Mathematics ib Diploma standard level book

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HAESE & HARRIS PUBLICATIONS Specialists in mathematics publishing Mathematics for the international student Mathematical Studies SL second edition Mal Coad Glen Whiffen Marjut Mäenpää Mark Humphries James Foley Michael Haese Sandra Haese for use with IB Diploma Programme IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_00\001IB_STSL-2_00.CDR Wednesday, 24 February 2010 4:09:29 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100Graphics calculator instructions Contents: A Casio fx-9860G B Texas Instruments TI-84 Plus C Texas Instruments TI-nspire IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_00b\015TSL-2_00b.CDR Friday, 26 March 2010 11:59:06 AM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 10016 GRAPHICS CALCULATOR INSTRUCTIONS In this course it is assumed that you have a graphics calculator. If you learn how to operate your calculator successfully, you should experience little difficulty with future arithmetic calculations. There are many different brands (and types) of calculators. Different calculators do not have exactly the same keys. It is therefore important that you have an instruction booklet for your calculator, and use it whenever you need to. However, to help get you started, we have included here some basic instructions for the Casio fx-9860G, the Texas Instruments TI-84 plus and the Texas Instruments TI-nspire calculators. Note that instructions given may need to be modified slightly for other models. The instructions have been divided into three sections, one for each of the calculator models. CASIO FX-9860G A BASIC FUNCTIONS GROUPING SYMBOLS (BRACKETS) ( ) The Casio has bracket keys that look like and . Brackets are regularly used in mathematics to indicate an expression which needs to be evaluated before other operations are carried out. For example, to evaluate 2£(4+1) we type 2 £ ( 4 + 1 ) EXE . We also use brackets to make sure the calculator understands the expression we are typing in. 2 ¥ ( + ) For example, to evaluate we type 2 4 1 EXE . 4+1 2 ¥ + If we typed 2 4 1 EXE the calculator would think we meant +1. 4 In general, it is a good idea to place brackets around any complicated expressions which need to be evaluated separately. POWER KEYS The Casio has a power key that looks like . We type the base first, press the power key, then enter the index or exponent. 3 For example, to evaluate 25 we type 25 3 EXE . 2 Numbers can be squared on the Casio using the special key x . 2 2 For example, to evaluate 25 we type 25 x EXE . IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\016TSL-2_00b.CDR Tuesday, 5 January 2010 3:27:27 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100GRAPHICS CALCULATOR INSTRUCTIONS 17 ROOTS To enter roots on the Casio we need to use the secondary function key SHIFT . 2 We enter square roots by pressing SHIFT x . p 2 For example, to evaluate 36 we press SHIFT x 36 EXE . If there is a more complicated expression under the square root sign you should enter it in brackets. p 2 ( ¥ ) For example, to evaluate 18¥2 we press SHIFT x 18 2 EXE . ( Cube roots are entered by pressing SHIFT . p 3 ( For example, to evaluate 8 we press SHIFT 8 EXE . Higher roots are entered by pressing SHIFT . p 4 For example, to evaluate 81 we press 4 SHIFT 81 EXE . INVERSE TRIGONOMETRIC FUNCTIONS ¡1 ¡1 ¡1 The inverse trigonometric functions sin , cos and tan are the secondary functions of sin , cos and tan respectively. They are accessed by using the secondary function key SHIFT . ¡ ¢ 3 ¡1 3 For example, if cosx = , then x = cos . 5 5 ( ¥ ) To calculate this, press SHIFT cos 3 5 EXE . SCIENTIFIC NOTATION If a number is too large or too small to be displayed neatly on the screen, it will be expressed k in scientific notation, which is the form a£10 where 16a 10 and k is an integer. 3 To evaluate 2300 , press 2300 3 EXE . The answer 10 displayed is 1:2167e+10, which means 1:2167£10 . 