-Tailed Probability Chi-squared) which will UTPC give the critical X probability for a supplied number of degrees of freedom and a value. In this case we would like the value for a given probability so we will enter the formula into the Solve aplet.
Page 149 Bear in mind that if you use this program to create a column containing hundreds or even thousands of values then the program will take a long time to complete. In the case of thousands of values you may even exceed the calculator’s memory in the MAKELIST...
Page 150: The Linear Solver Aplet
As you may be able to see, this is a pair of parallel lines and so has no solution. To see this on the calculator we must first change to the 2x2 view, meaning 2 equations in 2 unknowns. At the bottom of the screen...
Page 151: Example 3
‘spindle’ of planes in 3-space as shown in the diagram above right. This situation allows infinite solutions anywhere along the line of intersection of the three planes. As you can see right, the calculator has correctly indicated the situation.
Page 152: The Triangle Solve Aplet
This aplet allows you to solve for missing sides and angles in a triangle, either right angled or not. Unlike most aplets it does not have a view but only the dual view discussed below. PLOT When you first start the aplet you will be presented with one of two views.
Page 153: Example 2
Since this is not a right triangle, the first step is to ensure that α β δ the three angles can be used to represent the 115 δ In this case I will use for no other reason than that it is at the top of the illustration, just as it is in the diagram of the triangle.
Page 154: Example 3
Solve the triangle shown right. This is an example of a triangle that has two possible solutions, generally referred to as “The Ambiguous Case”. The calculator will give both possible solutions. Begin by setting the calculator into Change into the...
Page 155: The Finance Aplet
For example, that a dollar invested today can generate more money than the same dollar invested later. The calculator manual contains a lengthier explanation including cash flow diagrams for those who need it, as does any high school or college textbook.
Page 156 (positive) and no payments (withdrawals) are made during the period of the investment. Five years of monthly payments means that is 60. The view on the right shows the problem on the calculator. The button has been pressed to give a future value...
Page 157 Annuities An engineer retires with $650,000 available for investment. She invests the money in a portfolio which is expected to have an average return of 5% per annum. She wants to have the account pay a monthly income to her and asks the accountant to assume that the income must last for 20 years.
Page 158 Amortization The second page of this aplet allows amortization calculations in order to determine the amounts applied towards the principal and interest in a payment or series of payments. Suppose we borrow $20,000 at an interest rate of 6.5% and make monthly payments of $300.
Page 159: The Quad Explorer Teaching Aplet
Rather than being a multi-purpose aplet, this is a teaching aplet specialized to the single use of exploring graphs of quadratics. As such it does not have the normal multi-purpose view. Objectives Using the Quadratic Explorer aplet, the student will investigate the behavior of the graph of y as the values of a, h and v change.
Page 160 ‘step size’ of the movements on the screen. Possible values for the increment are 0.5, 1 and 2. Pressing on the calculator, or the screen key labeled SYMB change the emphasis from the graph to the equation in the right hand half of the screen.
Page 161 If you go to HP’s website you can download a worksheet for use with your class. It takes the student through the process of deducing the effects of each of the coefficients on the shape of the graph, requiring them to...
Page 162: The Trig Explorer Teaching Aplet
Rather than being a multi-purpose aplet like most of the others covered so far, this is a teaching aplet specialized to the single use of exploring the graphs of trigonometric functions. As such it does not have the normal SYMB PLOT have meanings but not the normal ones.
Page 163 The operation of the two modes is summarized below. The PLOT mode The underlying concept in PLOT equation. The user has control of the graph via two manipulation points (see above and below) and any changes to the graph are reflected in the equation at the top of the screen.
Page 164 The c coefficient is shown as a multiple of coefficient is highlighted and can be changed using the up/down arrow keys in increments of 0.1 for the coefficients a, b and d. π The default increment for c is π π...
Page 165: The Math Menus
‘unknown user function’ error. Since they are not relevant for both calculators, the CAS commands are covered in the section on the CAS for the hp 40gs. key. MATH...
Page 166: Accessing The Math Menu Commands
The mechanics of accessing the MATH illustrate the process using the Polynomial function an extremely useful one. Change into the key. MATH When you do you will see the screen on the right. The menu always first appears with the Real functions highlighted. We could use the arrow keys to scroll down to the Polynomial functions but it is far faster to simply press the key labeled with the letter ‘...
Page 167 39gs & hp 40gs. If you need the higher level commands then consult the manual.
Page 168: The Phys Menu Commands
menu is divided up into three sections by learning area. PHYS These sections are: Chemistry • • Physics • Quantum Physics The contents are simply the numerical values of various physical constants that are useful in calculations and formulae. Chemistry Avogadro’s number •...
Page 169: The Math Menu Commands
- functions used in probability calculations. Some of these functions have little application at school level and will not be covered here. Others will be covered to varying depths. Anyone needing those not covered will find them in the manual that comes with the calculator.
Page 170: The 'Real' Group Of Functions
‘ ’ ‘ ’ CEILING(<num>) This is a ‘rounding’ function but different in that it always rounds up to the integer above. Mainly of interest to programmers. Eg. CEILING(3.2) = 33 CEILING(32.99) CEILING((12+ 6)/7) Note: = -2 not -3. The CEILING(-2.56) to the next integer above, which is -2.
Page 171 FNROOT(<expression>,<variable>,<guess>) This function is like a mini version of the Solve aplet. If you feed it an algebraic expression and an initial guess it will start from your guess and find the value which makes the expression zero. Don’t bother. It’s a lot easier to use the Solve aplet.
Page 172 This function works with time and angles. It converts degrees, minutes and seconds to degrees, and also hours, minutes and seconds to decimal time. The calculator can convert a value such as the form 45.2317 and then use the ′ ′′...
Page 173 INT(<num>) This function is related to the FLOOR those two, which consistently move down or up respectively, the function simply drops the fractional part of the number. Eg. INT(3.786) = -5 INT(-5.84) See also: FLOOR CEILING ROUND MANT(<num>) This function returns the mantissa (numerical part) of the number you feed it when transformed into scientific notation.
Page 174 MIN(num1,num2) As with , this function is used mainly by programmers. It returns the smaller of the two numbers entered. Eg. MIN(3,5) See also: <num> MOD <divisor> For those not familiar with modulo arithmetic, it will suffice to say that this function gives you the remainder when one number is divided by another.
Page 175 %CHANGE(<num1>,<num2>) This function calculates the percentage change moving from the formula . It can be used to calculate (for example) 100(Y-X)/X percentage profit and loss. Eg. I buy a fridge for $400 and sell it for $440. What is my profit as a percentage? Use: %CHANGE(400,440) I sell a toy for $5.95 that normally sells for $6.50...
Page 176 ROUND(<num>,<dec.pts>) This function rounds off a supplied number to the specified number of decimal places (d.p.). Eg. Round 66.65 to 1 d.p. Use: ROUND(66.65,1) Round 34.56784 to 2 d.p. Use: ROUND(34.56784,2) This function is also capable of rounding off to a specified number of significant figures (s.f.).
