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SCIENTIFIC
CALCULATOR
OPERATION GUIDE
SCIENTIFIC
CALCULATOR
OPERATION GUIDE
< EL-W531TL / W531TH / W531TG / W506T >
2
Arc trigonometric functions 38
CONTENTS
HOW TO OPERATE
Read Before Using
Functions and Key Operations
ON/OFF, entry correction keys 8
Data entry keys 9
Random key 10
Modify key 11
Basic arithmetic keys, parentheses 12
Percent 13
Inverse, square, cube, xth power of y,
square root, cube root, xth root 14
Key layout 4
Reset switch/Display pattern 5
Display format and decimal setting function 5-6
Exponent display 6
Angular unit 7
10 to the power of x, common logarithm,
logarithm of x to base a
18
Binary, pental, octal, decimal, and
hexadecimal operations (N-base) 44
e to the power of x, natural logarithm 22
Exponential, logarithmic 19-21
Permutations, combinations 24-26
Differentiation calculation 45-46
Integration calculation 47-49
Polynomial equation 53-56
Simultaneous calculation 50-52
Complex calculation 57-58
Power and radical root 15-17
Factorials 23
Time calculation 27
Fractional calculations 28
Memory calculations 29
Last answer memory 30
User-defined functions 31
Absolute value 32
Trigonometric functions 33-37
Hyperbolic functions 39-42
Coordinate conversion 43
Statistics functions 59
Data input for 1-variable statistics 59
“ANS” keys for 1-variable statistics 60-61
Data correction 62
Data input for 2-variable statistics 63
“ANS” keys for 2-variable statistics 64-66
~
~
x
d/dx x
i
Matrix calculation 67-68
dx
DATA INS-D STAT
3
Arc trigonometric functions 38
CONTENTS
HOW TO OPERATE
Read Before Using
Functions and Key Operations
ON/OFF, entry correction keys 8
Data entry keys 9
Random key 10
Modify key 11
Basic arithmetic keys, parentheses 12
Percent 13
Inverse, square, cube, xth power of y,
square root, cube root, xth root 14
Key layout 4
Reset switch/Display pattern 5
Display format and decimal setting function 5-6
Exponent display 6
Angular unit 7
10 to the power of x, common logarithm,
logarithm of x to base a
18
Binary, pental, octal, decimal, and
hexadecimal operations (N-base) 44
e to the power of x, natural logarithm 22
Exponential, logarithmic 19-21
Permutations, combinations 24-26
Differentiation calculation 45-46
Integration calculation 47-49
Polynomial equation 53-56
Simultaneous calculation 50-52
Complex calculation 57-58
Power and radical root 15-17
Factorials 23
Time calculation 27
Fractional calculations 28
Memory calculations 29
Last answer memory 30
User-defined functions 31
Absolute value 32
Trigonometric functions 33-37
Hyperbolic functions 39-42
Coordinate conversion 43
Statistics functions 59
Data input for 1-variable statistics 59
“ANS” keys for 1-variable statistics 60-61
Data correction 62
Data input for 2-variable statistics 63
“ANS” keys for 2-variable statistics 64-66
~
~
x
d/dx x
i
Matrix calculation 67-68
dx
DATA INS-D STAT
4
How to Operate
Mode key
1. KEY LAYOUT (EL-W531TL)
Read Before Using
This operation guide has been written mainly based on the EL-W531TL/W531TH/W531TG
model. And some functions described here are featured on the EL-W506T model only.
Note that key operations and symbols on the display may differ according to the model.
2nd function, ALPHA keys
Pressing these keys will enable
the functions written in orange
(2nd F) or green (ALPHA) above
the calculator buttons.
This calculator can operate in four different modes as follows.
[
NORMAL mode
]•
Mode = 0; normal mode for
performing normal
arithmetic and function
calculations.
ON/C, OFF key
<Power on>
<Power off>
Written in orange
above the ON/C key
[STAT mode]
•Mode = 1; mode for
performing 1- or 2-variable
statistical calculations. To
select the sub-mode, press
the corresponding number
key after .
HOME key
Pressing this key will return to
NORMAL mode.
Single variable statistic calculation
Linear regression calculation
Quadratic regression calculation
Euler Exponential regression calculation
Logarithmic regression calculation
Power regression calculation
Inverse regression calculation
Exponential regression calculation
[DRILL mode]
•Mode = 3; mode for
performing drill calculations.
To select the drill sub-mode,
press the corresponding
number key
after .
(MATH): Math drill
(TABLE): Multiplication table drill
[TABLE mode]
•Mode = 2; mode for
showing the changes in
values of a function in table
format.
NOTE:
The EL-W506T model has another modes (Complex, Equation, Matrix, Vector and
Distribution modes).
5
3.
DISPLAY PATTERN
NOTE:
The actual display does not appear like this.