3 ¥ To evaluate , press 3 20000 EXE . The answer 20000 ¡4 displayed is 1:5e¡04, which means 1:5£10 . You can enter values in scientific notation using the EXP key. 14 2:6£10 ¥ For example, to evaluate , press 2:6 EXP 14 13 13 13 EXE . The answer is 2£10 . IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\017TSL-2_00b.CDR Tuesday, 5 January 2010 3:27:36 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 10018 GRAPHICS CALCULATOR INSTRUCTIONS SECONDARY FUNCTION AND ALPHA KEYS The shift function of each key is displayed in yellow above the key. It is accessed by pressing the SHIFT key followed by the key corresponding to the desired shift function. p p 2 For example, to calculate 36, press SHIFT x ( ) 36 EXE . The alpha function of each key is displayed in red above the key. It is accessed by pressing the ALPHA key followed by the key corresponding to the desired letter. The main purpose of the alpha keys is to store values which can be recalled later. MEMORY Utilising the memory features of your calculator allows you to recall calculations you have performed previously. This not only saves time, but also enables you to maintain accuracy in your calculations. SPECIFIC STORAGE TO MEMORY Values can be stored into the variable letters A, B, ...., Z. Storing a value in memory is useful if you need that value multiple times. Suppose we wish to store the number15:4829 for use in a number of calculations. To store this number in variable A, type in the I number then press ALPHA X,µ,T (A) EXE . + We can now add 10 to this value by pressing ALPHA X,µ,T 10 EXE , or cube this value by pressing ALPHA X,µ,T 3 EXE . ANS VARIABLE The variable Ans holds the most recent evaluated expression, (¡) and can be used in calculations by pressing SHIFT . For example, suppose you evaluate 3£4, and then wish to subtract ¡ (¡) this from 17. This can be done by pressing 17 SHIFT EXE . + ¡ If you start an expression with an operator such as , , etc, the previous answer Ans is automatically inserted ahead of the operator. For example, the previous answer can be halved ¥ simply by pressing 2 EXE . JI If you wish to view the answer in fractional form, press F D . IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\018TSL-2_00b.CDR Tuesday, 5 January 2010 3:27:41 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100GRAPHICS CALCULATOR INSTRUCTIONS 19 RECALLING PREVIOUS EXPRESSIONS Pressing the left cursor key allows you to edit the most recently evaluated expression, and is useful if you wish to repeat a calculation with a minor change, or if you have made an error in typing. p p Suppose you have evaluated 100+ 132. If you now want to evaluate 100+ 142, instead of retyping the command, it can be recalled by pressing the left cursor key. Move the cursor between the 3 and the 2, then press DEL 4 to remove the 3 and change it to a 4. Press EXE to re-evaluate the expression. LISTS Lists enable us to store sets of data, which we can then analyse and compare. CREATING A LIST Selecting STAT from the Main Menu takes you to the list editor screen. To enter the dataf2, 5, 1, 6, 0, 8g into List 1, start by moving the cursor to the first entry of List 1. Press 2 EXE 5 EXE .... and so on until all the data is entered. DELETING LIST DATA To delete a list of data from the list editor screen, move the cursor to anywhere on the list you wish to delete, then press F6 (B) F4 (DEL-A) F1 (Yes). REFERENCING LISTS Lists can be referenced using the List function, which is accessed by pressing SHIFT 1. For example, if you want to add 2 to each element of List 1 and display the results in List 2, + move the cursor to the heading of List 2 and press SHIFT 1 (List) 1 2 EXE . Casio models without the List function can do this by pressing OPTN F1 (LIST) F1 + (List) 1 2 EXE . STATISTICS Your graphics calculator is a useful tool for analysing data and creating statistical graphs. We will first produce descriptive statistics and graphs for the data set: 52336 45375 71895. Enter the data into List 1. To obtain the descriptive statistics, press F6 (B) until the GRPH icon is in the bottom left corner of the screen, then press F2 (CALC) F1 (1VAR). IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\019TSL-2_00b.CDR Tuesday, 5 January 2010 3:27:47 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 10020 GRAPHICS CALCULATOR INSTRUCTIONS To obtain a boxplot of the data, press EXIT EXIT F1 (GRPH) F6 (SET), and set up StatGraph 1 as shown. Press EXIT F1 (GPH1) to draw the boxplot. To obtain a vertical bar chart of the data, press EXIT F6 (SET) F2 (GPH 2), and set up StatGraph 2 as shown. Press EXIT F2 (GPH 2) to draw the bar chart (set Start to 0, and Width to 1). We will now enter a second set of data, and compare it to the first. Enter the data set 9623557567 6344584 into List 2, then press F6 (SET) F2 (GPH2) and set up StatGraph 2 to draw a boxplot of this data set as shown. Press EXIT F4 (SEL), and turn on both StatGraph 1 and StatGraph 2. Press F6 (DRAW) to draw the side-by-side boxplots. STATISTICS FROM GROUPED DATA To obtain descriptive statistics for the Data Frequency data in the table alongside, enter the data 2 3 values into List 1, and the frequency 3 4 values into List 2. 4 8 5 5 Press F2 (CALC) F6 (SET), and change the 1 Var Freq variable to List 2. Press EXIT F1 (1Var) to view the statistics. TWO VARIABLE STATISTICS Consider the data x 1 2 3 4 5 6 7 y 5 8 10 13 16 18 20 To find s and s , the standard deviations of x and y, enter x y the x values into List 1 and the y values into List 2 using the instructions on page 19. IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\020TSL-2_00b.CDR Tuesday, 5 January 2010 3:28:10 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100GRAPHICS CALCULATOR INSTRUCTIONS 21 Press F2 (CALC) F2 (2VAR) to obtain the two variable statistics. s is given by x¾ =2. Use the H key to scroll x n down to find s , which is given by y¾ ¼ 5:08 . y n FINDING¾ AND THE LINE OF BEST FIT We can use our graphics calculator to find the line of best fit connecting two variables. We can also find the values of Pearson’s correlation coefficientr and the coefficient of determination 2 r , which measure the strength of the linear correlation between the two variables. We will examine the relationship between the variables x and y for the previous data. Enter the x values into List 1 and the y values into List 2. To produce a scatter diagram, press F1 (GRPH) F6 (SET), and set up StatGraph 1 as shown. Press EXIT F1 (GPH 1) to draw the scatter diagram. To find the line of best fit, press F1 (CALC) F2 (X). We can see that the line of best fit is y¼ 2:54x+2:71, and that r ¼ 0:998, indicating a very strong positive correlation between x and y. Press F6 (DRAW) to view the line of best fit. 2 CALCULATING 2 To calculate  for the data alongside, A A 1 2 select STAT from the Main Menu, then B 21 28 1 press F3 (TEST) F3 (CHI). B 13 24 2 Press F2 ( I MAT), then enter the values of the table into matrix A, using the instructions given on page 24. 2 Press EXIT twice to return to the  Test screen, highlight Execute, then press EXE . 2 So,  ¼ 0:526 . The p-value and the degrees of freedom are also given. The expected values are stored in matrix B. IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_00b\021TSL-2_00b.CDR Friday, 26 March 2010 10:58:55 AM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 10022 GRAPHICS CALCULATOR INSTRUCTIONS SOLVING EQUATIONS SOLVING LINEAR EQUATIONS To solve the linear equation 3:2x¡4:2=7:4, select EQUA from the Main Menu, then press F3 (SOLV). ¡ . Enter the equation by pressing 3:2 X,µ,T 4:2 SHIFT (=) 7:4 EXE . Highlight X= with the cursor and press F6 (SOLV) to solve the equation. So, the solution is x=3:625 . SOLVING QUADRATIC EQUATIONS 2 To solve the quadratic equation 3x +2x¡ 11 = 0, select EQUA from the Main Menu. Press F2 (POLY) then F1 (2). Enter the coefficients 3, 2 and¡11 and then press F1 (SOLV) to solve the equation. So, the solutions are x¼¡2:28 or 1:61 . WORKING WITH FUNCTIONS GRAPHING FUNCTIONS Selecting GRAPH from the Main Menu takes you to the Graph Function screen, where you can store functions to graph. Delete any unwanted functions by scrolling down to the function and pressing F2 (DEL) F1 (Yes). 