TI-nspire CAS Replacement calculator screens for the Core material

Transcription

1 TI-nspire CAS Replacement calculator screens for the Core material Chapter 1: Univariate data Page 30 Using the calculator to find the mean, median and mode Use the calculator to find the mean and median of the following data : 2, 3, 5, 4, 3, 6, 5, 7, 3, 8, 1, 7, 5, 5, 9 Key c andopena3:lists & Spreadsheets page. Use column 1; check that the cursor is in the first position of column 1 and type 2 into the first position Key or > to move to the next position in the list. Note : I have named the list of values page30 To find the descriptive statistics for the data Key b > 4 :Statistics >1:Stat Calculation >1:One-Variable Statistics Complete the screens as shown, using the e key to move between fields. All the available descriptive statistics for this variable appear on the screen :- On this screen the first statistic,, isthemean. The mean of the data is (to 3 decimal places) The second statistic: x =73 indicates that the sum of all the data values as 73; the Greek symbol meaning sum of. The next three statistics we will consider in section 1E. n : 15 refers to the fact that there are 15 data values in the set. MedianX : 5 means the median is 5 The other statistics on this part of the screen are the statistics of the five-number summary which is covered in section 1F 1

2 Page 34 Using the calculator to find the range and inter-quartile range: Key the data into a column. The data does not have to be ordered. Key b > 4 :Statistics >1:Stat Calculation > 1 : One-Variable Statistics The screen at right shows all the statistics for the data. The range is MaxX MinX =14 3=11 The inter-quartile range (IQR) = Upper quartile lower quartile =Q 3 X Q 1 X =9 5 =4 Page 38 Using the calculator to find the standard deviation Key the data into a column. Key b > 4 :Statistics >1:Stat Calculation > 1 : One-Variable Statistics The screen at right shows all the statistics for the data. The sample standard deviation that we use is X The sample standard deviation is Note: The variance is not given on the screen. The variance can be found by squaring the standard deviation. Page 38 Standard deviation for grouped data Key the data into the columns using column A for the data values and column B for the frequency. Key b > 4 :Statistics >1:Stat Calculation>1:One-Variable Statistics Complete the pop-up screens as shown. The sample standard deviation that we use is X The sample mean is 3.38 and the sample standard deviation is Page 47 boxplot. Using the graphing calculator to find descriptive statistics and construct a 2

3 T Tap on Using the data from the example above: 2, 3, 5, 4, 3, 6, 5, 7, 3, 8, 1, 7, 5, 5, 9 Key the data into a column. Key b > 4 :Statistics >1:Stat Calculation > 1 : One-Variable Statistics The screens below shows all the statistics for the data; arrow down and the second screen shows the 5-number summary necessary to draw a boxplot. Move the cursor into the data column (column 1), key b > 3 : Data> 5 Quick graph and the data is graphed as a dotplot. Key b > 1 : Plot Type >2:BoxPlot Alternatively : Key c andopena5:data & Statistics page. Move the cursor to the bottom of the screen Click to add variable ; a dotplot appears. Key b > Plot Type > 2:Box Plot. Use the arrows to move the cursor over the graph to see the values of the 5-number summary. Page 50 and 51 Using the calculator to draw the boxplot in Example 20. Key the data into a column. Key b > 4 :Statistics >1:Stat Calculation > 1 : One-Variable Statistics The second screen shows the 5-number summary necessary to draw a boxplot. 3

4 Key c andopena5:data & Statistics page. Move the cursor to the bottom of the screen Click to add variable ; a dotplot appears. Key b > Plot Type > 2:Box Plot. Use the arrows to move the cursor over the graph to see the values of the 5-number summary. Page 55 Using the random number generator on the calculator: Key c and open a 1 : Calculator page. Key b > 5 : Probability >4:Random >2:Integer randint( will appear on the screen. Type 1,45) and a random number between 1 and 45, inclusive, is given; 43 in this case Other random numbers in the same range can be obtained by keying repeatedly. To obtain a list of six random numbers in the range 1 to 45 inclusive Key c> 1 : Calculator Key b > 5 : Probability >4:Random >2:Integer Type in 1, 45, 6) ; the last number indicating the number of random numbers required. Close the bracket and press A list of six random numbers in the range 1 to 45 will be generated. Other sets of random numbers in the same range can be found by tapping repeatedly EXE Page 56 Example 21 Key c> 1: Calculator Key b > 5 : Probability >4:Random >2:Integer Use the twice to copy the randint( instruction. Use the to edit the numbers. Core Chapter 2 : Bivariate data Page 63 Using the calculator to graph parallel 4

5 boxplots. Key c and open a 3 :Lists and Spreadsheets page In the first column type the mode (of transport). Type car in the cell a1 and fill down (b >3:Data>3:Fill down) to cell a14; there are 14 times for travel by car. Complete the car times in column B. Repeat the process for the train times and the tram times (15 of each) Label the columns mode and time. Opena5:Data and Statistics page All the data will appear on the page. Change the y variable to mode Change the x variable to time. Key b > 1 : Plot Type >2:Boxplot. The parallel boxplots are plotted on the same axes. 5