This illustration is for explanatory purposes only.
10000 8.1
Appears
when the
entire
equation
cannot be
displayed.
Hyperbolic
symbol (HYP)
Alphabet
symbol
(ALPHA)
Angular unit
indicator
(DEG/RAD/GRAD)
2ndF symbol
Appears when the
entire equation
cannot be displayed.
Display format
indicator
(FIX, SCI, ENG,
N1, N2)
WriteView mode
(View as it is written)
Equation display
Answer display
Memory
symbol
For convenient and easy operation, this model can be used in one of five display modes.
The selected display status is shown in the upper part of the display (Display format indicator).
Note: If more 0’s (zeros) than needed are displayed when the ON/C key is pressed, check
whether or not the calculator is set to a Special Display Format.
• Floating decimal point format 1/2 (N1/N2 is displayed)*1
Valid values beyond the maximum range are displayed in the form of [10-digit
(mantissa) + 2-digit (exponent)]
• Fixed decimal point format (FIX is displayed)
Displays the fractional part of the calculation result according to the specified
number of decimal places.
• Scientific notation (SCI is displayed)
Frequently used in science to handle extremely small or large numbers.
• Engineering scientific notation (ENG is displayed)
Convenient for converting between different units.
2.
RESET SWITCH
Reset switch
RESET
If the calculator fails to operate
normally, press the reset switch on
the back to reinitialize the unit. The
display format and calculation mode
will return to their initial settings.
NOTE:
Pressing the reset switch
will erase any data stored
in memory.
4. DISPLAY FORMAT AND
DECIMAL SETTING FUNCTION
<Example>
*1 The calculator has two settings for displaying a floating point number:
NORM1 (default setting) and NORM2. In each display setting, a number is
automatically displayed in scientific notation outside a preset range:
• NORM1: 0.000000001 <
_
x <
_
9999999999
• NORM2: 0.01 <
_
x <
_
9999999999
Let’s compare the display result of
[10000 8.1 =] in each display format.
(NORM1 mode)
(Mixed fractions)
Display format indicator
6
5. EXPONENT DISPLAY
<Example>
0.32 191.6
(FIX mode, TAB = 3)
(SCI mode)
(ENG mode)
(NORM1 mode)
The distance from the earth to the sun is approx. 150,000,000 (1.5 x 108) km. Values
such as this with many zeros are often used in scientific calculations, but entering the
zeros one by one is a great deal of work and it’s easy to make mistakes. In such
cases, the numerical values are divided into mantissa and exponent portions,
displayed and calculated.
What is the number of electrons flowing in a conductor when
the electrical charge across a given cross-section is 0.32 coulombs.
(The charge on a single electron = 1.6 x 10-19 coulombs).
(Improper fractions)
(Decimal numbers)
NOTE:
In EL-W506T,
Use instead of .
SET UP
7
<Example>
6. ANGULAR UNIT
Operation Display
90
Angular values are converted from DEG to RAD to GRAD with each push
of the DRG►
key (2nd function of ). This function is used when doing calculations related to
trigonometric functions or coordinate geometry conversions.
Degrees (DEG is shown at the top of the display)
A commonly used unit of measure for angles. The angular measure of a circle
is expressed as 360
°
.
Radians (RAD is shown at the top of the display)
Radians are different from degrees and express angles based on the circumfer-
ence of a circle. 180
°
is equivalent to π radians. Therefore, the angular mea-
sure of a circle is 2π radians.
Grads (GRAD is shown at the top of the display)
Grads are a unit of angular measure used in Europe, particularly in France. An
angle of 90 degrees is equivalent to 100 grads.
The relationships between the three types
of angular units can be expressed as right:
π
2
90° (DEG) =
π/2 (RAD) =
100 (GRAD) =
Check to confirm 90 degrees equaling π/2 radians
equaling 100 grads. (π=3.14159...)
(DEG)
8
Turns the calculator on or clears the data. It also clears the contents of the
calculator display and voids any calculator command; however, statistics, as
well as values stored in the memory, are not erased.
Clears all internal values, including the last answer (ANS) and statistics. Values
stored in memory are not erased.
These arrow keys are useful for Multi-Line playback, which lets you
scroll through calculation steps one by one.
Turns the calculator off.
These keys are useful for editing equations. The key moves the
cursor to the left, and the key moves the cursor to the right.
The key deletes the symbol/number at the left of the cursor,
and the key deletes the symbol/number at the cursor.
ON/OFF, Entry
Correction Keys
Functions and Key Operations
9
Data Entry Keys
<Example>
0 to 9
Operation
21.496 8
Display
Decimal point key. Enters a decimal point.
Numeric keys for entering data values.
Pressing this key switches to scientific notation data entry.