2 To graph the function y =x ¡3x¡5, move the cursor to Y1 2 ¡ ¡ and press X,µ,T x 3 X,µ,T 5 EXE . This stores the function into Y1. Press F6 (DRAW) to draw a graph of the function. IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_00b\022TSL-2_00b.CDR Wednesday, 27 January 2010 9:18:21 AM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100GRAPHICS CALCULATOR INSTRUCTIONS 23 To view a table of values for the function, press MENU and select TABLE. The function is stored in Y1, but not selected. Press F1 (SEL) to select the function, and F6 (TABL) to view the table. You can adjust the table settings by pressing EXIT and then F5 (SET) from the Table Function screen. ADJUSTING THE VIEWING WINDOW When graphing functions it is important that you are able to view all the important features of the graph. As a general rule it is best to start with a large viewing window to make sure all the features of the graph are visible. You can then make the window smaller if necessary. The viewing window can be adjusted by pressing SHIFT F3 (V-Window). You can manually set the minimum and maximum values of the x and y axes, or press F3 (STD) to obtain the standard viewing window ¡106x6 10, ¡106y6 10: FINDING POINTS OF INTERSECTION It is often useful to find the points of intersection of two graphs, for instance, when you are trying to solve simultaneous equations. 12¡x We can solve y=11¡3x and y = simultaneously 2 by finding the point of intersection of these two lines. Select GRAPH from the Main Menu, then store 11¡ 3x into Y1 12¡x and into Y2. Press F6 (DRAW) to draw a graph of 2 the functions. To find their point of intersection, press F5 (G-Solv) F5 (ISCT). The solution x=2, y=5 is given. If there is more than one point of intersection, the remaining points of intersection can be found by pressing I . FINDINGx-INTERCEPTS In the special case when you wish to solve an equation of the form f(x)=0, this can be done by graphing y =f(x) and then finding when this graph cuts the x-axis. 3 2 To solve x ¡ 3x +x+1 = 0, select GRAPH from the 3 2 Main Menu and store x ¡3x +x+1 into Y1. Press F6 (DRAW) to draw the graph. To find where this function cuts thex-axis, press F5 (G-Solv) F1 (ROOT). The first solution x¼¡0:414 is given. Press I to find the remaining solutions x=1 and x¼ 2:41 . IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\023TSL-2_00b.CDR Tuesday, 5 January 2010 3:28:21 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 10024 GRAPHICS CALCULATOR INSTRUCTIONS TURNING POINTS 2 To find the turning point or vertex of y =¡x +2x+3, select GRAPH from the Main 2 Menu and store ¡x +2x+3 into Y1. Press F6 (DRAW) to draw the graph. From the graph, it is clear that the vertex is a maximum, so to find the vertex press F5 (G-Solv) F2 (MAX). The vertex is (1, 4). FINDING THE TANGENT TO A FUNCTION 2 To find the equation of the tangent to y = x when x=2, we first press SHIFT MENU (SET UP), and change the 2 Derivative setting to On. Draw the graph of y = x , then press SHIFT F4 (Sketch) F2 (Tang). Press 2 EXE EXE to draw the tangent at x=2. The tangent has gradient 4, and equation y=4x¡4. FINANCIAL MATHEMATICS Suppose we invest5000 at 7:2% p.a. compounded annually. To find the value of the investment after 10 years, select TVM from the Main Menu and press F2 (CMPD) to display the compound interest screen. Set up the screen as shown: Note: All money being invested is considered as outgoings and is entered as a negative value. There are no payments into the account during the term of the investment, so PMT is set to 0. Press F5 (FV) to find the future value. So, the investment amounts to 10021:16. MATRICES STORING MATRICES 0 1 23 A To store the matrix 14 , select RUN¢MAT from the 50 Main Menu, and press F1 (IMAT). This is where you define matrices and enter their elements. To define the matrix as matrix A, make sure Mat A is highlighted, and press F3 (DIM) 3 EXE 2 EXE EXE . This indicates that the matrix has 3 rows and 2 columns. IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_00b\024TSL-2_00b.CDR Wednesday, 27 January 2010 9:19:02 