6 Page 69 Using the calculator to construct a scatterplot Key c and open the Lists & Spreadsheets application Key the data into the columns; the independent variable (height) in column 1 and the dependent variable (weight) in column 2 Key b > 4 :Statistics>1:Stat Calculation > 4 : Linear Regression (a + bx) OR 3 : Linear Regression (mx+b) The coefficients for the linear regression equation are in the third screen above. The equation is Weight = Height. Keyc andopena5:data & Statistics page. Random points will appear on the screen. Move the cursor down to the bottom of the screen to the Click to add variable area. Select Height. Move the cursor across to the left of the screen to the Click to add variable area. Select Weight. The completed scatterplot is below. The cursor and the mouse can be used to find the co-ordinates of individual points. 6

7 Enter Page 75 Using the calculator to find Pearson s correlation coefficient Key c and open the Lists & Spreadsheets application Key the data into the columns; the independent variable (x) in column 1 and the dependent variable (y) in column 2 Produce a scatterplot for the data; it will reveal any errors made in entering the data and any outliers. It will also indicate whether the data is linear. The scatterplot indicates strong, positive, linear correlation between the variables. Key / to return to the Lists... page: Key b > 4 :Statistics >1:Stat Calculation > 4 : Linear Regression (a + bx) OR 3 : Linear Regression (mx+b) Select the variables : The linear regression co-efficients for this data are shown above. Scrolling down will reveal the value of Pearson s correlation coefficient r ; for this example. This r value indicates a strong, positive correlation as indicated by the scatterplot. Page 78 Calculation of the coefficient of determination. r 2 is also found on the linear regression screen of the calculator. r 2 = for this example Alternatively, if the value of r is known, then this can simply be squared. 7

8 Core Chapter 3 : Linear regression Page 88 Using the calculator to find the three median regression line. Key c and open the Lists & Spreadsheets application Enter the data into the columns and construct a scatterplot to make sure the data is entered correctly. Keyc andopena5:data & Statistics page. Random points will appear on the screen. Move the cursor down to the bottom of the screen to the Click to add variable area. Select x. Move the cursor across to the left of the screen to the Click to add variable area. Select y. Key b > 4 : Analyze >6:Regression >3:Show Median-Median. The three-median regression line will appear on the scatterplot as well as the equation of the threemedian regression line. The equation in this case is Page 91 Using the calculator to find the least squares regression line. Key c and open the Lists & Spreadsheets application Enter the data into the columns and construct a scatterplot to make sure the data is entered correctly. Keyc andopena5:data & Statistics page. Random points will appear on the screen. Move the cursor down to the bottom of the screen to the Click to add variable area. Select x. Move the cursor across to the left of the screen to the Click to add variable area. Select y. Key b >4 :Analyze >6:Regression > 2 : Show Linear (a + bx) OR select 1 : Show Linear (mx + b) The three-median regression line will appear on the scatterplot as well as the equation of the three-median regression line. 8

9 Use the cursor and the mouse to show the co-ordinates of specific points. The equation is Page 98 Using the calculator to find and plot the residuals. Using the data from Page 91 and continuing from the previous graphs and calculation Move the cursor to the click to add variable box on the left of the screen and choose stat.resid. The residuals plot will appear. Use the cursor to find the values of individual points. Alternatively: Using the spreadsheet: Type the regression equation into cell C1 using A1 for the variable x. (see bottom of screen). Fill down. In column D type = B1 C1 into D1.(residuals = actual predicted) Fill down and call the column Resid. Key c and open the Lists & Spreadsheets application Select the variables as x and resid as shown on the screen at right. 9

10 Page 104 Example 4 a. Key c and open the Lists & Spreadsheets application Enter the data into the columns and construct a scatterplot. The data does not appear to be linear. Go back to the lists and b >4 :Statistics>1:Stat Calculations.. >4 Linear Regression (a + bx) OR 3 : Linear Regression (mx+b) Select the variables. Go back to the graph page (/ ) and select b > 4 : Analyze > 6 : Regression > 2 Show Linear (a +bx) Construct a residuals plot. The residual plot shows a pattern suggesting that the linear model is not the best fit. b. We can apply the x 2 transformation by adding an extra column containing x 2 value. Go back to the Lists.. page ; in column F type A1 2 (see bottom of screen) Key b > 3 : Data >.3 : Fill Down and then move the cursor down to fill the column. Call the column xsquar Go back to the graph page (/ ) and change the variable on the horizontal axis to x squar. The scatterplot of y versus x 2 appears to be linear. Apply a linear regression line and a residual analysis to confirm that the relationship is linear:- 10

11 The equation connecting the variables is (or ) Page 109 Example 5 a. Enter the data and construct a scatterplot. The data is clearly not linear and has the general shape. audience size day We therefore consider the four transformations where x represents the day and y represents the audience size. For each of the transformations we add the appropriate column to the data, construct a scatterplot including a least squares regression line, and a residual plot. log x : log y : : 11