Page 177 TRUNCATE(<num>) This function operates similarly to the the extra digits instead of rounding up or down. It is somewhat similar in effect to the function but the FLOOR any number of decimal places or significant figures instead of always dropping to the nearest lower integer value. TRUNCATE(3405.6375,-6) TRUNCATE(32.889,1) See also:...
Page 178: The 'Stat-Two' Group Of Functions
If these conditions are satisfied then the function produce a predicted y (dependent) value for the x (indep.) value of Calculator Tip The line of best fit used in the function last plotted. It is up to you to ensure that this is in fact the one you want...
Page 179: The 'Symbolic' Group Of Functions
= sign is simply used in exactly the way that you would expect it to be, mainly in the Solve aplet. It’s easier to obtain the = sign directly from the keyboard. The reason that it is found in the calculator, the hp 38g, had less keys on the keyboard and had no room for it. It was added in later models. ISOLATE(<expression>,<var-name>) This function will rearrange a formula so that its subject is another variable.
Page 180 − ± − 4 ac to give both ’ formal variable to represent the where were memory values, in order to tell the calculator that the active variable was ± = 5 or −1 tools. ± 4 − 6...
Page 181 If you are fortunate enough to have an hp 40gs rather than an hp 39gs then you can do all this far more easily in the CAS.
Page 182: The 'Tests' Group Of Functions
These two groups of functions cover the Trigonometry functions, plus others, which are less commonly used and which have consequently not been given their own keys on the face of the calculator. Use them in the same way as the normal...
Page 183 Some further functions are available in the Hyperbolic group of functions. They are duplicates of functions available on the face of the calculator but give more accurate answers. They would primarily be of use to those people, such as architects and engineers, for whom high accuracy is paramount. These are: EXP(<num>)
Page 184: The 'Calculus' Group Of Functions
LNP1(<num>) As in the previous function, this is supplied to supplement the function and gives a more accurate value when x is near zero. Again, this is not something which would normally be of concern at school level. ‘ ’ ‘...
Page 185 The <expression> supplied is approximated with respect to <var_name> by terms of a polynomial up to <num> power. The screen shot on the right shows the calculator deriving the Taylor polynomial for sin(x) up to the 7 approximated by taking terms from the polynomial: sin( x) = −...
Page 186: The 'Complex' Group Of Functions
‘ ’ Complex numbers on the hp 39gs & hp 40gs can be entered in either of two ways. Firstly, in the same way as they are commonly written in mathematical workings: a + bi. Secondly, as an ordered pair: (a,b).
Page 187 In addition to the trig functions, there are other functions that take complex arguments. ABS(<real>) or ABS(<complex>) The absolute function, which is found on the keyboard above the left bracket key, returns the absolute value of a real number. returns a value of 3. ABS(-3) When you use the absolute function on a complex number a + bi it returns the magnitude of the complex number as...
Page 188 As mentioned earlier, a very useful function ( on the keyboard as the function for SHIFT number in (r, ) form as shown right, then the calculator will display it in θ θ the form r cos( ) + r sin( ) from, and as an (a,b) ordered pair.
Page 189: The 'Constant' Group Of Functions
, are far more easily obtained via the keyboard. π The first, , is available via a key on the face of the calculator above the 3 key. The other two , easily obtained as lowercase letters via the ‘...
Page 190: The 'List' Group Of Functions
‘ ’ ‘ ’ CONCAT(< list1>, <list2>) This function concatenates two lists - appending one on to the end of the other in the order that you specify. Lists must be enclosed in curly brackets unless list variables are used. L 1={2,5,-2,10,3.75} L2={1,2,3,4,5} = {2,5,-2,10,3.75,1,2,3,4,5}...
Page 191 Eg. 1 MAKELIST( X2,X,1,10,2) L1 produces { 1, 9, 25, 49, 81 } as X goes from 1 to 3 to 5 to … and also stores the result into Eg. 2 MAKELIST(RANDOM,X,1,10,1) in this case serves only as a counter since it does not appear in the expression. Eg.
Page 192 SIZE(<list>) or SIZE(<matrix>) This function returns the size of the list or matrix specified. Since normal users would probably know anyway, and could find out easily via the list catalog, this is clearly another of those functions which are of more use to programmers (who won’t know when they write their program just how long the list you will ask it to deal with will be when you eventually run the program).
Page 193: The 'Loop' Group Of Functions
‘ ’ ‘ ’ This is a group of functions that may be of use for students studying discrete functions and sequences but are primarily of use to programmers. ITERATE(<expression>,<var_name>,<num>,<num>) This function evaluates an expression in terms of a variable, starting with a supplied initial value, for a specified number of iterations.
Page 194 ‘i’, ‘j’ & ‘k’ are simply the letters traditionally used in mathematical problems involving summation. When working in the hp 40gs CAS it is not possible to use ‘i’ because the CAS will interpret it as i rather than as a variable name.
Page 195: The 'Matrix' Group Of Functions
‘ ’ ‘ ’ This group of functions is provided to deal with matrices. The scope of functions & abilities covered in this group is in fact vastly greater than would be required by the average high school student or teacher. In many cases supplying an explanation in more detail than the manual of what the function is used for would occupy many pages to no real useful gain.
Page 196 DET(<matrix>) This function finds the determinant of a square matrix. See page 213 for an example of its use in finding an inverse matrix. ⎡ 2 3 ⎤ A = ⎢ then find det(A). ⎥ ⎣ − 1 5 ⎦ Ans: det (A) = 2x5-3x(-1) = 13...
Page 197 Some people write the inverse matrix as a fraction (one over the determinant) multiplied by a matrix, so as to avoid decimals and fractions within the inverse matrix. The calculator does not do this. If you want the matrix with the determinant factored out,...
Page 198 LSQ(<matrix1>,<matrix2>) The least squares function displays the minimum norm least squares matrix (or vector). LU(<matrix>) This Decomposition function is similar to the into three matrices, returning them in the form of a list variable. {[[lower triangular]],[[upper triangular]],[[permutation]]} The upper triangular has ones on its diagonal. The matrices can be separated in the same method outlined for the function.
Page 199 ROWNORM(<matrix>) Finds the row norm of a matrix: the maximum, over all rows contained in the matrix, of the absolute values of the sum of the elements in each row. ⎡ ⎢ Eg. For the matrix M 1 ⎢ ⎢ ⎣...
Page 200 For example, suppose we use the system of equations below, in which the third equation is a linear combination of the first two but the constant is not consistent with this - ie no solution. If we solve this in the same way as before, the matrix ⎧...