AM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100GRAPHICS CALCULATOR INSTRUCTIONS 25 Enter the elements of the matrix, pressing EXE after each entry. Press EXIT twice to return to the home screen when you are done. TEXAS INSTRUMENTS TI-84 PLUS B BASIC FUNCTIONS GROUPING SYMBOLS (BRACKETS) ( ) The TI-84 Plus has bracket keys that look like and . Brackets are regularly used in mathematics to indicate an expression which needs to be evaluated before other operations are carried out. £ ( + ) For example, to evaluate 2£(4+1) we type 2 4 1 ENTER . We also use brackets to make sure the calculator understands the expression we are typing in. 2 ¥ ( + ) For example, to evaluate we type 2 4 1 ENTER . 4+1 2 ¥ + If we typed 2 4 1 ENTER the calculator would think we meant +1. 4 In general, it is a good idea to place brackets around any complicated expressions which need to be evaluated separately. POWER KEYS The TI-84 Plus has a power key that looks like . We type the base first, press the power key, then enter the index or exponent. 3 For example, to evaluate 25 we type 25 3 ENTER . 2 Numbers can be squared on the TI-84 Plus using the special key x . 2 2 For example, to evaluate 25 we type 25 x ENTER . ROOTS To enter roots on the TI-84 Plus we need to use the secondary function key 2nd . 2 We enter square roots by pressing 2nd x . p 2 For example, to evaluate 36 we press 2nd x 36 ) ENTER . The end bracket is used to tell the calculator we have finished entering terms under the square root sign. IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\025TSL-2_00b.CDR Tuesday, 5 January 2010 3:28:27 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 10026 GRAPHICS CALCULATOR INSTRUCTIONS p 3 Cube roots are entered by pressing MATH 4: ( . p 3 ) To evaluate 8 we press MATH 4 : 8 ENTER . p Higher roots are entered by pressing MATH 5: x . p 4 To evaluate 81 we press 4 MATH 5 : 81 ENTER . INVERSE TRIGONOMETRIC FUNCTIONS ¡1 ¡1 ¡1 The inverse trigonometric functions sin , cos and tan are the secondary functions of SIN , COS and TAN respectively. They are accessed by using the secondary function key 2nd . ¡ ¢ 3 3 ¡1 For example, if cosx = , then x = cos . 5 5 ¥ ) To calculate this, press 2nd COS 3 5 ENTER . SCIENTIFIC NOTATION If a number is too large or too small to be displayed neatly on the screen, it will be expressed k in scientific notation, which is the form a£10 where 16a 10 andk is an integer. 3 To evaluate 2300 , press 2300 3 ENTER . The answer 10 displayed is 1:2167e10, which means 1:2167£10 . 3 ¥ To evaluate , press 3 20000 ENTER . The answer 20000 ¡4 displayed is 1:5e¡4, which means 1:5£10 . You can enter values in scientific notation using the EE function, , which is accessed by pressing 2nd . 14 2:6£10 , ¥ For example, to evaluate , press 2:6 2nd 14 13 13 13 ENTER . The answer is 2£10 . SECONDARY FUNCTION AND ALPHA KEYS The secondary function of each key is displayed in blue above the key. It is accessed by pressing the 2nd key, followed by the key corresponding to the desired secondary function. p p 2 ) For example, to calculate 36, press 2nd x ( ) 36 ENTER . The alpha function of each key is displayed in green above the key. It is accessed by pressing the ALPHA key followed by the key corresponding to the desired letter. The main purpose of the alpha keys is to store values into memory which can be recalled later. MEMORY Utilising the memory features of your calculator allows you to recall calculations you have performed previously. This not only saves time, but also enables you to maintain accuracy in your calculations. IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\026TSL-2_00b.CDR Tuesday, 5 January 2010 3:28:31 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100GRAPHICS CALCULATOR INSTRUCTIONS 27 SPECIFIC STORAGE TO MEMORY Values can be stored into the variable letters A, B, ...., Z. Storing a value in memory is useful if you need that value multiple times. Suppose we wish to store the number15:4829 for use in a number of calculations. To store this number in variable A, type in the I number, then press