User Manual - Ryan Harrison



User ManualNumerical Methods HelperRyan HarrisonContents –Front CoverContentsIntroductionMain FeaturesInstallation InstructionsGetting StartedThe Graphing View in more detailThe Integration TabThe Parameters, Worked Examples and Results Section – A past exam question The Differentiation TabExplanation of resultsThe Extrema TabAn exampleThe Iterations TabA past exam questionCopy to ClipboardIntegrationDifferentiationExtremaIterationsMathematical Expressions as ParametersSupported Operators and ConstantsBinary OperatorsUnary OperatorsConstantsForming an Expression or Function Common Error MessagesRecovering from an error Known BugsIntroduction – Numerical Methods Helper is a learning tool for numerical algorithms in AS and A2 Mathematics modules. The program supports methods of numerical integration, root finding through iteration, local extrema, and tangents and normal lines to functions.The progress of each algorithm is drawn onto a freely movable and interactive graph to allow students the most engaging learning experience. All parameters for each algorithm are completely customisable, and the results and graphs will be re drawn to accommodate the new values.Not only does Numerical Methods Helper give a final result to an algorithm, but it also gives the calculations needed for students to get the final answer for themselves. For example, all integration algorithms are accompanied with list of ordinates used in the algorithm, along with a final formula that can be typed into any calculator to yield the final same final result.Functions for use in any of the algorithms can be freely customised and changed using a number of built operators including trigonometry and logarithms. Not only can the program be used for the tuition of numerical methods, but the graphs can also be utilised for other topics such as transformations of graphs.The main purpose of Numerical Methods Helper is to provide a more engaging learning experience for a variety of mathematical topics. The interactive graphs and customisable functions can be used to demonstrate solutions to a wide variety of exam questions both inside and outside of the numerical methods modules.Main Features – Freely scrollable and zoomable graphing viewWorked examples of all numerical algorithms and AS and A2 Maths including – The Trapezium ruleThe Mid – Ordinate RuleSimpsons RuleRoot Finding by iterationTeaching aids for other topics such as local extrema and tangents and normal lines to functionsEvery parameter for each algorithm is fully customisable to meet a wide range of questionsFunctions for use in any algorithm can be changed using a wide range of built in operatorsGraphing views can be copied to the clipboard to paste into other applications for annotation and printingMain parameters for algorithms can be modified by dragging points on the graph itself. This allows for a more interactive and visual approach to the progress and inaccuracies of the algorithmNumerical data used in the calculations of the results of the algorithm is also provided to allow students themselves to reach a solutionInstallation Instructions – To install the program, make sure that the two files - ‘MathEngine.dll’ and ‘Numerical Methods Helper.exe’ - are in the same folder on your computer. For example – 90170205105To open the program, navigate to the folder where both files are stored, then double click on the file named ‘Numerical Methods Helper.exe’.Getting started – First of all open the program named ‘Numerical Methods Helper.exe’. If the installation process was successful, you should be greeted with a window that looks like this – 54321The main user interface is split up into these main sections –The tool strip – contains a button to close the application along with an option to copy the current graphing view (along with algorithm parameters) to the clipboard to allow for pasting to other applications.The tabs – these tabs allow you to switch between the different mathematical modules supported by the application – Integration – numerical integration algorithmsDifferentiation – tangents and normal lines to functionsExtrema – search for local minimums and maximums to a functionIterations – search for the root of a function using iterationsThe graphing view – the graphing view draws the current function and algorithm. The view is fully interactive. Drag the mouse to scroll the view. Scroll the mouse wheel to zoom in and out. Hit the spacebar to reset the graph to the default view.Algorithm parameters, worked examples and results for the tab currently selectedThe current function for use in all tabs. To change the current function, type a mathematical expression into the textbox (in the form f(x) = ...) and hit enter. All graphs and result will be automatically updated. Refer to the ‘Forming an Expression or Function’ page for a list of operators that can be used in the function.The Graphing View in more detail - Here the graph of f(x) = abs(x^2 + 8*x + 12) has been drawn (see the ‘Forming an Expression or Function page for more information) - 1905020320To scroll the graphing view, drag the mouse over the graph in the direction you want to scroll in. The can be used to get a better view of the current graph. If you scroll out of the view of the X and Y axis’, they will still appear at the edge of the graphing view to inform you of the current coordinates.641985100965To zoom in and out of the graphing view, scroll the mouse wheel depending on which direction you want to zoom in. This can be used to get a better view of a certain feature of the graph. In this