Page 201 SVD(<matrix>) This function performs a Singular Value Decomposition on an m × n matrix. The result is two matrices and a vector: {[[m × m square orthogonal]],[[n × n square orthogonal]],[real]}. SVL(<matrix>) This function returns a vector containing the singular values of the supplied matrix. TRACE(<matrix>) This function finds the trace of a square matrix.
Page 202: The 'Polynomial' Group Of Functions
‘ ’ ‘ ’ This group of functions is provided to manipulate polynomials. We will use the function shown right to illustrate some of the tools in the Polynomial group. Its equation is: ( ) = − 2)(x + 3)(x −1) = POLYCOEF([root1,root2,…]) This function returns the coefficients of a polynomial with roots x x x vector form means in square brackets.
Page 203 POLYFORM(<expression>,<var_name>) This is a very powerful and useful polynomial function. It allows algebraic manipulation and expansion of an expression into a polynomial. The expected parameters for the function are firstly the expression to be expanded, and secondly the variable which is to be the subject of the resulting polynomial.
Page 204 As you can see in the screen shot, the roots of 2, -3 and 1 have been correctly found. See page 309 for details on finding roots of real and complex polynomials using the CAS on the hp 40gs. − 7x + 6 we can enter the coefficients as...
Page 205: The 'Probability' Group Of Functions
‘ ’ ‘ ’ This group of functions is provided to manipulate and evaluate probabilities and probability distribution functions (p.d.f.’s). COMB(<n>,<r>) This function gives the value of Eg. Find the probability of choosing 2 men and 3 women for a committee of 5 people from a pool of 6 men and 5 women.
Page 206 It is important to realize that the values produced by the Inside the calculator is a mathematical procedure (an algorithm) which uses a ‘seed’ number to produce them. Unfortunately, when taken straight out of the box, two calculators will produce exactly the same sequence of “random”...
Page 207 UTPN(<mean>,<variance>,<value>) This function, the ‘Upper-Tail Probability (Normal)’, gives the probability that a normal random variable is greater than or equal to the value supplied. Note that the variance must be supplied, NOT the standard deviation. Eg. 1. Find the probability that a randomly chosen individual is more than 2 meters tall if the population has a mean height of 1.87m and a standard deviation of 10.4cm σ...
Page 208 SYMB key is provided in the SYMB Final answer… 47.06% and 82.94% are the cut-offs. Calculator Tip The normal order for the arguments in the UTPN(mean, variance, value) probability. probability instead. Fortunately the function can easily be adapted for this.
Page 209: Working With Matrices
The hp 39gs & hp 40gs deal very well with matrices. They offer many powerful tools as well as a special with full editing facilities. MATRIX Catalog The MATRIX Catalog is entered by pressing (located above MATRIX Catalog MATRIX the 4). It allows storage of ten matrices ( ) which can be M1,M2,..M9,M0...
Page 210 Enter the numbers 4, 5 and 6 and you will find that the calculator automatically drops down to row three without the need to use the down arrow key again, since it now knows how many columns the matrix is to contain.
Page 211 The method for doing this on the calculator is as follows… Step 1. Enter the MATRIX Catalogue matrices if desirable.
Page 212 The third option is actually identical to the first. The original hp 38g only had the later models.
Page 213 Finding an inverse matrix Eg. 2 Find the inverse matrix The first step is to store the matrix A into inverse into you will find, depending on the determinant, that the result is probably a collection of decimal values (see right). This answer is correct and we could stop there.
Page 214 This substitutes to give a solution of: On the calculator, the functions magnitude respectively, when fed with vectors. The calculator writes vectors as row matrices. a = (3, 4) For example would be written as The calculations are shown in the two screen shots on the right.
Page 215: Working With Lists
A list in the hp 39gs or hp 40gs is the equivalent of a mathematical set. As with a set, it is written as numbers separated by commas and enclosed with curly brackets. {2,5,-2,10,3.75} The list variables Using the view these lists can be stored in special list variables.
Page 216 Lists can be sent from one calculator to another using the infra-red link on the hp 39gs or using the supplied mini-serial cable with the hp 40gs. The procedure is the same as that for sending aplets from one calculator...
Page 217: Working With Notes & The Notepad
The hp 39gs & hp 40gs provide access to Notes which can either be attached to an aplet or exist independ ently. The notes belonging to the standard aplets are blank unless you add to them, but copies you transfer from a computer or another calculator may have had notes added to them as instructions on how to use them.
Page 218 An example is shown above right of a sketch from one of the aplets available from HP’s web site called ‘Periodic’. If you installed this aplet on your calculator then this sketch could be viewed via the...
Page 219: Independent Notes And The Notepad Catalog
These notes can be sent to (received from) another calculator or from a computer via the 39gs this is done via the infra-red link. On the hp 40gs there is no infra-red capacity and the transfer is done via the supplied mini-serial cable shown right. More information on cables and their use can be found on page 237.
Page 220: Creating A Note
) and you will see the an existing Notes to or from another calculator (or a computer). A key will delete all Notes in the catalog. SHIFT CLEAR menu. However, if the CHARS key.
Page 221 The original hp 38g only had the function INVERSE(…) and the added for convenience sake in the hp 39g, released in 2000. To do this the creators had to borrow one of the existing unused characters, the...
Page 222: Working With Sketches
Facilities provided in the Sketch view are good for a bit of fun, but very primitive when you try to do anything at all complex. This is not meant as a criticism of the calculator. It does an extremely good job at what it was designed for - working with numbers - but it was never designed to compete with a computer drawing package.
Page 223: The Draw Menu
There are two font sizes available via the size being large. If you press the . Although there is no apparent change when you are typing in the text, the font will become smaller when it appears in the window. Only uppercase is available in this small font.
Page 224 CIRCLE The circle command is similar to the box command. You should position the cursor at the center of the proposed circle. Pressing the cursor outwards from the center, forming a radius. As you do so you will see a small arc appear, giving you an indication of the curvature of the circle.
Page 225 If you’re intending to do this to produce a set of ‘cheat notes’ for your next test or exam, you would do better to spend the time studying! Calculator Tip The screen capture facility demonstrated here can be used to capture...
Page 226: Copying & Creating Aplets On The Calculator
No programming is necessarily required at the lowest level and so, unless you want to learn about the programming language of the hp 39gs & hp 40gs, there is no reason to worry about it unless you want to produce highly enhanced aplets.
Page 227: Different Models Use Different Methods To Communicate
39gs. The mini-serial port is still present though because it is anticipated that it may be used in the future to connect to data-loggers and other possible peripherals.
Page 228: Sending/Receiving Via The Infra-Red Link Or Cable
Any aplet, note, program, matrix or list can be copied from one calculator to another via the infra-red link at the top of the calculator on an hp 39gs or via the supplied cable on an hp 40gs. A sketch can be transferred by sending the aplet to which it belongs.
Page 229 The process is essentially: • Press the key on the sending calculator and the • Choose the option for your particular calculator. On an hp 39gs this will be “HP39 (IrDA)” and on an hp 40gs it will be “HP39/40 (Ser)”. • Press on both calculators and the transmission process will begin.