Provided the earth is moving around the sun in a circular orbit,
how many kilometers will it travel in a year?
Circumference equals diameter x
π
; therefore,
1.496 x 108 x 2 x
π
* The average distance between the earth and the sun being
1.496 x 108 km.
Enters π (3.14159...).
The constant π, used frequently in function calculations, is the ratio of the
circumference of a circle to its diameter
Enters the minus symbol.
The subtraction key is not used for entering negative numbers.
10
Random Key
<Example>
0. *** (A random number is generated.)
[Random Dice]
To simulate a die-rolling, a random integer between 1 and 6 can be generated by
pressing . To generate the next random dice number, press .
[Random Coin]
To simulate a coin flip, 0 (heads) or 1 (tails) can be randomly generated by pressing
. To generate the next random coin number, press .
[Random Integer]
You can specify a range for the random integer with “R.Int(”.
　R.Int(minimum value, maximum value)
For example, if you enter 1 99 , a random integer from 1 to
99 will be generated. To generate the next random integer, press .
Generates random numbers.
Random numbers are three-decimal-place values between 0.000 and 0.999. Using this
function enables the user to obtain unbiased sampling data derived from random
values generated by the calculator.
APPLICATIONS:
Building sample sets for statistics or research.
NOTE:
Using LINE mode is preferable, since the numbers are generated by fractions in
W-VIEW mode. In W-VIEW mode, press to convert it to decimal form.
(LINE mode)
11
<Example>
5.0
0.6
0.6
5.4
59
9
5 9
9
Modify Key
Function to round calculation results.
Even after setting the number of decimal places on the display, the calculator
performs calculations using a larger number of decimal places than that which
appears on the display.
By using this function, internal calculations will be performed using
only the displayed value.
FIX mode TAB = 1 (normal calculation)
Rounded calculation (MDF)
(internally, 0.5555...)
(internally, 0.5555...)
(internally, 0.6)
APPLICATIONS:
Frequently used in scientific and technical fields, as well as business,
when performing chained calculations.
(In W-VIEW mode, press to show the answer in decimal.)
(In W-VIEW mode, press to show the answer in decimal.)
(In W-VIEW mode, press twice to show the answer in decimal.)
12
The four basic operators. Each is used in the same way as a standard
calculator:
+ (addition), – (subtraction), x (multiplication), and ÷ (division).
Used to specify calculations in which certain operations have precedence.
You can make addition and subtraction operations have precedence over
multiplication and division by enclosing them in parentheses.
Finds the result in the same way as a standard calculator.
Basic Arithmetic
Keys, Parentheses
13
125 10
Percent
125 20
125 15
125 5
For calculating percentages. Four methods of calculating percentages
are presented as follows.
1) $125 increased by 10%…137.5
2) $125 reduced by 20%…100
3) 15% of $125…18.75
4) When $125 equals 5% of X, X equals…2500
NOTE:
In EL-W506T, when “(%)” is specified immediately after a value, the
value is treated as a percentage. “(%)” is specified by .
14
<Example>
Operation Display
Inverse, Square, Cube,
xth Power of y, Square Root,
Cube Root, xth Root
2 2 2 2
4 16
2 4
Calculates the inverse of the value.
Squares the value.
Cubes the value.
Calculates the xth power of the value.
Calculates the square root of the value.
Calculates the cube root of the value.
Calculates the xth root of the value.
15
<Example 1> Design a shaft that bears a torque T (= 9,550 Nm).
is a constant that is determined by the material of the shaft,
and is taken to be = 20 N/mm2.
Operation Display
16
9550 20
Power and Radical root
d = 16T
3
16
If the principal is a ($), the annual interest rate is r (%),
and the number of years of interest accumulation is x (years),
the final amount y ($) is given by the following equation:
(1) Find the final amount when a principal of $400,000 is
deposited for three years at an annual interest rate of 5%
and the interest is compounded annually.
(2) When a principal of $300,000 is deposited for five years
and the interest is compounded annually, the final amount is
$
339,422. The annual interest rate r is given by the equation below.
Find the annual interest rate r.
(1)
(2)
x
r = 100 - 1( )
y
a
5- 1
339422
300000
y = 400000 1 +
( )
5
100
3
r = 100 ( )
5
100
300000
y = a ( 1 + r / 100 )x
400000 1
3
100 5
339422
1
Operation Display
<Example 2>
Power and Radical root
17
The musical note A is 440 Hz.
Calculate the frequencies of the notes in (1) to (3).
(1) "C" of A, A# (B ), B, C
(2) "C" of A, G, F, E, D, C
(3) "A" one octave higher
(1)
(2)
12
2
440
3
12
(3)
12 2
440
12
3
2
2
440 x (12 2)3
440 x (12 2)12
440 x (12 2)3
2
440
Operation Display
Power and Radical root
<Example 3>
18
<Example>
1000
3
Operation Display
10 to the Power of x,
Common Logarithm,
Logarithm of x to Base a
Calculates the value of 10 raised to the xth power.