case the root at X = -2. The graphing view will zoom in or out in relation to the crosshair located in the middle of the graphing view. When zooming, the axis labels will automatically update to give a better coordinate scale.89979599695Currently the maximum zoom level is in increments of 000005.Currently the minimum zoom level is in increments of 1000To reset the graphing view to the default zoom level and origin, hit the spacebar when the graphing view is selected.Notes – When very complex expressions are inserted, it is possible that portions of the graph will not appear due to performance reasons.The Integration Tab – The integration tab provides access to algorithms for numerical integration. Supported algorithms include – The Trapezium ruleThe Mid – Ordinate RuleSimpsons RuleEach algorithm will be drawn onto the graphing view to allow a visual representation of the accuracy of the particular algorithm selected.-135255100965Here the function f(x) = sin(x) is used for the mid ordinate rule algorithm between 10 and 0 with 6 iterations.19685126365Here the function f(x) = sin(x) is used for the trapezium rule algorithm between 10 and 0 with 6 iterations.10033082550Here the function f(x) = sin(x) is used for the Simpsons rule algorithm between 10 and 0 with 6 iterations.To interactively move the current Upper and Lower bounds drag the blue and green points located on the X axis of the graphing view. You will see that the bounds, graph and result will automatically update for each change – The green point represents the Upper BoundThe blue point represents the Lower Bound163830469905054603810(upper at 10)16383018224531178541910(upper at 7)1234567The Parameters, Worked Examples and Results Section – -135255117475The Upper bound of integrationThe Lower bound of integrationThe number of Strips to in the algorithm selectedThe Result of the algorithm between the Upper and Lower bounds with the specified number of StripsThe current numerical integration algorithm selected. To select a different algorithm, simply click in the circle next to the corresponding nameThe list of Ordinates used in the algorithmThe Formula for the algorithm. This uses the list of ordinates to result in an expression to give the final result. The formula could be typed into any calculator to yield the same final result The formula will change depending on which algorithm is current selectedTo change any of the integration parameters, type in a new value into the relevant textbox, and press the enter key. The graph and results set will automatically update to show the new parameters.A past exam question – Use the mid – ordinate rule with four strips to find an estimate for the integral of30988041275To answer this exam question, first open Numerical Methods Helper and make sure the integration tab is open. Then type the new function into the function textbox and press enter to update the graphing view. In this case the function will be “cos(sqrt(3*x + 1))” (refer to the Forming an Expression or Function page for more information on forming functions). The graph view should now look something like this after the enter key has been pressed to update the function used - 1905038735Now that the new graph has been drawn, we need to update the integration parameters to reflect the exam question we wish to answer. In this case the Upper bound will be 0.4, the lower bound will be 0 and the number of Strips will be 4. As the Mid – Ordinate Rule is already selected, we do not need to change this property, however if another algorithm was already selected, change the method correspondingly. After the new parameters have been inserted, hit the enter key in one of the textboxes to update the graph and results. The view should not look like this – 196215501015As you can see the final result of 0.122 is displayed which in this case is the correct answer. A list of ordinates is also displayed on the right hand side. Finally the final formula used to calculate the result is also displayed. The formula can be typed into any calculator to yield the same final result.The Differentiation Tab – The differentiation tab can be used to plot the tangents and normal lines to a function at a specified target point. The gradients of the tangents and normal lines, along with their equations at the target point are also displayed. With the default function f(x) = sin(x), the differentiation tab will look something like this – Here the target point is set at 1.5. This is where the tangent and normal lines will cross the function.To change the target point, type in a new value into the target point textbox and hit the enter button. The graph, gradients and equation will be automatically updated with new results.On the graph the red line represents the tangent to the function and the grey dashed line represents the normal line.The blue point on the graph can be dragged to interactively change the target point. The tangent and normal lines will be automatically updated, along with the results section at the bottom of the tab.Explanation of results - 54321The target point of differentiation. This will be the point where the tangent and normal lines will cross the target functionThe gradient of the tangent line at the target pointThe gradient of the normal line at the target pointThe equation of the tangent line at the target point in the form ‘y – y1 = m(x – x1)’The equation of the normal line at the target point in the form ‘y – y1 = m(x – x1)’Notes –Due to the fact that derivatives are calculated using numerical approximation, the gradients for the tangent and normal may sometimes be