STO ALPHA MATH (A) ENTER . We can now add 10 to this value by pressing ALPHA MATH + 10 ENTER , or cube this value by pressing ALPHA MATH 3 ENTER . ANS VARIABLE The variable Ans holds the most recent evaluated expression, (¡) and can be used in calculations by pressing 2nd . For example, suppose you evaluate 3£ 4, and then wish to ¡ subtract this from 17. This can be done by pressing 17 (¡) 2nd ENTER . + ¡ If you start an expression with an operator such as , , etc, the previous answer Ans is automatically inserted ahead of the operator. For example, the previous answer can be halved ¥ simply by pressing 2 ENTER . If you wish to view the answer in fractional form, press MATH 1 ENTER . RECALLING PREVIOUS EXPRESSIONS The ENTRY function recalls previously evaluated expressions, and is used by pressing 2nd ENTER . This function is useful if you wish to repeat a calculation with a minor change, or if you have made an error in typing. p p Suppose you have evaluated 100+ 132. If you now want to evaluate 100+ 142, instead of retyping the command, it can be recalled by pressing 2nd ENTER . The change can then be made by moving the cursor over the 3 and changing it to a 4, then pressing ENTER . p If you have made an error in your original calculation, and intended to calculate1500+ 132, again you can recall the previous command by pressing 2nd ENTER . Move the cursor to the first 0. You can insert the5, rather than overwriting the0, by pressing 2nd DEL (INS)5 ENTER . IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\027TSL-2_00b.CDR Tuesday, 5 January 2010 3:28:35 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 10028 GRAPHICS CALCULATOR INSTRUCTIONS LISTS Lists are used for a number of purposes on the calculator. They enable us to store sets of data, which we can then analyse and compare. CREATING A LIST Press STAT 1 to access the list editor screen. To enter the dataf2, 5, 1, 6, 0, 8g into List 1, start by moving the cursor to the first entry of L1. Press 2 ENTER 5 ENTER .... and so on until all the data is entered. DELETING LIST DATA To delete a list of data from the list editor screen, move the cursor to the heading of the list you want to delete then press CLEAR ENTER . REFERENCING LISTS Lists can be referenced by using the secondary functions of the keypad numbers 1-6. For example, suppose you want to add 2 to each element of List1 and display the results in + List2. To do this, move the cursor to the heading of L2 and press 2nd 1 2 ENTER . STATISTICS Your graphics calculator is a useful tool for analysing data and creating statistical graphs. We will first produce descriptive statistics and graphs for the data set: 52336 45375 71895. Enter the data set into List 1. To obtain descriptive statistics of the data set, press STAT I 1:1-Var Stats 2nd 1 (L1) ENTER . To obtain a boxplot of the data, press 2nd Y= (STAT PLOT) 1 and set up Statplot1 as shown. Press ZOOM 9:ZoomStat to graph the boxplot with an appropriate window. To obtain a vertical bar chart of the data, press 2nd Y= 1, and change the type of graph to a vertical bar chart as shown. Press ZOOM 9:ZoomStat to draw the bar chart. Press WINDOW and set the Xscl to 1, then GRAPH to redraw the bar chart. IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\028TSL-2_00b.CDR Tuesday, 5 January 2010 3:28:38 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100GRAPHICS CALCULATOR INSTRUCTIONS 29 We will now enter a second set of data, and compare it to the first. Enter the data set 96235575676 344584 into List 2, press 2nd Y= 1, and change the type of graph back to a boxplot as shown. Move the cursor to the top of the screen and select Plot2. Set up Statplot2 in the same manner, except set the XList to L2. Press ZOOM 9:ZoomStat to draw the side-by-side boxplots. STATISTICS FROM GROUPED DATA To obtain descriptive statistics for the Data Frequency data in the table alongside, enter the data 2 3 values into List 1, and the frequency 3 4 values into List 2. 