Calculates the logarithm, the exponent of the power to which 10 must be
raised to equal the given value.
Calculates the logarithm of x to power a.
3 45
19
Operation Display
logE = 4.8 + 1.5M
logE - 4.8
1.5
M =
(1)
(2)
1.5 1
1.5 2
If E (units: joules) is the amount of energy released by an
earthquake and M is the magnitude, the relation
holds.
If E' is the energy when the magnitude increases by N,
holds.
(1) When the magnitude increases by 1, by what factor does
the energy increase?
(2) When the magnitude increases by 2, by what factor does
the energy increase?
(3)
The amount of energy in 20,000 tons of TNT is 8 x 1013 joules.
When this energy is converted to a magnitude,
holds. Find the magnitude M.
= 101.5N
E'
E
Exponential, Logarithmic
<Example 1>
(3)
4.8
8
13
1.5
20
Operation Display
1000000
1000000
101000
1000000
101000
0.01
1000000 0.01
0.434
<Example 2> Air is held inside a cylinder of volume V1 (= 0.01 m3) at a
pressure P1 (= 1,000,000 Pa) at 27°C with a piston.
Find the quantity of thermal energy Q needed to expand the air
at constant temperature to a pressure of P2 (= 101,000 Pa).
Q = p1V1In p2
p1
p2
p1
log
p1V1
0.434
Exponential, Logarithmic
21
Operation Display
Find the pH of hydrochloric acid HCl at a concentration of
1.0 x 10-8 mol/L
* pH = 7 (neutral), pH < 7 (acidic), pH > 7 (alkaline)
pH = -log10 2
- aa2+4x10-14
a+
( )
1.0
10
14
8
4
2
<Example 3>
Enter the value of a
Calculate the pH
Exponential, Logarithmic
22
<Example>
10
5
Operation Display
e to the Power of x,
Natural Logarithm
Calculates powers based on the constant e (2.718281828).
Computes the value of the natural logarithm, the exponent of the power
to which e must be raised to equal the given value.
23
<Example 1> Operation Display
7
Factorials
The product of a given positive integer n multiplied by all the lesser positive
integers from 1 to n-1 is indicated by n! and called the factorial of n.
cf.
n! = 1
x
2
x
3
x
…
x
n
APPLICATIONS:
Used in statistics and mathematics. In statistics, this function is used
in calculations involving combinations and permutations.
Operation Display
<Example 2> How many arrangements exist of cards of three colors:
red, blue, and yellow?
3! = 3 x 2 x 1 = 6
3
24
<Example 1>
Operation Display
6 4
6 4
Permutations, Combinations
This function finds the number of different possible orderings in selecting
r objects from a set of n objects. For example, there are six different
ways of ordering the letters ABC in groups of three letters—ABC, ACB,
BAC, BCA, CAB, and CBA.
The calculation equation is 3P3 = 3 x 2 x 1 = 6 (ways).
This function finds the number of ways of selecting r objects from a set of
n objects. For example, from the three letters ABC, there are three ways
we can extract groups of two different letters—AB, AC, and CB.
The calculation equation is 3C2.
APPLICATIONS:
Used in statistics (probability calculations) and in simulation hypotheses
in fields such as medicine, pharmaceutics, and physics. Also, can be used
to determine the chances of winning in lotteries.
25
(2)
5 3
3
35
Operation Display
<Example 2> (1)
When three cards are selected from five cards numbered
1 to 5 and placed in a row, how many possible orderings of
the cards are there?
(2)
When three cards are selected from five cards numbered
1 to 5, how many ways of selecting the cards are possible?
5P3 = 5 x 4 x 3
Let the number of ways of selecting the cards be C.
There are 3! possible orderings of the cards, and thus
when ordered in a row
Therefore C is
(1)
53
Permutations, Combinations
*This is written as 5C3.
C x 3! = 5P3
C = 5P3 ÷ 3!
26
13 x 4C2
12C3 x 43
43
52 5
<Example 3>
Find the probability of drawing one pair when 5 cards are
drawn from a deck of 52 cards.
No jokers are included in the deck.
Probability of drawing one pair =
Ways of selecting one pair Ways of selecting 5 cards
Ways of selecting one pair =
Ways of selecting two cards to make a pair x Ways of selecting
3 remaining cards
Ways of selecting two cards to make a pair
Ways of selecting the number: 13 possibilities from 1 to 13 (King)
Ways of selecting the suit: Two suits selected from four, 4C2
Hence
Ways of selecting five cards
52C5
The probability of drawing one pair is
(13 x 4C2) x (12C3 x 43) 52C5
Ways of selecting remaining three cards
Ways of selecting the number: Three types are selected from
(13 - 1) types (13-1)C3
Ways of selecting the suit: For each number on the three cards,
there are 4 types of suit 43
Hence
Operation Display
413
122
Permutations, Combinations
27
<Example>
Operation Display
24 28 35
Time Calculation
Converts a sexagesimal value displayed in degrees, minutes, seconds to
decimal notation. Also, converts a decimal value to sexagesimal notation
(degrees, minutes, seconds).