Page 230: Creating A Copy Of A Standard Aplet
In either of those two cases, the solution is to make a copy of the aplet concerned. You can make as many copies of any of the standard aplets as you wish. The only limit is the calculator’s memory. Depending on what you put in them the calculator’s memory is normally sufficient to store anything from 30 to 100 aplets.
Page 231 This saved aplet can now be downloaded to all the students’ calculators using the infra-red link in the case of an hp 39gs or using the cable in the case of the hp 40gs. This ensures that each student has exactly the required data sets, with views pre-set to the teacher’s needs.
Page 232: Some Examples Of Saved Aplets
The Triangles aplet In the view, the Solve aplet and APLET name of “Triangles”. Now The theta character can be obtained from the keyboard on the zero button using . Change into the ALPHA use another mode). By changing into the Some users choose to use the letter The Prob.
Page 233 Use it by substituting whatever function is in use for the one currently entered. As this formula involves the integration function, each use of the solve process will require the calculator to perform multiple integrations. Because of this the solving process will be relatively slow.
Page 234 Equation This equation gives P a ≤ ≤ calculate P x a just find P x a ≥ Equations Finally, equations concern the Normal distribution, with giving P a allowing calculation of questions such as “what distance either side of the ≤...
Page 235 So… how does this aplet work? The formulas in the view form the key to the process by allowing SYMB the calculator to fetch values from the matrices, with the values fetched being determined by the settings in the For example, as runs from...
Page 236 TRange values in the use a shapes that is highly symmetrical, like a square, as it makes it harder to recognize transformations. Calculator Tip It is probably faster to have the class set it up themselves instead of sending it via cable or infra-red.
Page 237: Storing Aplets & Notes To The Pc
PC and is safe from accidental loss. In addition to this, one of the nice features of the hp 39gs and hp 40gs is that when you switch it on it resumes operation in exactly the state you left it. Not all calculators will do this and it is clearly highly desirable but it also has its downside.
Page 238: Software Is Required To Link To A Pc
The connectivity software for the hp 39gs and hp 40gs was being rewritten at the time when this book was being published. The version on the CD which came with your calculator may not be the most recent version. For the latest version of the software for your calculator you should consult Hewlett Packard’s web site (http://www.hp.com/calculators) or the author’s website at...
Page 239: Sending From Calculator To Pc
This mini-USB cable is used for communication with a PC and the same cable is used for both the hp 39gs and the hp 40gs via the USB cable on the PC.
Page 240 Before beginning you should install the Connectivity software. This can be found on the CD that came with your calculator but it is best to download a fresh version from the web so as to obtain the most recent version (see page 237).
Page 241 PC. This can be one of the standard ones or one that you have saved under another name. Press the button on the calculator to see the view shown right, which lists the choices on an hp 39gs. On an hp 40gs the “ HP39+ (IrDA) Ensure that the “...
Page 242 The reason for the contraction is that the original hp 38g from which the hp 39gs and hp 40gs are derived was released for an earlier version of Windows that did not allow long filenames. This has never been changed on later models.
Page 243 The HP HOME view (at http://www.hphomeview.com). Normally you do not need to worry about this, since the calculator knows they belong with the aplet and will automatically transmit them with it. This can greatly increase the transmission time and it is important that you don’t interrupt the process early.
Page 244: Receiving From Pc To Calculator
The process of retrieving objects that have been stored to the PC is almost identical to that of sending them in the first place. Connect the calculator, run the software and choose the folder in which the aplets, notes, or other objects are stored. Then press the button and again choose the “...
Page 245: Aplets From The Internet
The Hewlett-Packard site is one possible starting point and can be found at http://www.hp.com/calculators. From that point you can follow the links to collections of material for the hp 39gs and/or hp 40gs as well as to software and utilities. Other sites can be found that have been created by enthusiasts. One of the most extensive is that of the author, The HP HOME view, found at http://www.hphomeview.com.
Page 246 You may notice separate download icons for the 38G and for the 39G, 40G and 39g+ with no mention of the new hp 39gs and hp 40gs. This will change as the sites update the contents to reflect the new models.
Page 247 Having downloaded an aplet from the web to the computer, we now have the task of transferring it from the computer to the calculator via the HP Connectivity Software. To do this, of course, you need the mini-USB cable that came with your calculator (see page 239) plus some software.
Page 248 The process of transferring the newly downloaded aplet from the PC to the calculator is exactly the same as it is for an aplet which you have saved to the PC yourself. The instructions for this can be found on page 244.
Page 249: Using Downloaded Aplets
It is worth reading the information contained in Appendix B: Teaching or Learning Calculus on page 314. key on your hp 39gs or hp 40gs then menu VIEWS . When you delete the...
Page 250: Deleting Downloaded Aplets From The Calculator
“Coin Tossing”. .COIN Simply position the highlight on each of the programs in turn and press key. If your calculator only contains one aplet with programs linked, then it is faster to use SHIFT CLEAR once.
Page 251: Capturing Screens Using The Connectivity Kit
Once the image is captured it can be copied to the clipboard and pasted into another application. The software also allows it to be saved as a .BMP image file. All the images of calculator screens in this book were captured in this way.
Page 252: Editing Notes Using The Connectivity Software
Software will also allow you to edit it once it is on the PC. On previous models this was done using a separate piece of software called the ADK but with the release of new software with the hp 39gs and hp 40gs this ability was integrated into one program.
Page 253 The changes you make are automatically saved as you type, just as they are on the calculator itself. When you’ve finished editing you can use the Connectivity Software to transfer the result back to the calculator.
Page 254 The names used to record the Notes on the PC are not terribly imaginative, as can be seen to the right. You must not change these names! files HP39DIR.CUR and HP39DIR.000 and the calculator will expect to find them under those names. Highlighting an existing Note and pressing the Copy button will produce a copy of the existing Note with the title “Copy of …………”.
Page 255: Programming The Hp 39Gs & Hp 40Gs
Although you can choose to simply create programs, it should be remembered that the whole point of working on the hp 39gs or hp 40gs is to use aplets. Working with an aplet means that you inherit its abilities such as...
Page 256 To use this software you must be able to send to and receive from a computer, and for models before the hp 39gs & hp 40gs this means buying a cable. For the hp 39gs & hp 40gs the mini-USB cable required is included in the package with the calculator.
Page 257: Planning The Views Menu
When the ‘helper’ program terminates the calculator drops into whatever view you as the designer choose. For example, a ‘helper’ program might set up axes based on the data entered and then drop the user into the view.
Page 258 Another example of an existing aplet is shown right. It is called “Tangent Lines” and it draws a tangent line onto a graph and then lets the user move it around, displaying the gradient as it does so. This aplet has the Function aplet as its parent because of the need to graph functions.
