Page 1: Table Of Contents
When you are working on problems in the mathematical chapters, you can refer to this chapter for extra help with your GDC if you need it. Instructions for the Casio fx-9860GIl calculator Use this list to help you fi nd the topic you need Table of contents...
Page 2: Functions
Casio fx-9860GII Before you start You should be familiar with: Important keys on the keyboard: SHIF T DEL MENU ● X,i,T OPTN EXIT For a reminder of how to ● The home screen perform the basic operations ● Using the Main Menu to enter the mode you want have a look at your GDC manual.
Page 3: Finding Information About The Graph
Casio fx-9860GII Finding information about the graph The GDC can give you a lot of information about the graph of a function, such as the coordinates of points of interest and the gradient (slope). 1.2 Finding the gradient (slope) of a line The correct mathematical notation for gradient (slope) is .
Page 4: Simultaneous And Quadratic Equations
Casio fx-9860GII To draw graphs y = 10 − 2x and y = x − 2: Press MENU and choose 5: GRAPH and press The default graph type is Function, so the form Y= is displayed. Type 10 – 2x and press and x –...
Page 5: Solving Simultaneous Linear Equations In Three Unknowns
Casio fx-9860GII Choose equations in two unknowns and press Note: This is how you will use the linear equation solver in your examinations. In your project, you might want to solve a more complicated system with more equations and more variables.
Page 6 Casio fx-9860GII Example 5 Solve the equations 2x − 3y + 4z = 1 x − y − z = −1 −x + 2y − z = 2 Press MENU . You will see the dialog box as shown on the right.
Page 7 Casio fx-9860GII Example 6 Solve the equations 2x + 4y + 2z = 8 x + 2y + z = 4 3x − y + z = −9 Press MENU . You will see the dialog box as shown on the right.
Page 8 Casio fx-9860GII Type the coefﬁ cients from the three equations into the matrix, pressing after each number. Press The reduced row echelon form gives the solutions to the equations of the line if the ﬁ nal row of the matrix is all zeroes.
Page 9: Quadratic Functions
Casio fx-9860GII The GDC solves the equations by an inverse matrix method. In this example there is an error because the matrix is singular, which means that there will be no unique solution. To ﬁ nd the solution you can use the reduced row echelon form method instead.
Page 10: Solving Quadratic Equations
Casio fx-9860GII Adjust the window to make the exponential curve ﬁ t the screen better. For help with changing axes, see your GDC manual. 1.7 Solving quadratic equations When solving quadratic equations in an examination, you do not need to show any method of solution.
Page 11: Finding A Local Minimum Or Maximum Point
Casio fx-9860GII 1.8 Finding a local minimum or maximum point Example 10 Find the minimum point on the graph of y = x − 2x + 3. Draw the graph of y = x − 2x + 3 (See Example 8).
Page 12 Casio fx-9860GII OPTN Press GLINK. When you scroll through the table the cross moves along the curve. The minimum value shown in the table is 2 when x = 1. Look more closely at the values of the function around x = 1.
Page 13 Casio fx-9860GII Example 11 Find the maximum point on the graph of y = −x + 3x − 4. Press MENU and choose 5: GRAPH and press The default graph type is Function, so the form Y = is displayed.
Page 14 Casio fx-9860GII Press to view the table. Press G·CON to display the graph alongside the table. OPTN Press GLINK. When you scroll through the table the cross moves along the curve. The minimum value shown in the table is –2 when x = 1 and x = 2.
Page 15: Complex Numbers
Casio fx-9860GII The calculator displays the maximum point on the curve at (1.5, −1.75). Complex numbers 1.9 Operations with complex numbers Example 12 Evaluate the following expressions − (2 + 3i ) ⋅ (3 − 4i ) − (1 − i )
Page 16: Conjugate, Modulus And Argument