12 See the text for the remainder of Example 5. Core Chapter 4:Time series Page 136 Example 6 To graph the deseasonalised data open the List and Spreadsheets application and enter the time period 1 to 48 in column 1 and the deseasonalised data in column 2. The time period data, 1 to 48, can be entered into column 1 using a sequence: Keyb >3:Data>1:Generate Sequence. Complete the screen as shown. Key c and open a 5 :Data and Stat... page. Enter the variables and the graph will appear as a scatterplot. Key b > 1 : Plot Type>6:XYLine Plot. 12

13 Select b > 4 : Analyze >6:Regression > 2 : Show Linear (a +bx) to paste a least-squares regression line and equation on the screen. 13

14 TI-nspire CAS Replacement calculator screens for Module 1 : Number Patterns Chapter 5 : Arithmetic and Geometric Sequences Page 157 Finding terms of a sequence using the calculator. Method 1 Key c and open a 1 : Calculator page. Key b >1 : Define. Type the function as shown. Type t(25) to find the value of the 25 th term. Method 2 Key c and open a 3 : Lists and spreadsheets page. In any cell in column A, key b > 1 : Generate sequence Complete the screen as shown Scroll down column A to find the 25 th term. Page Displaying a sequence using a calculator Key c and open a 1 : Calculator page. The seq( can be typed as shown or it can be found in the catalogue. The instructions are seq(rule, variable, starting term, ending term) Page 159 Example 2 Key c and open a 1 : Calculator page. a. Type and the answer will be given as a fraction. b. Type in the two subtractions and it is seen that the answers, given as fractions, are not the same value so the sequence is not arithmetic. 1

15 Page 160 Example 3 Key c and open a 3 : Lists and spreadsheets... page. In any cell in column A key b > 1 : Generate sequence.complete the screen as shown Scroll down to term 25. Page 162 Example 5 Key c and open a 3 : Lists and spreadsheets... page. In any cell in column A key b > 1 : Generate sequence Complete the screen as shown Scroll down the table until you come to the value 927 Term 186 has the value 927. Page 162 Example 6 Key c and open a 3 : Lists and spreadsheets... page. In any cell in column A key b > 1 : Generate sequence Complete the screen as shown Scroll down the table until you come to the first value that is greater than Term 24 is the first term that is greater than Page 164 Graphing Arithmetic Sequences. Key c and open a 3 : Lists and spreadsheets... page. In any cell in column A > b > 1 : Generate sequence Complete the screen as shown This will produce the values of n in the first column. 2

16 In any cell in column B, key b > 1 : Generate sequence Complete the screen as shown The sequence is in column B Label both of the columns. Key c and open a 5 : Data and Stat...page. Move the cursor to the bottom of the screen and add the variable n Move the cursor to the left of the screen and add the variable seq1 Key to obtain the graph. Page 167 Example 9 Method 1 The sum can be found by typing the expression on the screen at right on a 1 : Calculator page. Method 2 Key c and open a 3 : Lists and spreadsheets... page. In any cell in column A, key b > 1 : Generate sequence Complete the screen as shown 3

17 In column B type the first term into B1 and in the second cell, B2, type the formula = b1 + a2 The value 20 will be calculated. In cell B2 key b > 3 : Data > 3 : Fill Down (the cell will have the edge highlighted) Use the down arrow to fill down the column. This will produce the sum of n terms, next to the n th term in column B. Scroll down to find the sum of 25 terms. Page Example 11 Using the method in example 9 above typing the formula = b1 + a2 into cell B2. Scroll down to find the sum of 25 terms. Page 168 Example 12 Using the method in example 9 above typing the formula = b1 + a2 into cell B2. Scroll down to find the number of terms whose sum is first more than 50000; 26 terms. 4

18 Page 174 Example 15 Key c and open a 3 : Lists and spreadsheets... page. In any cell in column A, key b > 1 : Generate sequence Complete the screen as shown Scroll down to find the 10 th term. Page 175 Example 16 b. Key c and open a 3 : Lists and spreadsheets... page. Generate a sequence for n and the geometric sequence in columns A and B respectively. Open a 5: Data and Stat... page. Move the cursor to the bottom of the screen and add the variable n Move the cursor to the left of the screen and add the variable seq1 Key b > 5 : Window/Zoom> Window Settings. Change the window settings as shown in the screen below. The graph will be displayed. c. Scroll down the table until you find the first value that is greater than Term 13 (n = 13) is the first term that is greater than