Convert 24
°
28’ 35” (24 degrees, 28 minutes, 35 seconds)
to decimal notation. Then convert 24.476
°
to sexagesimal
notation.
Inputs values in sexagesimal notation (degrees, minutes, seconds).
Convert to decimal notation
Repeat last key operation to return to the previous display.
APPLICATIONS:
Used in calculations of angles and angular velocity in physics, and
latitude and longitude in geography.
28
<Example>
Operation Display
5
Fractional Calculations
Inputs proper or improper fractions which consist of a numerator and
denominator.
Inputs a mixed fraction.
1
2
5
7
Add 3 and , and convert to decimal notation.
Convert to an improper fraction
Convert to decimal notation
APPLICATIONS:
There is a wide variety of applications for this function because
fractions are such a basic part of mathematics. This function is useful
for calculations involving electrical circuit resistance.
3 1 2
7
29
<Example 1>
25 27
7 3
Operation Display
~
~
<Example 2>
Operation Display
Calculates $/¥ at the designated exchange rate.
$1 = ¥110 ¥26,510 = $? $2,750 = ¥?
110
26510
2750
Memory Calculations
Stores displayed values in memories A~F, X, Y, M.
Recalls values stored in A~F, X, Y, M.
Temporary memories
Independent memory
Adds the displayed value to the value in the independent memory M.
Subtracts the displayed value from the value in the independent memory M.
0
(Enter 0 for M)
30
<Example>
y = 4 xandx = 2 + 3
Operation Display
2 3
4
Last Answer Memory
Recalls the last answer calculated by pressing
Solve for x first and then solve for y using x.
31
<Example>
Operation Display
User-Defined Functions
Recall a function that was defined by the user.
APPLICATIONS:
Functions that you have previously defined, including those using
common 2nd Function buttons, can be stored in D1~ D3 for
later use, thus saving time on keystrokes.
26
~
~
32
<Example>
Operation
Absolute Value
Returns an absolute value.
Display
3
-
4
33
[DEG mode]
<Example 1>
Operation Display
sin
θ
=b
a
tan
θ
=b
c
cos
θ
=c
a
a
c
b
θ
Trigonometric functions determine the ratio of three sides
of a right triangle. The combinations of the three sides are
sin, cos, and tan. Their relations are:
Calculates the sine of an angle.
Calculates the cosine of an angle.
Calculates the tangent of an angle.
The angle from a point 15 meters from
a building to the highest floor of the
building is 45°. How tall is the building?
45 15
15
View point
APPLICATIONS:
Trigonometric functions are useful in mathematics and various engineering
calculations. They are often used in astronomical observations, civil
engineering and in calculations involving electrical circuits, as well as in
calculations for physics such as parabolic motion and wave motion.
Trigonometric Functions
34
<Example 2>
Find the length of the side of the
following triangle.
<Angle setting " " (DEG)>
30
20
A
B
B
CC
A
ax
y
2
17
b
a = 20 sin 30
b = 20 cos 30
x = tan17
2
y = sin17
2
Operation Display
(DRG)
(DEG)
Trigonometric Functions
NOTE:
In EL-W506T,
Use instead of .
SET UP
35
20 30
20
17
2
30
17
2
Trigonometric Functions
36
<Example 3>
The instantaneous value V of the AC voltage is expressed by the
equation below.
Find the instantaneous value of the AC voltage at time t = 2.000,
2.002, 2.004, 2.008, 2.012, 2.016
<Angle setting "rad" (RAD)>
Operation Display
(DRG)
(RAD)
2
2
2
100
2.00060
Root mean square value Ve = 100 [V]
Frequency f = 60 [Hz]
V = 2Vesin(2 ft) [V]
Trigonometric Functions
NOTE:
In EL-W506T,
Use instead of .
SET UP
37
4
8
12
16
Trigonometric Functions
38
<Example 1>
Operation Display
θ
=
sin
-1 b
a
θ
= cos
-1 c
a
θ
= tan-1 b
c
c
a
b
θ
100
80
Arc Trigonometric Functions
Arc trigonometric functions, the inverse of
trigonometric functions, are used to determine an
angle from ratios of a right triangle.
The combinations of the three sides are sin-1, cos-1,
and tan-1. Their relations are;
(arc sine) Determines an angle based on the ratio
b/a of two sides of a right triangle.