Casio fx-9860GII 1.10 Conjugate, modulus and argument Example 13 Let z z ∗ Find arg (z) Press MENU . You will see the dialog box as shown on the right. Choose 1: RUN·MAT and press OPTN Press CPLX | Conj Enter the complex number in brackets.
Page 17: Polar Form
Casio fx-9860GII From the menu, choose Polynomial and press If there are previous equations in the memory, EXIT press until you return to this menu. You will see the dialog box as shown on the right. Choose equations of degree 3 and press Another dialog box opens for you to enter the equation.
Page 18: Exponential Functions
Casio fx-9860GII a b / c ∠ π Enter SHIF T SHIF T and then press OPTN CPLX | a + bi Press The ∠ symbol is above the X,i,T You can also change the mode that the calculator uses to display complex results in settings.
Page 19: Finding A Horizontal Asymptote
Casio fx-9860GII 1.14 Finding a horizontal asymptote Example 17 Find a horizontal asymptote to the graph of y = 3 + 2. Draw the graph of y = 3 + 2 (See Example 16). You can look at a table of the values of the function.
Page 20: Logarithmic Functions
Casio fx-9860GII Press OPTN GLINK. As the value of x gets smaller, Y gets closer and closer to 2. Press EXIT and then press SET. Change the minimum value of x to –12. OPTN G·CON GLINK. Eventually the value of Y displayed in the table reaches 2.
Page 21: Finding An Inverse Function
Casio fx-9860GII 1.16 Finding an inverse function The inverse of a function can be found by interchanging the x and y values. Geometrically this can be done by reﬂ ecting points in the line y = x. Example 19 Show that the inverse of the function y = 10 is y = log x by reﬂ...
Page 22: Drawing A Logarithmic Graph
Casio fx-9860GII 1.17 Drawing a logarithmic graph Example 20 Draw the graph of y = 2log x + 3. Press MENU . You will see the dialog box as shown on the right. Choose 5: GRAPH and press X,i,T Press...
Page 23: Drawing Trigonometric Graphs
Casio fx-9860GII 1.19 Drawing trigonometric graphs Example 22 Draw the graph of 2sin Press MENU . You will see the dialog box as shown on the right. Choose 5: GRAPH and press 2sin Type and press X,i,T Press SHIF T...
Page 24: More Complicated Functions
Casio fx-9860GII More complicated functions 1.20 Solving a combined quadratic and exponential equation To solve the equation, fi nd the point of intersection between the quadratic function y − 2x + 3 and the exponential function = 3 × 2 −x...
Page 25: Sequences And Series
Casio fx-9860GII Sequences and series 1.21 Summation of a series Example 24 Find the sum of the ﬁ rst 20 terms of the arithmetic sequence 4, 7, 10, 13, … The kth term of an arithmetic sequence is u + (k − 1)d In this example u = 4, d = 3 and n = 20.
Page 26 Casio fx-9860GII Example 26 + ... are needed before their sum exceeds 5.5? How many terms of the series 2 1 = 2, r = In the example u and n is to be found. − ⎛ ⎞ ∑ ⋅...
Page 27: Modelling
Casio fx-9860GII Modelling 1.22 Using sinusoidal regression x, cos x, tan x, … is a mathematical convention that has little algebraic The notation sin meaning. To enter these functions on the GDC, you should enter (sin(x)) , etc. However, the...
Page 28: Drawing A Piecewise Function
Casio fx-9860GII 1.23 Drawing a piecewise function Example 28 ì ï 4 +3, ï ï í Draw the function ï ï ï î Press MENU . You will see the dialog box as shown on the right. Choose 5: GRAPH and press The default graph type is Function, so the form Y= is displayed.
Page 29: Drawing A Tangent To A Curve
Casio fx-9860GII Adjust the window to make the cubic curve ﬁ t the screen better. For help with changing axes, see your GDC manual. Press SHIF T (SET UP). Set Derivative to On. Press DRAW to return to the graph.
Page 30: Finding Maximum And Minimum Points
Casio fx-9860GII Press Sketch. Press Tang. The calculator displays the trace screen. (–) Press Press The equation of the tangent is y = −3.25x + 5.75. 2.3 Finding maximum and minimum points Example 31 Find the local maximum and local minimum points on the cubic curve.