Page 259: The Setviews Command
Part of the job of the ‘helper’...
Page 260 Special entries in the SETVIEWS command In addition to the lines which form the menu for your aplet, there are some special entries which are treated differently. If you include entries called “Start” or “Reset”, then the ‘helper’ programs associated with those •...
Page 261 Shown below is a program which illustrates this for an aplet with Function as its parent… SETVIEWS The behaviour will be: • Choose “ ” Opt. 1 • Choose “ ” Plot-Table • Choose “ ” Auto Scale • Choose “ ”...
Page 262: Example Aplet #1 - Displaying Info
This example uses the SETVIEWS totally useless) aplet, which illustrates a few of the concepts useful in programming the hp 39gs or hp 40gs. We’ll call it the ‘Message’ aplet and create it as a child of the Function aplet. Change into the...
Page 263 ERASE a message on lines 4 DISP and 5 of the screen. The calculator then s waiting learning a new language! FREEZE for a key to be pressed. .MSG.FN This is the most complex of the programs.
Page 264 Having created all of the programs that make up the aplet ‘Message’, we can now run the program , severing the aplet’s link to its current .MSG.SV aplet, and substituting this new menu. Before you do this, check that you are still in the correct aplet. Press key and check that the title at the top still says “...
Page 265 The next option in the menu is ‘Input value’. Choosing this option will create an input screen. The statement controlling this was: INPUT N; "MY TITLE"; "Please enter N.."; "Do as you're told."; 20: Examine the snapshot on the right and notice the connection between the various parts of the statement and their effect.
Page 266 The final option is ‘Show function’. The program this runs is a little more complex than the ones shown so far and illustrates a useful technique. The line: '((X+2)^3+4)/(X-2)' x + 2 stores the expression Notice the way the function is in single quotes so that the algebraic expression itself is used rather than its value when evaluated using the current contents of memory X.
Page 267 Finally the commands commands are used to draw an oblique line across the screen and a LINE box near the center. LINE Xmin; Ymin; Xmax; Ymax: BOX 3; 3; -2; -2: Notice the use of Xmin Xmax maximum limits of the current PLOT appear the same even if the screen were to be re-sized in the hand, would change size if the screen size were to be changed in...
Page 268: Example Aplet #2 - The Transformer Aplet
Xmin and then changes it back. In the original version the user had to press draw. This technique fools the calculator into thinking that view. PLOT SETUP and therefore forces a re draw without the need to press a key. It also re- ed and ready for use.
Page 269 .TRANSF.SHAPE (continued…) Since the default contents of any variable is zero and there is no zero’th option on a list this means a program bug waiting to happen unless you preset the value. Options 1 and 2 load preset matrices while option 3 allows the user to edit their own.
Page 270: Designing Aplets On A Pc
Connectivity Kit” but this too may have changed. Look for the latest version on HPs website or on The HP HOME view (at http://www.hphomeview.com). In the next example we will use The Connectivity Kit to create small program and then to re-create the same ‘Transformer’...
Page 271 So… if there is a “Save” button or an option on the File menu of “Save” then use this now to save your code. If there’s no such option then go on to the next step. Change back to the Folders/Transfers and download your program to the calculator for use. The result is shown in operation below.
Page 272: Example Aplet #3 - Transformer Revisited
– – Run the Connectivity Kit and use the File menu to create a new folder called “Transformer”, and highlight that folder to hold your aplet. In my experience it is a very good idea to store each aplet in a separate folder but this is not strictly necessary.
Page 273 As you enter each triplet, the boxes will blank ready for the next menu item to be added. You can construct the entire menu at one time OR you can edit the code for the program before proceeding. In many ways it is better to design the entire menu structure before beginning to code but that may not be the way you prefer to work.
Page 274: Example Aplet #4 - The Linear Explorer Aplet
As can be seen in the example below, CHARS there is no attempt to place line breaks at the end of words. The view on the calculator will certainly not match that on the PC so why bother?
Page 275 If you have done this correctly then your menu have three entries VIEWS shown right when it is transferred to the calculator. The text for the ‘helper programs’ associated with each menu entry is shown below:...
Page 276 It will probably be easier to understand how the aplet works if you see it in action first so you may wish to download the aplet from The HP HOME View The first option on the menu plots a set of axes and must be VIEWS chosen first.
Page 277 The second and third lines insert a function into done, of course, if the parent aplet is parent is another aplet then the code will still execute but the function will be inserted into the real Function The reason for inserting this particular function is that we need a function when the axes are plotted or the normal error message will be displayed (see above right).
Page 278 Still referring to the code on the previous page, you will see that it refers to . The sketches in the PageNum calculator’s view are numbered 1, 2, 3…etc. Sketch number 1 is always present but after that only SKETCH sketches that have been created are available and the program will crash if you try to access one that does not exist.
Page 279 Any labels off the edge of the screen will be ignored by the calculator so you don’t need to check for that in the program. This is a very nice feature of the HP.
Page 280 This aplet illustrates most of the commonly used programming techniques. If you would like to gain experience then I suggest that you do as I did when I first began programming the HP - download aplets and pull them apart to see how they work.
Page 281: Alternatives To Hp Basic Programming
HP Basic might take 50 lines or even more in sRPL. However it is quite likely that those 50 lines will execute fifty to sixty times as fast as the five lines of HP Basic. The drawback is that they will also have a much greater risk of containing a bug –...
Page 282 The HPG-CC Programming language The hp 39g+ was the first of this family of calculators which didn't use the Saturn 5 as its ROM chip. Up to that point the HP38G, HP39G & HP40G had all shared the same chip along with others in the HP48 family.
Page 283 At the time of writing this information in early 2006, HPG-CC had reached Version 2.0 but subsequent to this version the support for the hp 39g+, hp 39gs & hp 40gs had been discontinued. This is not to say that it won’t work on those calculators, just that the developers are no longer fixing bugs discovered by...
Page 284: Flash Rom
In normal ROM the contents of the chip are burned in at the factory and can’t be altered. The difference with the hp 39gs & hp 40gs is that the ROM used is a ‘flash’ ROM. This is a special chip where, although the contents are preserved when the batteries are removed or a reset is performed, the contents are not frozen permanently.
Page 285 40gs. The operating system of an hp 40gs is quite a bit larger than that of the hp 39gs due to the inclusion of the CAS. It will not fit onto the ROM chip of an hp 39gs and any attempt to try will result in an expensive paperweight.
Page 286: Programming Commands
As was explained in a previous chapter, the hp 39gs and hp 40gs are supplied with a simple and easy to use programming language called HP Basic. All programming commands can by typed in by hand but, as with the commands, can also be obtained from a menu.
Page 287: The Branch Commands
IF <test> THEN <true clause> [ELSE <false clause>] END Note the need for a double sign when comparing equalities. Any number of statements can be placed in the true and false sections. Enclosing brackets are not required, as they are in some other languages.