(arc cosine) Determines an angle based on the ratio
c/a for two sides of a right triangle.
(arc tangent) Determines an angle based on the
ratio b/c for two sides of a right triangle.
At what angle should an airplane climb in order
to climb 80 meters in 100 meters?
<Angle setting " " (DEG)>
(DRG)
(DEG)
NOTE:
In EL-W506T,
Use instead of .
SET UP
39
Hyperbolic Functions
The hyperbolic function is defined by using natural exponents in
trigonometric functions.
Arc hyperbolic functions are defined by using natural logarithms in
trigonometric functions.
APPLICATIONS:
Hyperbolic and arc hyperbolic functions are very useful in electrical
engineering and physics.
40
The length L of rope that creates this sag is
expressed by the following equation.
When a = 0.846 and b = 2, find the rope sag D and
the rope length L.
* The value a is called the catenary factor, and
determines the shape of the curve.
D = acosh 2aa
b-
L = 2asinh 2a
b
Operation Display
0.846
0.846
0.846
2 2
<Example 1>
The curve that forms when a rope hangs from two fixed points is called a "catenary",
and the sag D of the rope can be expressed using a hyperbolic function.
0.846
0.846
2
2 2
Catenary
b (width between fixed points)
Sag D
Hyperbolic Functions
41
A = 6.82
B = 1.44
(A and B are constants determined by a raindrop diameter of 1 mm and the
physical properties of air.)
Find the fall velocity at time t = 0, 1, 2, 5, 10, 15.
*As the calculations are continued, v approaches 6.82. Therefore, the
velocity of a raindrop is about 6.82 m/s (24.6 km/h) when it reaches the ground.
Note: The fall distance from time t = 0 to 15 [s] is given by the following equation.
(Calculation of integral)
v = AtanhBt [m/s]
1. Enter Atanh(BX) (use the characters A, B, and X to enter)
[DEG mode]
Answer
Operation Display
<Example 2>
A drop of rain falls against an air resistance proportional to the square of the fall
velocity. The velocity v at time t seconds after the start of the fall is given by the
following equation:
B X
(6.82tanh(1.44x))dx = 99.01718518
15
0
x
0
1
2
5
10
15
v
0
6.0950185
6.777153851
6.819992397
6.82
6.82
Hyperbolic Functions
NOTE:
This example is solved by the Simulation calculation (ALGB)
.
EL-W506T has the Simulation calculation (ALGB)
.
This function is convenient for repeated calculations using varying values of X.
(This example is for EL-W506T only.)
42
2. Enter the Simulation calculation.
<Simulation calculation>
For example,
4. Enter the value of B
3. Enter the value of A
5. Enter the value of X
6. The answer is obtained.
Repeat 2 to 6
(If 6.82 appears, press only the key)
6.82
(If 1.44 appears, press only the key)
1.44
1
Hyperbolic Functions
ALGB
43
Rectangular coordinates
P (x,y)
y
x
o
y
x
y
P (r,
θ
)
x
o
r
Polar coordinates
θ
<Example>
[DEG mode]
Operation Display
7 3
7.6 23.2
Coordinate Conversion
θ
Converts rectangular coordinates to polar coordinates (x, y r,
)
Converts polar coordinates to rectangular coordinates (r, x, y)
θ
Splits data used for dual-variable data input.
Determine the polar coordinates (r, ) when the
rectangular coordinates of Point P are (x = 7, y = 3).
θ
APPLICATIONS:
Coordinate conversion is often used in mathematics and engineering, espe-
cially for impedance calculations in electronics and electrical engineering.
44
Binary, Pental, Octal,
Decimal, and Hexadecimal
Operations (N-Base)
This calculator can perform conversions between numbers expressed in binary, pental, octal,
decimal, and hexadecimal systems. It can also perform the four basic arithmetic operations,
calculations with parentheses and memory calculations using binary, pental, octal, decimal,
and hexadecimal numbers. In addition, the calculator can carry out the logical operations
AND, OR, NOT, NEG, XOR, and XNOR on binary, pental, octal, and hexadecimal numbers.
Conversion is performed on the displayed value when these keys are pressed.
Operation Display
<Example 1> HEX(1AC) BIN PEN OCT DEC
1011 AND 101 = (BIN) DEC
<Example 2> Operation Display
1AC
1011
101
Converts to the binary system.
"BIN" appears.
Converts to the pental system.
"PEN" appears.
Converts to the octal system.
"OCT" appears.
Converts to the hexadecimal system.
"HEX" appears.
Converts to the decimal system.
"BIN", "PEN", "OCT", and "HEX"
disappear from the display.
45
d/dx x
Differentiation calculation
<Example 1>
D = - 24
P
25920
If the demand curve is expressed by
find the price elasticity of demand when P=360 (D=48).