Page 31: Finding A Numerical Derivative
Casio fx-9860GII SHIF T Press G-Solv. Press MIN to ﬁ nd the minimum point. The calculator displays the local minimum at the point (2.23, −7.24). 2.4 Finding a numerical derivative Using the calculator it is possible to ﬁ nd the numerical value of any derivative for any value of x.
Page 32: Graphing A Numerical Derivative
Casio fx-9860GII 2.5 Graphing a numerical derivative Although the calculator can only evaluate a numerical derivative at a point, it will graph the gradient function for all values of x. Example 33 If y , draw the graph of MENU Press .
Page 33 Casio fx-9860GII Example 34 − Find the values of x on the curve y 1 where the gradient is 3. MENU Press . You will see the dialog box as shown on the right. Choose 5: GRAPH and press Press CALC.
Page 34: Using The Second Derivative
Casio fx-9860GII Press to select the second point. The GDC shows a point of intersection at (2, 3). The curve has gradient 3 when x = –4 and x = 2. 2.6 Using the second derivative The calculator can ﬁ nd ﬁ rst and second derivatives. The second derivative can be used to determine whether a point is a maximum or minimum point.
Page 35: Integral Calculus
Casio fx-9860GII Press OPTN CALC | to enter the second derivative template. Enter Y1 in the template as the function using the same procedure to enter the Y. Enter the value of x as 0. Repeat for the second derivative when x = 3.
Page 36: Finding The Area Under A Curve
Casio fx-9860GII Enter the upper and lower limits, the function and x in the template. Press The value of the integral is 21.5 (to 3 sf ). 3.2 Finding the area under a curve Example 37 y = 3x − 5 Find the area bounded by the curve, the x-axis and the lines x = –1 and x = 1.
Page 37: Vectors Scalar Product
Since the area lies below the x-axis in this case, the integral is negative. 4 Vectors There is no scalar product function on the Casio GDC, Scalar product but you can fi nd the result by multiplying the vectors 4.1 Calculating a scalar product as lists and then fi nding the sum of the terms in the list.
Page 38: Calculating The Angle Between Two Vectors
Casio fx-9860GII Press OPTN LIST | Sum. Enter the vectors as lists using curly brackets { }. Separate the terms of the vectors using commas. Multiply the two vector lists together. Close the bracket and press ENTER ⎛ ⎞ ⎛...
Page 39: Vector Product
The angle between 2 Vector product 4.3 Calculating a vector product The Casio fx-9860GII does not have the ability to perform cross products of vectors. 5 Statistics and probability You can use your GDC to draw charts to represent data and to calculate basic statistics such as mean, median, etc.
Page 40: Entering Data From A Frequency Table
Casio fx-9860GII 5.2 Entering data from a frequency table Example 41 Number 1 2 3 4 5 Enter the data in the table: Frequency 3 4 6 5 2 MENU Press . You will see the dialog box as shown on the right.
Page 41: Drawing A Frequency Histogram From A Frequency Table
Casio fx-9860GII Press (Trace). SHIF T Use the key to move to each of the bars and display their value and frequency. You should now see a frequency histogram for the data in the list 1, 1, 3, 9, 2.
Page 42: Drawing A Box And Whisker Diagram From A List
Casio fx-9860GII 5.5 Drawing a box and whisker diagram from a list Example 44 Draw a box and whisker diagram for this data: 1, 1, 3, 9, 2. Enter the data in List 1 (see Example 40). Press GRPH and SET.
Page 43: Calculating Statistics
Casio fx-9860GII Press SHIF T (Trace). Use the key to move the cursor over the plot to see the quartiles, Q1 and Q3, the median and the maximum and minimum values. Calculating statistics You can calculate statistics such as mean, median, etc. from a list, or from a frequency table.