Page 288 RUN <program name> This command runs the program named, with execution resuming in the calling program afterwards. If a particular piece of code is used repeatedly then this can be used to reduce memory use by placing the code in a separate program and calling it from different locations. See command for information on how to link a program to SETVIEWS an aplet when it does not appear on the primary menu.
Page 289: The Drawing Commands
ARC <x-center>;<y-center>;<radius>;<start angle>;<end angle> This command draws an arc on the screen. It uses the current values in the PLOT SETUP coordinates and the settings in the the angle format. This command is unfortunately quite slow. BOX <x >;<y >;<x >;<y >...
Page 290 This means that it can be used to erase previously drawn lines. If you would like to see this command in action, download the aplet called “Sine Define” from the author’s website The HP HOME view (at http://www.hphomeview.com). This aplet contains extensive use of this command.
Page 291: The Graphics Commands
See the chapter “Programming the hp 39gs & hp 40gs” beginning on page 255 for examples illustrating some of the graphics commands that I have used regularly. Consult the manual for more. FOR <variable> = <start value> TO <end value> [STEP <increment>] <statements> END For those familiar with the Basic language in other forms, this is a standard FOR…NEXT command, except without the ‘NEXT’.
Page 292: The Matrix Commands
There is no situation in which a GOTO statement cannot be replaced by one of the loops or conditional branches available in the HP Basic language. EDITMAT <matrix var> This command pops up a window in which the user can edit or input a matrix with an bottom.
Page 293: The Print Commands
39g family. This printer was originally designed for the first calculator in this family, the hp 38g, released in 1995. Very few of them sold because it was far easier to simply use the connectivity software to capture screens and images and then just paste them into a document on the PC.
Page 294: The Prompt Commands
BEEP <frequency>;<duration> This will use the piezo crystal in the calculator to create a sound of the specified frequency for the specified duration (in seconds). The resulting frequency is not terribly accurate, varying by up to 5% from one calculator to the next and depending also on the temperature. If you want accuracy then use a piano! The volume is also not very loud because of concerns with interruptions to tests and examinations.
Page 295 DISPXY <xpos>;<ypos>;<font>;<expression> This command displays the text/object/result contained in <expression> at the screen position specified using the font specified. An extensive example can be found in the chapter “Programming the hp 39g & hp 40gs” on page 274. The xpos and ypos values refer to the top left corner of the text when displayed.
Page 296 DISPTIME This command pops up a box displaying the calculator’s internal time and date. These can be set by storing values to the variables Time . Suppose the current time is 3:46:29 pm on the 1 of December, Date 1998 then you would store 15.4629 to and 12.011998 to...
Page 297 WAIT <duration> This command pauses execution for the specified number of seconds. Execution resumes at the next statement after the command. WAIT...
Page 298: Appendix A: Some Worked Examples
The examples which follow are intended to illustrate the ways in which the calculator can be used to help solve some typical problems. In some cases more than one method is shown. In some cases the method is chosen more to illustrate the capabilities of the calculator than because it is necessarily the most efficient method.
Page 299: Finding Complex Solutions To A Complex Equation
This can make the real roots harder to isolate and use. However, it is still the best method. Method 3 - Using the CAS on the hp 40gs See page 309 for an example of finding roots of real and complex polynomials using the CAS on the hp 40gs.
Page 300: Finding Critical Points And Graphing A Polynomial
( ) = x For the function find the intercepts. (ii) find the turning points. (iii) draw a sketch graph showing this information. (iv) find the area under the curve between the two turning points. Step 1. Enter the function into the it is available for plotting.
Page 301 Step4. Because I know that part (iv) of the question requires me to re use these extremum values in an integration (which I would like to be as accurate as possible), I am going to ‘save’ the extremum value just found. I change into the store it as shown in memory Note: You MUST store the point of interest before moving the cursor in the...
Page 302: Solving Simultaneous Equations
Solve the systems of equations below: − 3 y = − 7 ⎫ y = 2 ⎭ Method 1 - Graphing the lines Because the first set of equations is a 2x2 system it can be graphed in the Function aplet. To do this it is necessary to re-arrange the functions into the form y = ……...
Page 303 If you are fortunate enough to own an hp 40gs rather than an hp 39gs then you can use the CAS for this. See page 346 for an example.
Page 304: Expanding Polynomials
Expand the expressions below. (i) 2x + 3 (ii) 3a − 2b (i) POLYFORM((2X+3)^4,X) Use the key to examine the result. Result: 16x + 96 x + 216x (ii) POLYFORM((3A-2B)^5,B) as a function of . Then use the polynomial function again, ing the result from the first expansion and expanding this time as a function of used to view it, using the left and right arrows to scroll the...
Page 305: Exponential Growth
Predict the number of bacteria colonies after 15 hours. (iii) How long does it take for the number of colonies to double? (i) Find and k using the calculator. Step 1. Start up the Statistics aplet, set it to given. Change to the SYMB SETUP Exponential line of best fit for the data.
Page 306 (ii) Predict N for t = 15 hours. In the view, press up arrow to move the cursor onto the PLOT curve of best fit. Now press cursor will jump to the predicted value for x=15, which is currently off screen. Alternatively, change to the function.
Page 307: Solution Of Matrix Equations
⎟ , ⎝ − 1 5 ⎠ The algebraic calculation for this process is shown above right. Having done this, we will now see how to calculate this result on the calculator. Store the values of A and B into respectively.
Page 308: Finding Complex Roots
Find the complex roots of z The best way to do this is using POLYROOT the hp 39gs and hp 40gs can be complex vectors, not just real valued matrices. (i) The coefficients can be entered into a+bi or as (a,b). In this case the roots are integers so there is no need to store it into a matrix.
Page 309: Complex Roots On The Hp 40Gs
Find all roots of the complex polynomial f z Find the complex roots of z On the hp 40gs you can obtain exact roots for polynomials using the CAS function . The instructions following assume that the SOLVEVX CAS is in its default configuration. See page 324 for more details on the CAS.
Page 310: Analyzing Vector Motion And Collisions
Ship A is currently at position vector 21 velocity of -4 km/hr. Ship B is at 30j and traveling at 2i + 3j km/hr. If the ships continue on their present courses, show that they will not collide and find the distance between them at the time of their closest approach.
Page 311: Circular Motion And The Dot Product
I want to graph this function for the first six seconds but I am not sure what y scale to use so I will set XRng - Auto Scale. The result is shown right. view and then choose VIEWS Using Extremum, I find that the time of closest approach is at t = 3.4 hours (3:24 pm) with a separation at that time of d = 1.3416 km.
Page 312: Inference Testing Using The Chi 2 Test
A teacher wishes to decide, at the 5% level of significance, whether the performance in a problem solving test is independent of the students’ year at school. The teacher selected 120 students, 40 from each of Years 8, 9 & 10, and graded their performance in a test as either A or B.