*Price elasticity of demand:
A value that indicates how sensitive demand is to changes of price.
P
dP
D
dD
D
P
dP
dD
Price elasticity
of demand
Rate of demand
change
Rate of price
change
= = =
- - -
Operation Display
48
24
360
360
25920
d/dx
x
Find the following value when P=360 and D=48.
x = 360
- 24
x
25920
D
Pd( )
dx
-
(This example is for EL-W506T only.)
46
<Example 2>
The semicircle above is given by the equation
Find the slope of the tangent AB at point B (-1/2, 3/2) on the semicircle.
x = -
dx 2
1
d( )
Operation Display
1
1 2
d/dx
x
- x2
1
y =
- x2
1
A
O
120
1
B(-1/2, 3/2)
Differentiation calculation
(This example is for EL-W506T only.)
d/dx x
47
<Example 1>
Integration calculation
(1)
Let the demand curve of the overall market be D = 3000 - 10P, the supply curve be
S = 20P, the equilibrium price be 100, and the equilibrium output be 2000.
(1) Find the consumer surplus of the overall market.
(2) Find the producer surplus of the overall market.
(3) Find the total surplus of the overall market.
(3000 - 10x - 2000) dx
Operation Display
100
103000
2000
0
x
(2)
100
202000
x
100
0
(2000 - 20x) dx
100
0
(3000 - 10x - 20x) dx
100
0
(This example is for EL-W506T only.)
dx x
dx
0
dx
48
(3)
10
20
3000
x
Integration calculation dx x
x
1000
dx
49
<Example 2>
Operation Display
1
1
- x2
1
y =
- x2
1
y =
O 1
dx
The fan shaped curve at left is given by the equation
Find the area of the fan shape with radius 1 and central
angle 90 .
- x2
1
1
dx
0
Integration calculation dx x
(This example is for EL-W506T only.)
0
dx
x
50
<Example 1>
To produce one unit of product X, 3 kg of material A and 1 kg of material B
are required.
To product one unit of product Y, 1 kg of material A and 2 kg of material B
are required.
There are 9 kg of A and 8 kg of B in stock.
If the selling price of product X is 300 dollars/unit and the selling price of
product Y is 200 dollars/unit, how many units of product X and how many
units of product Y should be produced in order to maximize sales K?
(Do not include the cost of materials and production or other expenses)
If the quantities produced of each product are x and y, the sales K can be
expressed as
K = 3x + 2y
The following relations hold for the quantities in stock:
3x + y 9
x + 2y 8
x 0, y 0
Based on these conditions, find the values of x and y that maximize sales K.
The conditions can be graphed as shown above.
The sales K is a maximum where the line K = 3x + 2y passes through the
intersection point P of lines 3x + y = 9 and x + 2y = 8.
The intersection point P can be obtained from the following simultaneous equations:
3x + y = 9
x + 2y = 8
Solving these gives
x = 2, y = 3
and thus the maximum value of the sales K is
K = 3 x 2 + 2 x 3 = 12 (x 100) dollars (when x = 2 units and y = 3 units)
9
4
2
0 3
y
P
K=3x+2y
x
8
K
Simultaneous Calculation
(This example is for EL-W506T only.)
51
(1) Solve the following simultaneous equations.
3x + y = 9
x + 2y = 8
K = 3x + 2y
(2) Use the result of (1) to find the following value.
<Equation mode>
Set the mode to Equation
Set the mode to Normal
<Simultaneous linear equations
in two unknowns>
Enter the coefficients
a1 = 3 , b1 = 1 , c1 = 9
a2 = 1 , b2 = 2 , c2 = 8
(1)
(2)
Operation
3 1 9
1 2 8
3 2 23
Simultaneous Calculation
Display
(EQUATION)
(NORMAL)
(2-VLE)
52
When ethanol C2H5OH is completely combusted, carbon dioxide CO2 and
water H2O are created.
The chemical reaction formula of this reaction is expressed as follows:
x C2H5OH + 3O2 y CO2 + z H2O
Find the values of x, y, and z to complete the chemical reaction formula.
The numbers of C, H, and O before and after the reaction are equal, hence
Number of C: 2x = y
Number of H: 5x + x = 2z
Number of O: x + 6 = 2y+ z
As such, the following simultaneous equations are obtained:
2x - y + = 0
6x - 2z = 0
x - 2y - z = - 6
Solving these gives
x = 1, y = 2, z = 3
and the chemical reaction formula is
C2H5OH + 3O2 2CO2 + 3H2O
<Equation mode>
<Simultaneous linear equations
in three unknowns>
Enter the coefficients
Set the mode to Equation
a1 = 2 , b1 = -1 , c1 = 0 , d1 = 0
a2 = 6 , b2 = 0 , c2 = -2 , d2 = 0
a3 = 1 , b3 = -2 , c3 = -1 , d3 = -6
(3-VLE)
Operation
2 1 0 0
6 0 2 0
12 1 6
<Example 2>
Simultaneous Calculation
Display
(This example is for EL-W506T only.)