19 Page 176 Example 17 Key c and open a 3 : Lists and spreadsheets... page. Generate a sequence for n and the geometric sequence in columns A and B respectively. Open a 5: Data and Stat... page. Key b > 5 : Window/Zoom> Window Settings. Change the window settings as shown in the screen below. The graph will be displayed. Page 177 Example 19 Open a 1 : Calculator page. Key /l and type the numbers 4 and Page 179 Example 21 Key c and open a 3 : Lists and spreadsheets... page. Generate the geometric sequence in column B Create a sum column in column C by typing the first term (1) in C1 and the formula = c1 + b2 in cell C2. Key b > 3 : Data > 3 Fill Down to create the sum column. Alternatively, the sum of the 7 terms can be calculated by substituting in the formula on a 1 : Calculator page. The answer will be given as a fraction. 6

20 Page 180 Example 22 The first term is 5 and the common ratio is 2 so the formula for the nth term is 5 2 n-1 Generate the geometric sequence in column A Create a sum column in column B by typing the first term (5) in B1 and the formula = b1 + a2 in cell B2. Key b > 3 : Data > 3 Fill Down to create the sum column. The minimum number of terms that give a sum of at least is 11. Page 185 Example 24 c. Generate the sequence in column A and scroll down to the 7 th term. d. Find the first term whose value is greater than 20000: Scrolling down shows that it is term 12. Page 185 Example 25 Generate the sequence in column A, as shown. Scroll down until you come to a value that is greater than 6,000. The population will reach 6000 twelve years after 1990; in the year

21 Page 186 Example 26 d. They will first be making an investment of $2000 or more on Maria s 25 th birthday. e. Generate a sum column by typing the formula = b1 + a2 into cell B2. Fill down. Scrolling down the sum table to n = 21 shows the total sum of $ has been made up to and including Maria s 21 st birthday. Page 189 Example 28 e. Generate the sequence in column A and create a sum column in column B. Scrolling down the sum table to a sum of $12.83 show that this occurs when n = 20. A phonecall of 20 minutes costs $ Page 193 Example 30 d. Create a sequence for n and the simple and compound interest sequences. Label the columns as shown. Open a 5 : Data & Stat... page and enter the variable n at the bottom of the screen.( x axis) Move the cursor to the left and enter the variable simp on the y axis. 8

22 Key b > 2 : Plot Properties > 6 : Add Y Variable Select comp and both sequences will be graphed. e. Go back to the Lists and spreadsheets page. Scroll down the table until you come to the values where comp > simp. This occurs at the beginning of the 8 th year. Chapter 6 : Difference Equations Page 203 Example 4 Key c and open a 3 : Lists and spreadsheets... page. Generate the sequence in column B. The equation is t n+1 = 2t n 1 ; t 1 = 5 but it is entered on the screen as u(n) = 2u(n-1) 1; u(n-1) is the term before u(n) in the same way that t n is the term before t n+1 The sequence is 5, 9, 17, 33, 65, 129,... To graph the sequence: Insert a column for n and name both columns. Key c to open a 5 : Data &Stat... page Select the variables as shown. The window can be changed. b > 5 : Window/Zoom > 1 : WindowSetting The value of term one is 5 and the value of term seven is 257 so y- values are set to cover these. 9

23 Page 213 Example 10 c. Enter the sequence as shown. Include a column for n values. Key c to open a 5 : Data &Stat... page Select the variables as shown. The window can be changed. b > 5 : Window/Zoom > 1 : WindowSetting The value of term one is 500 and the value of term nine is 84 so y- values are set to cover these. d. From the table it can be seen that after week 9 the size of the colony will not be sufficient to provide 120 mice for research purposes. e. b > 3 : Data > 1 : Generate Sequence. Edit the sequence as shown. From the table it can be seen that the population is now increasing as the weeks increase. / to access the Data & Stat.. page The window will need to be changed to graph this sequence : b > 5 : Window/Zoom > 1 : WindowSetting 10

24 f. b > 3 : Data > 1 : Generate Sequence. Edit the sequence as shown. From the table it can be seen that the population is remaining constant at 500 as the weeks increase. / to access the Data & Stat.. page The graph shows that the population is stable at 500. Page 218 Using the calculator for second-order difference equations. To enter a second-order difference equation of the type we will use u(n) = 2u(n-1)+u(n-2); where u(n-1) is the term before u(n) and u(n-2) is the term before u(n-1). Complete the screen as shown To graph the first seven terms first choose an appropriate window observing the required values from the table. ( b > 5 : Window/Zoom > 1 : WindowSetting.) / to access the Data & Stat.. page 11

25 TI-nspire CAS Replacement calculator screens for Module 2 : Geometry and Trigonometry. Chapter 7: Geometry. No replacement screens. Chapter 8: Trigonometry Page 267 Example 1 Key c and open a 1 : Calculator page m46)m Note: The default option for measuring angles is Radians but for the calculations in Further Maths you will be working in Degrees. Key c and select 8 ; System Info > System Settings e down to Angle and change this to Degree. e down to Auto or Approx. and change this to Approximate so that the calculator gives you answers as decimals. The calculator will now assume all angles are in degrees and will give answers in degrees. It will also give all answers as decimals sin (46 o ) = correct to 4 decimal places. Page 267 Example 2 In a 1 : Calculator page key /n ) cos ( o ) = To change the answer into degrees, minutes and seconds (DMS): Key k then d, scroll down to DMS Ans DMS will appear on the screen. Key The answer will be given in degrees, minutes and seconds ie. 30 degrees, 7 minutes and 43 seconds. 1