Page 313 Changing into the Solve aplet we can enter a formula which will allow us to calculate values from the Chi distribution using the function. UTPC With a 3x2 contingency table the number of degrees of freedom are 2. To find the critical χ...
Page 314: Appendix B: Teaching Or Learning Calculus
‘nice’ values. This aplet can now be sent to each student’s calculator at the end of a lesson using the infra-red link. Accompanying questions should address the issues below, and students should be required to either hand in a short written response, or contribute to a verbal discussion the next day.
Page 315: Domains And Composite Functions
There are a number of ways that the calculator can help with this. Examples are given below but others will no doubt occur to experienced teachers. Rational functions can be investigated using the example, enter the functions Discussion will elicit the fact that they are ‘identical’ algebraically...
Page 316 ii. When discussing the concept of a domain, the very useful in developing this (see right). In the view, enter the functions shown right, un SYMB first two non-composite functions. In the used the view to set the scale to start at -1 and NUM SETUP increase in steps of 0.25.
Page 317: Gradient At A Point
This is best introduced using an aplet called “Chords” downloaded from The HP HOME View web site (at http://www.hphomeview.com), but you can also use the Function aplet. If you use the aplet you will find that there is a worksheet supplied with it.
Page 318: Gradient Function
The process can be done in a far more visual style using an aplet downloaded from The HP HOME View web site (http://www.hphomeview.com) called “Tangent Lines”. This aplet will add a moveable tangent line to an existing graph, as shown in the screen shot above, allowing the user to move it along the curve with the gradient displayed at the top left of the screen.
Page 319: The Chain Rule
If desirable, an aplet is available from The HP HOME View web site (at http://www.hphomeview.com), called “Chain Rule”, which will encourage the student to deduce the Chain Rule for themselves. It is pre-loaded with five sets of functions, of increasing complexity, the first three of which are shown right.
Page 320: Area Under Curves
One of the concepts which students find quite difficult to come to grips with is that of sketching a field of slopes from a derivative function and, from this, sketching a family of curves. An aplet from The HP HOME View web site (at http://www.hphomeview.com), called “Slope Fields”, will assist with this process.
Page 321: Inequalities
The topic of inequalities is one that is often included in calculus courses, particularly during the study of domains and this is usually extended to graphing intersecting regions such as Although the hp 39gs & hp 40gs do not have the in-built ability to plot inequalities, the process is easily handled using an aplet from The HP HOME View web site (at http://www.hphomeview.com) called “Inequations”.
Page 322: Piecewise Defined Functions
Through the Sequence aplet the hp 39gs & hp 40gs provide very flexible tools for the investigation of sequences. These can easily be adapted to investigate series as well. Information and worked examples of using the calculator for evaluation of sequences and series can be found in the chapter “The Expert –...
Page 323: Transformations Of Graphs
“Quad Explorer” and “Trig Explorer” and are built into the calculator. Both of these aplets allow the student to explore the effect of changing parameters on the shape of the graph, one using a quadratic and the other with the sine and cosine curves.
Page 324: Appendix C: The Cas On The Hp 40Gs
This appendix is intended to give a useful introduction and over view to the user who is new to an hp 40gs. It is not intended to fully cover the topic, nor is it intended to serve as a reference text for the advanced user.
Page 325 Renée De Graeve, Jean-Yves Avenard and Jean Tavenas, and again adapted for inclusion in the HP49g+, HP48GII and HP 40gs calculators. The HP CAS system offers the user a vast array of functions and abilities as well as an easy user interface which displays equations as they appear on the page.
Page 326 It also had two new aplets, the Triangle Solver and the Linear Solver aplets. If you own an hp 39gs and you are wondering if you can fool it into believing that it is an hp 40gs and thereby activate the CAS, then please don’t try it. If you disable the infra-red by, for example, cutting the internal wires then you won’t have an hp 40gs - you will simply have a crippled hp 39gs.
Page 327: Using The Cas
For information on this and the other contents of the calculator. It is not advisable to change entries in this view without being clear on what you are doing since it can alter the behavior of the CAS. Some functions don’t work if...
Page 328 Defining new variables In addition to the pre-defined variables you can also define your own using the new variables can have names that are more than one character long and can contain not only numbers but objects such as algebraic expressions. For example, STORE(X -1,FRED)
Page 329 Assume that we want to show working by evaluating the binomial expression separately. Press the subtract, then press expression. The screen should appear as shown right. Press to evaluate this portion of the expression by expanding the brackets without affecting the rest of the expression. iii.
Page 330 viii. Pressing this time will result in the screen shown below right which displays the two complex roots. ix. Press CLEAR in this case is the whole CAS editing screen. In the material on the following page we will explore this editing screen in more detail, looking particularly at why it is necessary to highlight expressions in some cases and not others, and how to move the highlight from one component of the expression to another and how to use special keys to manipulate the expression.
Page 331 Pressing down arrow at that point moved down the tree. The default is to move to the left-most node D. This meant that the it was that expression which was cubed. At that point the tree now appears as shown right with the CAS in ‘typing/editing mode’ on node G. Pressing up arrow three times placed the highlight successively on nodes G, then D, then B.
Page 332 After typing the 5, press up arrow once to highlight that node S. If you now press left arrow you will find that the highlight will jump horizontally to node A, highlighting the entire numerator. Pressing down arrow four times moves down through the tree from A to P to B to D to F. To access and change the power of 2, press up arrow twice to move up to node B, then press right arrow to move from node B to node Q.
Page 333 Special characters As in the view, special characters such as inequalities are HOME available from the view, although the appearance of the CHARS view is somewhat different as can be seen right. There are no CHARS page up/down buttons, which makes it more difficult to move through. The initial two rows are invalid characters that can’t be used –...
Page 334 Special editing commands – Undo, multi-select & swap Unlike most calculators the CAS editing screen has an undo function. If you have performed some operation that was incorrect then pressing SHIFT MEMORY have more memory to work with and so allow multiple levels of undo, this will only undo a single operation. However, this can be very convenient at times.
Page 335 ‘in-line’ as if you were entering it in the calculator’s normal view.
Page 336 Changing Font Although the default font is very easy to read, it is quite large and often makes parts of the expression or result extend off the screen. Changing to a small font can help with this, at the cost of making the characters a little more difficult to read.
Page 337 If you want to delete the entire expression then the simplest method is to press re-enter it with a blank screen. Alternatively you can highlight part or all of the expression and then press . The highlighted section will be cleared. SHIFT ALPHA CLEAR Cutting and pasting of all or part of an expression can be easily done using the...
Page 338 The PUSH and POP commands Occasionally it is desirable to transfer results from the normal view to the CAS screen or vice versa. This is done using the PUSH Suppose we have just expanded (2x+3) we press to exit the CAS and then type HOME then the result will be retrieved to the of the...