inaccurate.When the gradient of the tangent is zero (or there is a stationary point), the normal line will not be drawn. This is because the normal gradient will be complex infinity.Similarly when the gradient of the normal is zero, the tangent line will not be drawn. This is because the gradient of the tangent line will be complex infinity.The Extrema Tab – The extrema tab can be used to search for local minimums and maximums to a function. With the default equation f(x) = sin(x), the extrema tab looks like this – 21A list of local extrema found. For each extrema, the X coordinate, Y coordinate and type of extrema (minimum or maximum) is shown.A button to search for local extrema between the bounds of the current graphing viewAn example - Find all local minimums and maximums to the function f(x) = sin(x - 2) between -3 and 4.To start we need to update the function used with that provided in the question. In this case the function will be sin(x - 2) (see the ‘Forming an Expression or Function’ page for more information). After the function has been inserted, hit the enter button to update the graph. The new graphing view should look like this – Before we can start searching for local extrema, we need to define the searching bounds. To do this move the graphing view (refer to the ‘Graphing view in more detail’ page for more information) so that the graph is shown between -3 and 4. For some instances this may not be possible. In these cases get as close to both limits as possible. The new graph view should look something like this – Now the searching bounds have been set, press the ‘Find Local Extrema’ button to begin searching. You will see a blue point move across the function. This defines the current position local extrema is being searched for. When the blue point reaches the end of the current view, you should see this message – Press Ok. You should now see that the list of local extrema has been populated with the three local maximums and minimums of the function that lie between -3 and 4 – As you can see three local extrema have been found in the searching bounds. If you look at the corresponding graph, we can see that all three local extrema between the searching bounds have been obtained correctly.Notes – Due to the fact that derivatives are calculated using numerical approximations, the X values of local extrema could be inaccurate. Usually the values are correct to one or two decimal places.Currently the application does not recognise point of inflexion. The application will count these points as a local maximum or minimum.In some circumstances duplicate extrema values can appear in the results set. Again this is due to inaccuracies when calculating derivatives.The Iterations Tab –The iterations tab allows for root finding through iteration. With the default equation f(x) = sin(x), the iterations tab looks like this – 4321The iteration results – each entry contains the iteration number and the result of the iteration.The number of iterations to useThe value of the first iteration (must be provided)Button to calculate the root using the specified first iteration and number of iterations to use.To change the number of iterations to use, simply press the up or down arrows. Alternatively you can also type in a value and hit the enter key. The maximum number of iterations is 50, and the minimum number of iterations is 1. Upon the value changing, the results set will automatically be recalculated.To change the value of the first iteration, type a value into the x1 textbox and press the enter key. Upon this value changing, the results set will be automatically recalculated.A past exam question – 144018029845Use the iteration -with x1 = 0.5 to find the value of x2 and x3, giving your answers to three decimal places.First of all we need to change the function for use in the iteration. In this case the function will be ‘sin((1/4) * x + 1)’ (for more information refer to the page ‘Forming an Expression or Function’). Hit enter when the new function has been inserted to update the results. The results should now look like this – We now need to change the iteration parameters to reflect the question. In this case the number of iterations will be three and the value of x1 is already 0.5 so we do not need to change this value. Press the calculate button to update the results. The results should now look like this – To check our answers – X1 = 0.5X2 = sin( (1 / 4) * 0.5 + 1) = 0.9022676...X3 = sin( ( 1 / 4) * 0.9022676 + 1) = 0.94099...As you can see these results are correct to three decimal places.Copy to Clipboard – To copy the current results set and graph (if available) to the clipboard, simply click on the ‘Copy to Clipboard’ button located at the top of the window – Examples of clipboard output – Integration – Differentiation –Extrema –Notes – Currently the list of extrema is unavailable in the clipboard output on the extrema tab. Instead the basic image of the function is displayed.Iterations – f(x) = sinxx1 = 0.5Iterations = 101 0.52 0.479433 0.461274 0.445095 0.430546 0.417367 0.405358 0.394349 0.384210 0.37482Mathematical Expressions as Parameters – You can insert a mathematical expression into any of the parameter fields throughout the application. For example in the integration tab, you can insert a value of ‘pi’ into the upper bound and ‘7 – 14’ into the lower bound field. When you press the enter key, these expressions are calculated and the results are used in the algorithm - 32639000 Supported Operators and Constants –Binary Operators - + (addition)(f(x) = ) x + 3, (f(x) = ) -3 + 1- (subtraction)(f(x) = ) x – 