26 Page 270 Example 5 In a 1 : Calculator page key /n 26p38) Key k then d, scroll down to DMS Ans DMS will appear on the screen. Key The answer will be given as 46 degrees, 49 minutes and 35 seconds. Page 272 Example 6 Follow the same instructions as for Example 5 above. Page 272 Example 7 Follow the same instructions as for Example 5 above to find the angle then key 12 cos (ans). For ans key /v Alternatively You can use the calculator to solve the equation This means that you do not have to transform your equations to solve them. b > 3 : Algebra > 1 : Solve Key n/v ) = 12 px,x) Page 275 Example 8 There are two methods for doing this and other examples on the calculator. It is worthwhile trying both methods so that you can see which one suits you. Method 1 You will need to establish the equation but you will not need to transpose it to find the solution. a. b > 3 : Algebra > 5 : Numerical Solve Key mx )p15 =m 30) p 9,X ) Angle A is found by keying 180 ans (/v) - 30 b. In a similar process to part a. we can find the value of a :- 2

27 Method 2 The whole calculation can be done on the Calculator page however, the equations will have to be transposed as in the text. Use ans (/v) to carry over the answer value from the previous calculation. Page 277 Example 10 In a 1 : Calculator page :- Access nsolve by b > 3 : Algebra > 5 : Numerical Solve Use ans (/v ) to carry over the previous answer. Page 280 Example 11 In a 1 : Calculator page :- For the second calculation use /q/v using ans (/v ) to carry over the previous answer. Page 281 Example 12 In a 1 : Calculator page :- For the second calculation use /q/v using ans (/v ) to carry over the previous answer. 3

28 Page 281 Example 13 In a 1 : Calculator page :- For the second calculation use /q/v using ans (/v ) to carry over the previous answer. To use a previous answer that is not the immediate previous answer use the arrows to highlight the required answer and press Alternatively : Access nsolve by b > 3 : Algebra > 5 : Numerical Solve Page 288 Example 16 In a 1 : Calculator page :- Access nsolve by b > 3 : Algebra > 5 : Numerical Solve 4

29 Chapter 9 Applications of Geometry and Trigonometry Page 294 Example 2 In a 1 : Calculator page :- Access nsolve by b > 3 : Algebra > 5 : Numerical Solve Page 299 Example 5 a. and b. In a 1 : Calculator page :- Access nsolve by b > 3 : Algebra > 5 : Numerical Solve c. Use ans (/v ) to carry over the previous answer. d. The area calculation can be done as one calculation. Access the previous answers by using the arrow until the value is highlighted. Key and the value will be pasted in the present calculation. 5

30 TI-nspire CAS Replacement screens for Module 3 : Graphs and relations Chapter 10: Construction and interpretation of graphs. Page 339 Example 8 c > Open a 2 : Graphs and geometry page. At the bottom of the page the cursor will be flashing in the function box. Enter 5x 3 next to f1(x) and key.. To find the intercepts with the axes: Key b > 5 :Trace > 1 :Graph Trace Use the keys to move along the graph. When the cursor approaches the x-axis it will jump to the x-intercept (zero) Key and the coordinates(0.6, 0) will appear. Key 0 and the y-intercept (0, -3) will be shown. Page 347 Example 14 c > Open a 2 : Graphs and geometry page. At the bottom of the page the cursor will be flashing in the function box. Enter -2x+7 next to f1(x) and key.and 4x+5 into f2(x) Note : You may need to change the Window and the placement of the graph equations. Key b > 6 : Points and Lines > 3 : Intersection Point(s) Move the cursor to one of the graphs and and key when it shows the label name. Do the same for the other graph. The co-ordinates of the point of intersection (2, 3) appears on the graph screen. Key to paste the co-ordinates on the screen. (Press d to exit the Point of Intersection tool) 1

31 Page 349 Using the calculator to solve simultaneous equations. Method 1. This method requires you to transpose the equations to make y the subject and then entering them as shown on the screen. Key b > 3 : Algebra > 1 : Solve. Key /r and choose Type the equations and variables in as shown. Method 2. Key b > 7 : Matrices & Vectors > 5 : Simultaneous Key /r and choose the symbol; fill in the matrix with the co-efficients of the equations,. Key /r and choose the symbol; fill in the matrix with the constants of the equations. Key ) Page 371 Using the calculator for Example 28 Key c and open a 2 : Lists & Speadsheets page. Enter the data into columns A and B and label the columns x and y. Key c and open a 5 : Data and Statistics page Move the cursor to the bottom of the page and select the variable x. Move the cursor to the left of the page and select the variable y. To test the x 2 tranformation.: / to the Lists...page. Go to column 3, cell C3, and type in the formula = a1 2 b > 3 : Data > 3 : Fill Down and use the cursor to fill down the column. Name the column xsq / to the Data & Stats page and change the variable at the bottom of the screen to xsq. The y versus x 2 graph is at right. The points on the graph appear to be in a straight line 2