Page 339 If you choose the Function aplet then you will be asked to nominate a destination. The current contents of each function is shown to allow you to choose whether to overwrite or not. All you need then do is exit the CAS and enter the Function aplet.
Page 340 (see page 73 and the page following). A better alternative is to use the shown below, even though it still does not add the ‘+c’ (see page 73 for reasons). Calculator Tip The result of the and pressing...
Page 341: Examples Using The Cas
In these examples we will begin with exercises which demonstrate the basic abilities of the CAS to simplify expressions and then move on to the use of the functions available through the various menus. In the initial examples the exact keystrokes will be supplied but in later ones this may not always be the case. Example 1: Simplifying a fraction with working Suppose you are required to simplify the expression shown right, giving your answer as a proper fraction.
Page 342 ENTER ENTER iii. Finally, select the entire expression with SHIFT and press to simplify: ENTER iv. If you want the result as a decimal, press . Pressing SHIFT will cause the calculator to analyze the decimal and re-instate the surd.
Page 343 Functionally there is no difference so why it was added is not clear. ENTER ENTER 2 - 0 are the same. On the old model hp 40g the only function was ” as −∞ LIMIT...
Page 344 iii) Find 2 + 5 x→∞ Limits to infinity are also permitted using the function, with infinity entered using the shortcut SHIFT 0 scroll to and press ENTER ENTER 344...
Page 345 Example 4: Factorizing expressions If you highlight an expression such as (2x+3) the result extends beyond the screen we will scroll through it using the arrow keys. The results can be factorized again using the COLLECT In this example we will also illustrate the use of the CAS History to fetch a previous calculation.
Page 346 Example 5: Solving equations − = Solve the equation real solutions and complex solutions. From within the CAS, press SHIFT MODES menu and ensure that you are in real mode by, if necessary, , as shown right. Complex mode Now use the function, typing: SOLVEVX function assumes that the active variable is being used.
Page 347 function can also be used to solve problems with symbolic coefficients such as the one below. LINSOLVE Solve the system of equations: The command is LINSOLVE( 2.X+K.Y-1 AND (Q+3).X-Y-5, X AND Y) and it produces the results shown. Press to see the final solution in a VIEWS scrollable format.
Page 348 As can be seen above, the initial integration gives an equation involving a fraction. This can be simplified by multiplying both sides by 6, highlighting the entire equation first. Notice that when the final simplification is equal to zero, the calculator does not bother to include the ‘ zero unless otherwise specified.
Page 349 We can now use the LINSOLVE second linear equation is still highlighted, fetch the command from the menu. Then press highlight the linear expression again as shown right. Now, while the entire expression is highlighted, as shown above, press to obtain the ‘ ’.
Page 350 Example 8: Defining a user function function allows you to define your own functions, which are then available for use. In the example below it has been used to define Fermat’s prime function . Note that the sequence of keys is: ENTER ALPHA F ( SHIFT = 2 + 1 ENTER...
Page 351 We can now test to see if this is a prime number by using the function from the menu. This is found in the Integer section of the MATH CAS function list as shown right. It returns a value of (false) indicating that it is not a prime number.
Page 352 1. As in the previous case, the to jump to the first function that starts with , the letter on 2. The hp 40gs will probably ask if it should turn “Complex mode” on, assuming it is in its default configuration. One of the is to tell it “Yes”.
Page 353 And, having linearized it, we store it as a variable ALPHA M ENTER When the command is executed the STORE expression is echoed back to the screen. Press SHIFT ALPHA CLEAR At this point, any reference to image above is in small font purely to allow the entire expression to be seen. Your screen will not be unless you’ve selected small font earlier.
Page 354 vi. Clear the current contents of the screen using SHIFT ALPHA CLEAR imaginary part of ALPHA M MATH ENTER ENTER Note: As before, the function with that letter ( ENTER ALPHA Y 1 ( ALPHA SHIFT T SHIFT SHIFT vii. In order to show that the function is symmetrical about the x axis we need to show that equivalent algebraically to evaluate each side.
Page 355 When a function is sent to the Parametric aplet the real part is sent to the x and the imaginary part to the y component of the parametric equation selected. Calculator Tip Although earlier this has no relevance for the pasting into the Parametric aplet.
Page 356 If you press highlight each function in turn and press This seems to make the calculator accept that it is a valid function, since it has had the chance to examine the function as if you had typed it in yourself. Pressing produce the screen shown right.
Page 357 Example 10: First order linear differential equation In order to illustrate the use of the CAS help pages discussed on page 361 we will the example provided in them rather than making one up. The functions available for solving differential equations are LDEC Begin by pressing SHIFT SYNTAX...
Page 358: The Cas Menus
There are a variety of different places that functions are stored, often overlapping for greater convenience. The Screen menus On the main screen of the CAS you will see six labels of discussed beginning on page 335. The others each pop up a menu of functions that are the most commonly used ones in each category but there are others that you can access via the MATH button or the menu.
Page 359 The MATH menu Pressing the button in the CAS has a different effect than in the MATH screen. In the screen the result is as shown right. As was HOME HOME discussed on page 165, these commands are broken into broad groups of MTH (mathematical), CONS (constants) and CAS.
Page 360 The abilities of the functions listed here and on the previous page will not be discussed in this text beyond the examples given on the previous pages. The manual supplied with your calculator is generally quite clear on their use and abilities.
Page 361: On-Line Help
One of the most helpful features of the hp 40gs CAS is the on-line help provided by the button ( SYNTAX display the menu shown to the right. You can use the up or down arrow keys to scroll through this list but it is very extensive and it is far quicker to press the button corresponding to the first letter of the function on which you require information.
Page 362: Configuring The Cas
One method is via the configuration line ) at the top of each menu. The line shown right of means that the calculator is set to variable, and you are working in If you select this option of the menu then you will see the further menu shown to the right.
Page 363 Below the title bar you can see the first section of a series of alternatives which let you manipulate the configuration. Most alternatives are toggles having only two values. For example, choosing pressing will cause the menu to momentarily disappear and then re-display with the new setting of ENTER .
Page 364 + bi or in the form of an ordered pair (a,b). If Complex mode is not selected and an operation results in a complex number, you will be asked to switch to Complex mode. If you decline, the calculator will report an error or, in the case of factorizing, simply return the original expression. [Default: unselected.] When in Complex mode, the CAS is able to perform a wider range of operations than in non- complex (or real) mode, but it will also be considerably slower.
Page 365 When selected, it simplifies to ¼. I suggest that this is generally more desirable! Calculator Tip Changing the configuration can have a profound effect on the behavior of the CAS. is suggested that you read carefully the relevant information in the manual and/or the information in Renée de Graeve’s book mentioned...
Page 366: Tips & Tricks - Cas
& & • In CAS, angles are always expressed in radians and no other setting is possible. When you are the calculator screen, you can use the HOME the CAS. • mode might appear to be quite useful for students but is quite limited in what it actually Step by step displays.

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