4 8 Press STAT I 1 : 1-Var Stats 2nd 5 5 , 1 (L1) 2nd 2 (L2) ENTER . TWO VARIABLE STATISTICS Consider the data x 1 2 3 4 5 6 7 y 5 8 10 13 16 18 20 To find s and s , the standard deviations of x and y, enter x y the x values into List 1 and the y values into List 2 using the instructions on page 30. I , Press STAT 2:2-Var Stats, then 2nd 1 (L ) 2nd 2 1 (L ) ENTER to obtain the two variable statistics. 2 s is given by ¾ =2. Use the H key to scroll down to find x X s , which is given by ¾ ¼ 5:08. y y FINDINGr AND THE LINE OF BEST FIT We can use our graphics calculator to find the line of best fit connecting two variables. We can also find the values of Pearson’s correlation coefficientr and the coefficient of determination 2 r , which measure the strength of the linear correlation between the two variables. We will examine the relationship between the variablesx and y for the data above. Enter the x values into List 1 and the y values into List 2. To produce a scatter diagram, press 2nd Y= (STAT PLOT) 1, and set up Statplot 1 as shown. Press ZOOM 9: ZoomStat to draw the scatter diagram. IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\029TSL-2_00b.CDR Tuesday, 5 January 2010 3:28:42 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 10030 GRAPHICS CALCULATOR INSTRUCTIONS I We will now find r and the line of best fit. Press STAT 4:LinReg(ax+b) to select the linear regression option from the CALC menu. , , I Press 2nd 1 (L ) 2nd 2 (L ) VARS 11 (Y ). 1 2 1 This specifies the lists L and L as the lists which hold the 1 2 data, and the line of best fit will be pasted into the function Y . 1 Press ENTER to view the results. The line of best fit is given as y¼ 2:54x+2:71. If ther and 2 r values are not shown, you need to turn on the Diagnostic by pressing 2nd 0 (CATALOG) and selecting DiagnosticOn. The r value of ¼ 0:998 indicates a very strong positive correlation between x and y. You can view the line of best fit by pressing GRAPH . 2 CALCULATING 2 To calculate  for the data alongside, A A 1 2 enter the values of the table into matrix B 21 28 1 A, using the instructions given on page B 13 24 2 36. 2 Press STAT J , then select C: -Test... from the TESTS menu. Highlight Calculate, then press ENTER . 2 So,  ¼ 0:526. The p-value and the degrees of freedom are also given. The expected values are stored in matrix B. SOLVING EQUATIONS SOLVING LINEAR EQUATIONS To solve the linear equation 3:2x¡4:2=7:4, we rearrange the equation to 3:2x¡11:6=0, so that one side is equal to zero. Press MATH then 0 to select 0:Solver... Press N , then enter the equation into eqn:0= by pressing 3:2 ¡ X,T,µ,n 11:6 ENTER . Highlight X= with the cursor and press ALPHA ENTER (SOLVE) to solve for x. So, the solution is x=3:625. IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_00b\030TSL-2_00b.CDR Wednesday, 27 January 2010 9:21:42 AM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100GRAPHICS CALCULATOR INSTRUCTIONS 31 SOLVING QUADRATIC EQUATIONS The TI-84 Plus does not have a built-in method to solve quadratic equations. We must write our own program. 2 To solve the quadratic equation 3x +2x¡11 = 0, press PRGM J 1 to create a new program, label the program QUADRAT, then press ENTER . Enter the code for the program as shown alongside. The Disp and Prompt commands are accessed from the I/O menu of the PRGM screen. You can enter = by pressing 2nd MATH 1. Press 2nd MODE (QUIT) when you are done. To use the program, press PRGM , then 1:QUADRAT. Press ENTER to start the program. As prompted, enter the coefficients A, B and C, pressing ENTER after each one. So the solutions are x¼¡2:28 and 1:61. WORKING WITH FUNCTIONS GRAPHING FUNCTIONS Pressing Y= selects the Y= editor, where you can store functions to graph. Delete any unwanted functions by scrolling down to the function and pressing CLEAR . 