Page 44: Calculating Statistics From A Frequency Table
Casio fx-9860GII 5.8 Calculating statistics from a frequency table Example 47 Calculate the summary statistics for this data: Number 1 2 3 4 5 Frequency 3 4 6 5 2 Enter the data in List 1 and List 2 (see Example 41).
Page 45: Calculating The Interquartile Range
Casio fx-9860GII 5.9 Calculating the interquartile range Example 48 Calculate interquartile range for this data: The interquartile range is the Number 1 2 3 4 5 difference between the upper Frequency and lower quartiles (Q − Q 3 4 6 5 2...
Page 46: Use Of Ncr
Casio fx-9860GII Calculating binomial probabilities 5.11 The use of nCr Example 50 ⎛ ⎞ Find the value of ⎜ ⎟ ⎝ ⎠ MENU Press . You will see the dialog box as shown on the right. Choose 1: RUN·MAT and press Press PROB to enter probability menu.
Page 47: Calculating Binomial Probabilities
Casio fx-9860GII 5.12 Calculating binomial probabilities Example 52 X is a discrete random variable and X ~ Bin(9, 0.75). Calculate P(X = 5) ⎛ ⎞ P X = 0 75 0 25 ⎝ ⎜ ⎠ ⎟ The calculator can ﬁ nd this value directly.
Page 48 Casio fx-9860GII Enter the values 0,1,2,3,4,5,6,7 in List1. Press after each number. Press DIST | BINM | Bpd. Press List. Choose List1. Enter the number of trials, 7 and the probability 0.3. Press The calculator displays each of the probabilities.
Page 49: Calculating Poisson Probabilities
Casio fx-9860GII When using the lower bound probability to calculate other probabilities, it is easier to do this without the wizard. MENU Press . Choose 1: RUN·MAT and press P(5 ≤ X ≤ 15) = P(X ≤ 15) – P(X ≤ 4)
Page 50 Casio fx-9860GII The calculator shows that P(X = 2 ) = 0.0758 (to 3 sf) − × − × − × ( . ) ( . ) ( . ) ≤ The calculator can ﬁ nd this value directly. Press...
Page 51 If X ~ Po( μ ) ﬁ nd the value of μ , correct to 3 decimal places, given that P(X = 2) = 0.035. There is no inverse Poisson function on the Casio fx-9860GII, so instead you should use the numerical solver function to ﬁ nd a value of μ when you are given a probability.
Page 52: Calculating Normal Probabilities
Casio fx-9860GII The required value of μ is 0.309 (to 3 sf). Calculating normal probabilities 5.14 Calculating normal probabilities from X-values Example 57 A random variable X is normally distributed with a mean of 195 and a standard deviation of 20 or x ~ N(195, 20) .
Page 53: Calculating X-Values From Normal Probabilities
Casio fx-9860GII P(X < 190) = 0.401 (to 3 sf) P(x > 194) Press EXIT to return to the entry screen. Enter Lower Bound as 194, Upper Bound as 1E99, σ as 20 and μ as 195. P(X > 194) = 0.520 (to 3 sf) P(187 <...
Page 54 Casio fx-9860GII Example 58 A random variable X is normally distributed with a mean of 75 and a standard You are given a lower- deviation of 12 or X ~ N(75, 12 ). If P (X < x) = 0.4, ﬁ nd the value of x.
Page 55 Casio fx-9860GII Press DIST NORM InvN to use the Inverse Normal function. Press Var. You should enter the values: Area (probability), σ and μ , in order. ), Area as 0.2, σ as 12 and μ as 75. Enter Tail as Right ( Press So if P (X >...

Casio fx-9860GII Instructions Manual

Using a Graphic Display Calculator

Functions

Finding Information about the Graph

Simultaneous and Quadratic Equations

Quadratic Functions

Complex Numbers

Exponential Functions

Logarithmic Functions

Trigonometric Functions

More Complicated Functions

Sequences and Series

Modelling

Integral Calculus

Vectors Scalar Product

Vector Product

Statistics and Probability

Entering Data

Entering Data from a Frequency Table Example 41

Drawing Charts

Example 45

Number 1 2 3 4

Frequency 3 4 6 5

Example 46

Calculating Statistics

Calculating Poisson Probabilities

Calculating Normal Probabilities

Quick Links

Table of contents

Chapters

Related Manuals for Casio fx-9860GII

Summary of Contents for Casio fx-9860GII

Table of Contents