32 Repeating the process for the x 3 transformation: The points on the graph do not appear to be linear. If it is known that the relationship between x and y is of the form then the calculator can be used to find the values of k and n. change the variable at the bottom of the screen back to x. Key b > 4 : Analyze > 6 : Regression > 7 : Show Power The graph shows the original points with the regression line fitted. The equation is pasted on the screen:- Chapter 11: Linear programming There are no replacement screens for this chapter however it is possible to graph some inequalities on the calculator and solving simultaneous equations is simplified by using the calculator. It is not possible to graph inequalities of the form x < 5 on the TI-nspire. You will need to transpose any equations that do not have y as the subject. Key c and open a 2 : Graphs and Geometry page Constraint The cursor will be flashing in the functions box at the bottom of the screen.. Backspace. to clear the f1(x) = and type y>=0 (Y>=0 ) The region is graphed at right. Constraint You will need to transpose the inequality so that y is the subject; Delete the f1(x) = in the functions box and replace with y < = 8 x 3

33 Both inequalities are graphed on the screen. Constraint You will need to transpose the inequality so that y is the subject; Delete the f1(x) = in the functions box and replace with y > = ½ (12 x) All inequalities are graphed on the screen. The feasible region is the one that has the darker shade. Improvements can be made by changing the window (Screen 1)and by using the process of shadingout so that the feasible region is the unshaded region (Screen 2). Screen 1 Screen 2 4

34 TI-nspire CAS Replacement calculator screens for Module 4: Business Related Mathematics Chapter 12 : Financial transactions and asset values. Page 437 Example 13 Key c and open a 3 : Lists & spreadsheets page. Enter the numbers 1 to 7 in the first column and name the column year In cell B1 of the second column type. Key b > 3 : Data > 3 : Fill Down. Use the arrow to fill the column. Label the column inflate Key c and open a 5 : Data & Stat... page. Move to cursor to the bottom of the screen and change the variable to year Move the cursor to the left of the screen and change the variable to inflate. Page 443 Example 18 Key c and open a 1 : Calculator page. Key b > 3 : Algebra >1 : Solve Type in the equation, x ) The value x = is given. Rounding this value up we can say that after 6 years the value will be less than $4000. Chapter 13: Loans and Investment Pages Calculations for Simple interest can be done on a 1 : Calculator page. The solve application can be used if you want to avoid transposing the formula. Page 454 Example 4 a. Key b > 8 : Finance > 6 : Days between dates Key in the two dates as shown on the screen b. Calculate the interest. c. Calculate the total. 1

35 Page 463 Example 11 a. In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver N is the number of instalment periods; 5 years in this example. I% is the Annual interest rate; 5.5% in this example. PV is the present value or initial investment; $ in this example. (entered as as this is an outgoing amount) PMT is the amount repaid each period; $0 in this example. FV is the future value; the amount that needs to be calculated in this example. In this example there is 1 instalment period per year, PpY, and the interest is calculated once per year (CpY) e between fields and to get back to FV : Key and the future value will be calculated. b. Copy (/ C ) and paste (/V ) the future value back to a calculator screen and subtract the initial investment amount. Page 464 Example 12 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver a. b. Enter the N value as 5 4 c. Enter the N value as and PpY as 4 and PpY as 365 Page 465 Example 13 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver a. Interest is b. $ is in the account after 3 years (36 months) 2

36 c. Copy ( / C) the FV value from part b. and paste this (/ V )value onto a calculator page Complete the calculation as shown. Page Finding times and periods for compound interest. In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver The investment of $5000 (outgoing amount so PV = -5000) grows to $7500 (incoming amount so FV = 7500) in years. Page 469 Example 14 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver a. the minimum number of time periods is 41 b. 41 time periods (months) is 3 years, 5 months. Page 470 Example 15 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver It will take 321 days for the investment to grow from $1700 to $1800. Page Using the calculator to plot interest graphs. Open a 3 : List & Spreadsheets page and enter the numbers 0 to 10 in column A. Call this column year Go to cell B1 and type =. Key b Key b > 3 : Data > Fill Down. Use the cursor to fill down the column with the values. Call this column comp 3

37 Go to cell C1 and type = ) Key b Key b > 3 : Data > Fill Down. Use the cursor to fill down the column with the values. Call this column simple Key b open a 5 Data and Stats... page Add the year variable to the horizontal axis and the comp variable to the vertical axis. To graph both the compound interest and the simple interest values:- Key b > 2 : Plot Properties > 6 : Add Y Variable Add simple as a Y variable. Page 477 Example 17 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver a. N is the number of instalment periods; 5 12 months in this example. I% is the Annual interest rate; 6% in this example. PV is the present value or initial investment; -$1000 in this example (outgoing amount). PMT is the amount paid each period; -$250 in this example (outgoing amount). FV is the future value; the amount that needs to be calculated in this example. PpY and CpY are both 12 as there are 12 payments per year and 12 compounding period per year. Tap on the FV box to get the amount accumulated.($ ) Page 478 Example 18 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Complete the screen as shown. Key when the cursor is in the N field to find the number of months to save $ It will take 25 months to save $