2 To graph the function y = x ¡3x¡5, move the cursor to 2 ¡ ¡ Y1, and press X,T,µ,n x 3 X,T,µ,n 5 ENTER . This stores the function into Y1. Press GRAPH to draw a graph of the function. To view a table of values for the function, press 2nd GRAPH (TABLE). The starting point and interval of the table values can be adjusted by pressing 2nd WINDOW (TBLSET). IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_00b\031TSL-2_00b.CDR Wednesday, 27 January 2010 9:26:10 AM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 10032 GRAPHICS CALCULATOR INSTRUCTIONS ADJUSTING THE VIEWING WINDOW When graphing functions it is important that you are able to view all the important features of the graph. As a general rule it is best to start with a large viewing window to make sure all the features of the graph are visible. You can then make the window smaller if necessary. Some useful commands for adjusting the viewing window include: ZOOM 0:ZoomFit : This command scales the y-axis to fit the minimum and maximum values of the displayed graph within the current x-axis range. ZOOM 6:ZStandard : This command returns the viewing window to the default setting of ¡106x6 10, ¡106y6 10. If neither of these commands are helpful, the viewing window can be adjusted manually by pressing WINDOW and setting the minimum and maximum values for the x and y axes. FINDING POINTS OF INTERSECTION It is often useful to find the points of intersection of two graphs, for instance, when you are trying to solve simultaneous equations. 12¡x We can solve y=11¡3x and y = simultaneously 2 by finding the point of intersection of these two lines. 12¡x Press Y= , then store 11¡3x into Y1 and into 2 Y2. Press GRAPH to draw a graph of the functions. To find their point of intersection, press 2nd TRACE (CALC) 5:intersect. Press ENTER twice to specify the functions Y1 and Y2 as the functions you want to find the intersection of, then use the arrow keys to move the cursor close to the point of intersection and press ENTER once more. The solution x=2, y=5 is given. FINDINGx-INTERCEPTS In the special case when you wish to solve an equation of the form f(x)=0, this can be done by graphing y =f(x) and then finding when this graph cuts the x-axis. 3 2 For example, to solve x ¡ 3x +x+1=0, press Y= and 3 2 store x ¡3x +x+1 into Y1. Then press GRAPH . To find where this function first cuts the x-axis, press 2nd TRACE (CALC) 2:zero. IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_26\032TSL-2_00b.CDR Tuesday, 5 January 2010 3:28:52 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100GRAPHICS CALCULATOR INSTRUCTIONS 33 Move the cursor to the left of the first zero and press ENTER , then move the cursor to the right of the first zero and press ENTER . Finally, move the cursor close to the first zero and press ENTER once more. The solution x¼¡0:414 is given. Repeat this process to find the remaining solutions x=1 and x¼ 2:414 . TURNING POINTS 2 To find the turning point or vertex of y =¡x +2x+3, press 2 Y= and store ¡x +2x+3 into Y1. Press GRAPH to draw the graph. From the graph, it is clear that the vertex is a maximum, so press 2nd TRACE (CALC) 4:maximum. Move the cursor to the left of the vertex and press ENTER , then move the cursor to the right of the vertex and press ENTER . Finally, move the cursor close to the vertex and press ENTER once more. The vertex is (1, 4). FINDING THE TANGENT TO A FUNCTION 2 To find the equation of the tangent to y =x when x=2,we 2 first draw the graph of y =x . Press 2nd PRGM (DRAW) 5:Tangent, then press 2 ENTER to draw the tangent at x=2. The tangent has gradient 4, and equation y=4x¡4. FINANCIAL MATHEMATICS Suppose we invest5000 at 7:2% p.a. compounded annually. To find the value of the investment after 10 years, press (APPS) 1:Finance... to display the finance menu, then 1 to select 1:TVM Solver... Set up the TVM screen as shown: Note: All money being invested is considered as outgoings and is entered as a negative value. There are no payments into the account during the term of the investment, so PMT is set to 0. Highlight FV and press ALPHA ENTER (SOLVE) to find the future value. The investment amounts to 10021:16. IB_STSL-2ed cyan magenta yellow black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_00b\033TSL-2_00b.CDR Tuesday, 5 January 2010 4:35:05 PM PETER 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100 0 5 25 50 75 95 100

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