3, (f(x) = ) 2 – 4/ (division)(f(x) = ) x / 3, (f(x) = ) 3 / 2* (multiplication)(f(x) = ) 3 * x, (f(x) = ) 2 * 4^ (exponent)(f(x) = ) x ^ 2, (f(x) = ) 9 ^ 0.5Unary Operators – sin (sine function)(f(x) = ) sin(x), (f(x) = ) sin(45)cos (cosine function)(f(x) = ) = cos(x), (f(x) = ) cos(45)tan (tangent function)(f(x) = ) tan(x), (f(x) = ) = tan(45)asin (inverse sine function)(f(x) = ) asin(x), (f(x) = ) asin(0.5)acos (inverse cosine function)(f(x) = ) acos(x), (f(x) = ) acos(0.5)atan (inverse tangent function)(f(x) = ) atan(x), (f(x) = ) atan(0.5)abs (absolute value or modulus function)(f(x) = ) abs(x), (f(x) = ) abs(-3.25)sqrt (square root function)(f(x) = ) sqrt(x), (f(x) = ) sqrt(9)ln (natural logarithm function)(f(x) = ) ln(x), (f(x) = ) ln(1.25)e (e raised to the exponent of)(f(x) = ) e(x), (f(x) = ) e(2.5)Constants – pi(f(x) = ) sin(x) + pi, (f(x) = ) 3 + pieuler (e raised to the first power)(f(x) = ) sin(x) + e, (f(x) = ) 3 + eForming an Expression or Function – Due to the algorithm which is used to evaluate the input functions, it is likely that for some functions, the application will inform you that the function is invalid, when in fact there are no syntax errors. Some of the most common issues include – Multiplication with variables – Currently in the application, you must explicitly include the multiplication operator. For example the function ‘5x’ will become ‘5 * x’.Negatives – Currently the application does not support functions directly in the form ‘-x’ for example. To get around this issue, multiply the variable by -1. In this case the final function will be ‘x * -1’.Another example – a function like ‘16 - 4 * -3’ will become ’16 – (4 * -3)’.Negatives in polynomials – The application does support positive polynomials such as ‘x^2 + 8*x + 12’. However polynomials with negatives are not supported in this basic format. To get around this issue, surround each portion of the polynomial with a set of parenthesis. For example the incorrect polynomial ‘x^2 – 5*x’ must become ‘x^2 – (5 * x)’. Similarly if a new expression was added to the end, again a new set of parenthesis must be added. For example ‘x^2 – 8 *x – 12’ will become ‘(x^2 - (8*x)) – 12’.There may be more issues when evaluating custom functions. If this is the case, check through the function for any possible negatives and add a new set of parenthesis between every portion of the mon Error Messages – "Invalid first iteration" – The value of the first iteration on the Iterations tab is invalid. Make sure that the input is numerical or if a mathematical expression has been entered, refer to the page ‘Forming an Expression or Function’ for more information"Invalid Target Point" – The value of the target point on the Differentiation tab is invalid. Make sure that the input is numerical or if a mathematical expression has been entered, refer to the page ‘Forming an Expression or Function’ for more information"Iterations must be less than 99" – On the Integration tab, the number of iterations to use must be less than 99 for performance reasons."Number of iterations must be greater than zero" – On the integration tab, the number of iterations to use must always be greater than zero."Invalid Equation" – The function that has been entered is not valid. Try checking the function for any syntax errors or refer to the page ‘Forming an Expression or Function’ for more information."Unable to find extrema" – There has been an error when finding local extrema in the Extrema tab. Check that the function and the searching bounds are valid for the function. Refer to the page ‘Forming an Expression or Function’ for more information."The Function is invalid" – The function that has been entered is not valid. Try checking the function for any syntax errors or refer to the page ‘Forming an Expression or Function’ for more information.Recovering from an error –Every effort has been made to ensure that this application is as error free as possible. Most error messages are shown through message boxes that can usually be dealt with by the user. However in certain circumstances it is possible that the application may experience an unhandled error. In this circumstance an error message like this may appear – It is advisable to close the application under these circumstances. However under some instances the application will not ‘crash’. If this happens, click on any other tab to refresh the graphing view. You can then revert back to the tab that caused the error with a default graph.Under some circumstances the application may also appear to enter the ‘Not Responding’ state. Usually this is because the application is calculating a large number of iterations for an algorithm and the message will eventually disappear when the results have been calculated.Known Bugs – For some very complex functions, it is possible that the graph will sometimes disappear in certain viewing regions, and appear again when the graphing view has been moved.Some functions such as 16 - 4 * - 3 are incorrectly evaluated to 1612 instead of 28. Refer to the page on ‘Forming an expression or Function’ for more information.Sometimes the application will crash when a function is not defined at a specific point. For example when trying to find local extrema of the function ‘ln(x)’ with two positive bounds (as ln(x) is not defined for positive numbers)If you uncover any other bugs are problems with this software. Please don’t hesitate to contact me with the exact circumstances under which the problem occurs.Thank you for choosing this software. ................
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