38 Page 485 Examples 22 and 23 Calculations of the amount owing using the annuities formula can be done on the Main application. Page 488 Example 25 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver $10000 is borrowed (incoming amount, so positive) for 5 years and interest is charged at 7% p.a. compounding yearly. A repayment of $2000 (outgoing amount, so negative) is made each year. Key in the FV box to find the amount still owing after 5 years: $ This is shown as a negative value indicating that it is still owing (outgoing). Page 489 Example 26 a. In a 1 : Calculator page key b > 8 : Finance > 1 Finance Enter the instalment period (N) as There are 12 repayments and compounding periods per year. The amount still owing is $ b. Enter the instalment period (N) as There are 4 repayment and compounding periods per year. The amount still owing is $ Page 489 Example 27 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver The repayment amount (PMT) is required in this example. If the loan is paid out this means that the future value is zero. The monthly repayment is $ to pay out the loan in 5 years. Page 490 Example 28 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver The loan is for 20 years so there are = 240 instalment periods (N) The repayments (outgoing amount, so negative) are $1250 per month. 5

39 The loan is to be paid out in the 20 years so FV is zero. Tap on PV to find the amount that can be borrowed; $158, Page 491 Example 29 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Calculate the yearly repayment amount for the loan then dc and open a 3 : Lists & Spreadsheets page. In the first column generate a sequence to give you the years 1 to 25 : key b > 3 : Data > 1 :Generate Sequence You will need to enter the annuities formula in B1 box. Key b > 3 : Data > Fill Down. Use the cursor to fill down the column with the values. Call this column balance Key b open a 5 Data and Stats... page Add the n variable to the horizontal axis and the balance variable to the vertical axis. Page 491 Example 30 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Enter the values for I%, PV, PMT, FV, PpY and CpY as shown. Solve for N to find the number of instalment periods; will need to be rounded-up to 29. 6

40 Page 492 Solving for the principal, P In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Enter N as months and the other values as shown. solve for PV for the amount that can be borrowed; $ Page 492 Example 31 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Enter N as fortnights and the other values as shown. Solve for I% to give an interest rate of 6.10% Page 496 Example 32 In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Solving for N, for a loan of $ at 7.5% interest and repayments of $120, gives an error message. This occurs because $120 is less than the interest on the loan for a month so the loan will be increasing in value and will never reach zero. Page 496 Example 32 a. Solve for N entering the values for I%, PV, PMT, FV, PpY and CpY as shown. N = 158, rounding-up. Key b > 8 : Finance > 3 : Amortization3 : Interest paid. Complete the PM1 and PM2 values as 1 and 158 then the values for N,I%, PV, PMT, FV, PpY and CpY will need to be typed in. $ is paid in interest for this loan. 7

41 b. In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Solve for N entering the values for I%, PV, PMT, FV, PpY and CpY as shown. N = 61, rounding-up. Key b > 8 : Finance > 3 : Amortization3 : Interest paid. Complete the PM1 and PM2 values as 1 and 61 respectively. The values for I%, PV, PMT, FV, PpY be typed-in. and CpY will need to $ is paid in interest for this loan. c. In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Solve for N entering the values for I%, PV, PMT, FV, PpY and CpY as shown. N = 38, rounding-up. Key b > 8 : Finance > 3 : Amortization3 : Interest paid. Complete the PM1 and PM2 values as 1 and 38 respectively. The values for I%, PV, PMT, FV, PpY and CpY will need to be typed-in. $ is paid in interest for this loan. Page 498 Example 33 a. In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Solve for N entering the values for I%, PV, PMT, FV, PpY and CpY as shown. N = 14, rounding-up. Key b > 8 : Finance > 3 : Amortization3 : Interest paid. Complete the PM1 and PM2 values as 1 and 14 respectively. The values for I%, PV, PMT, FV, PpY and CpY will need to be typed-in. $ is paid in interest for this loan. 8

42 b. In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Solve for N entering the values for I%, PV, PMT, FV, PpY and CpY as shown. N = 53, rounding-up. Key b > 8 : Finance > 3 : Amortization > 3 : Interest paid. Complete the PM1 and PM2 values as 1 and 53 respectively. The values for I%, PV, PMT, FV, PpY and CpY will need to be typed-in. $ is paid in interest for this loan. c. In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Solve for N entering the values for I%, PV, PMT, FV, PpY and CpY as shown. N = 157, rounding-up. Key b > 8 : Finance > 3 : Amortization3 : Interest paid. Complete the PM1 and PM2 values as 1 and 157 respectively. The values for I%, PV, PMT, FV, PpY and CpY will need to be typed-in. $ is paid in interest for this loan. Page 499 Example 34 a. Calculate the repayment amount (PMT) per month; $ Key b > 8 : Finance > 3 : Amortization >3 : Interest paid. Complete the PM1 and PM2 values as 1 and 240 respectively The values for I%, PV, PMT, FV, PpY and CpY will need to be typed-in. The total cost of the loan is interest + fees = $ $8 = $

43 b. In a 1 : Calculator page key b > 8 : Finance > 1 Finance Solver Calculate the repayment amount (PMT) per month; $ entering the values for I%, PV, PMT, FV, PpY and CpY as shown. Key b > 8 : Finance > 3 : Amortization3 : Interest paid. Complete the PM1 and PM2 values as 1 and 240 respectively. The values for I%, PV, PMT, FV, PpY and CpY will need to be typed-in. The total cost of the loan is (interest only) $ Option A is a cheaper loan. Page 500 Example 35 a. Calculate the repayment amount; $ per month for option A For the simple interest loan: Interest = = The total amount that is to repaid is $ so the monthly repayments are $114000/96 = $ b. The total amount of interest paid for option A is $ and for option B it is $ The difference is $ ; more interest paid for option B. c. An equivalent reducing balance loan rate to option B can be found by using the repayment amount from option B and tapping on I%. An equivalent rate is 11.23% 10

44 TI-nspire CAS Replacement screens for Module 5: Networks and Decision Mathematics Chapter 14: Undirected graphs and networks No replacement screens for this chapter. Chapter 15: Directed graphs and networks Page 564 Finding connectivity matrices on a calculator. Key c and open a 1 : Calculator page Key/ r and select the symbol on screen. The screen to select the dimensions of the matrix is shown; in this case we want a 5 5 matrix so change the figures to 5 and 5 using the e key to move between fields. Key the elements into the 5 X 5 matrix keying e after each element. Store the matrix as a by keying /ha It is now possible to find powers of the matrix a. Key a ^ 2 (Aq) ) Key a + a^2 (A+Aq ) 1

45 TI-nspire CAS Replacement screens for Module 6: Matrices Chapter 16 Matrix representation and its application. Page 625 Using a calculator for matrix operations. Key c and open a 1 : Calculator page Key/ r and select the symbol on screen. The screen to select the dimensions of the matrix is shown; in this case we want a 3 3 matrix so change the figures to 3 and 3 using the e key to move between fields. Key the elements into the 3 X 3 matrix keying e after each element. Store the matrix as a by keying /ha Establish another 3 3 matrix as b To calculate a b : key ArB To calculate 2a + b: key 2rA+B 1

46 Page 630 Powers of matrices To find a 2 : key Aq To find a 3 : key Al3 Page 637 Using the calculator to find inverses. Set-up matrix a Key a -1 ( Alv1 ) to find the inverse of a. Check that a a -1 = I Page 638 Example 10 Set-up matrix b Key c and open a 1 : Calculator page Key/ r and select the symbol on screen. The screen to select the dimensions of the matrix is shown; in this case we want a 4 4 matrix so change the figures to 4 and 4 using the e key to move between fields. 2

47 Key the elements into the 4 X 4 matrix keying e after each element. Store the matrix as b by keying /hb Key b -1 ( Blv1 ) to find the inverse of b. with the elements as decimals. If the document settings are changed to EXACT then the elements of the inverse will be given as fractions. Page 640 Example 12 Set-up matrix a: Key the elements into the 3 X 3 matrix keying e after each element. Store the matrix as a by keying /ha Establish another 3 3 matrix as b To solve the simultaneous equations we need to calculate a -1 b (Key Alv1rB) 3

48 Chapter 17 Transition matrices Page 648 Example 1 Establish the transition matrix a and the initial state matrix b. c. The state matrix for Tuesday is a b The state matrix for Friday is a 4 b (Al4rB ) d. To see the progression to steady state, a 5, a 10 and a 15 have been calculated. The steady state proportion of students buying their lunch at the canteen is or 37.5%. Summary Page 657 Using the calculator for matrix operations. Establish matrices a and b. Key c and open a 1 : Calculator page Key/ r and select the symbol (for a 2 2 matrix)on screen. Store the matrix as a by keying /ha Addition: Key a + b 4

49 Multiplication by a scalar: Key 3a Key 4a 2b Multiplication Key a b (ArB ) Powers of a square matrix Tap a 2 (Aq ) Tap b^5 ( Bl5 ) Determinant of a matrix Key b > 7 : Matrices and Vectors > 2 : Determinant Key a) The determinant of matrix a is 10 Similarly the determinant of matrix b is 5. The inverse of a matrix Establish the matrix b. Check that the determinant is not zero. To find the inverse: Key b -1 (Blv1 )and the inverse is given with the elements as decimals. If the document settings are changed to EXACT then the elements of the inverse will be given as fractions. 5