(EQUATION)
53
<Example 1>
Let the hydrochloric acid concentration be c (= 1.0 x 10
-8
mol / ), and the
hydrogen ion concentration be x.
(1)
Solve the following quadratic equation to find the hydrogen ion concentration x:
x
2
- cx - Kw = 0
where
Kw = 1.0 x 10
-14
[mol / ] (ionic product of water)
(2) Use the result of (1) to find the pH (= - log x) of hydrochloric acid.
Save constants
(1)
pH = - log x (x>0)
1.0 14
(NORMAL)
Operation
1.0 8
B
C
Polynomial equation
Display
(This example is for EL-W506T only.)
54
Set the mode to Equation
<Quadratic equation>
(EQUATION) (QUAD)
0.000000105
Solve the equation (enter coefficients a, b, c)
1
C
B
Set the mode to Normal
(NORMAL)
(2)
Polynomial equation
55
<Example 2>
Let the acetic acid concentration be c (= 0.1 mol / ), and the hydrogen ion
concentration be
x
.
(1)
Solve the following quadratic equation to find the hydrogen ion concentration x:
x
3
+ Kax
2
- (cKa + Kw)x - KaKw = 0
where
Ka = 2.75 x 10
-5
[mol / ] (
ionization equilibrium constant of acetic acid
)
Kw = 1.0 x 10
-14
[mol / ] (ionic product of water)
(2) Use the result of (1) to find the pH (= - log x) of acetic acid.
Save constants
(1)
pH = - log x (x>0)
2.75 5
Operation
1.0
0.1
14 B
C
Display
Polynomial equation
(This example is for EL-W506T only.)
(NORMAL)
56
Set the mode to Equation
<Cubic equation>
0.001644619
Solve the equation (enter coefficients a, b, c, d)
1
C
B
B
Set the mode to Normal
(2)
Polynomial equation
(EQUATION) (CUBIC)
(NORMAL)
57
An AC sine wave voltage of 100 V, 50 Hz is applied to a circuit consisting of
a
resistor (R = 250 ) and capacitor (C = 20 x 10-6F) connected in parallel.
Find the impedance of this circuit.
Circuit impedance = Value of polar coordinate r
<Complex mode>
(Rectangular coordinates)
(Angle units: RAD)
(Polar coordinates)
(COMPLEX)
(DRG)
(RAD)
Operation
1
2
1 250
50
20 6 i
<Example 1>
Let R = 250, C = 20 x 10-6, and f = 50.
If the complex number Z = 1 ((1 R) + 2 fCi),
find the value of the complex number Z and the values of r.
Display
Complex Calculation
(This example is for EL-W506T only.)
i
SET UP
58
An AC sine wave voltage of 100V, 60Hz is applied to a circuit consisting of a resistor
(R = 120 ), coil (L = 4 H), and capacitor (C = 3 x 10
-6
F) connected in series.
(1) Find the impedance of the circuit.
(2) Find the phase difference between the current and the voltage.
Circuit impedance = Value of polar coordinate r
(rectangular coordinates)
(Angle units: DEG)
(Polar coordinates)
(DEG)
Operation
120
3
2
2
60
1
6
4
60
i
i
Phase difference = Polar coordinate
Let R = 120, L = 4, C = 3 x 10-6, and f = 60. If the complex number
Z = R + 2 fLi + 1 (2 fCi), find the value of the complex number Z and
the values of r and .
Display
Complex Calculation
<Example 2>
i
(This example is for EL-W506T only.)
<Complex mode>
(COMPLEX)
(DRG)
SET UP
59
<Example 1>
Operation Display
Data table 1
No.
Score
No. of pupils
1 2 3 4 5 6 7 8
30 40 50 60 70 80 90 100
2 4 5 7 12 10 8 2
Select single-variable statistics mode
(The input table is displayed.)
Statistics Functions
The statistics function is excellent for analyzing qualities of an event. Though primarily
used for engineering and mathematics, the function is also applied to nearly all other
fields including economics and medicine.
DATA INPUT FOR 1-VARIABLE STATISTICS
Close/display the input table.
Splits data used for X and FRQ data input (or X, Y, and FRQ data input).
Here is a table of examination results. Input this data
for analysis.
Insert a line in the input table for data insertion.
.
.
.
30 2
100 2
DATA INS-D
DATA
INS-D
STAT
STAT
Statistical values can be calculated from the STAT menu.
60
NOTE:
1. Sample data refers to data selected randomly from the population.
2. Standard deviation of samples is determined by the sample data
shift from an average value.
3. Standard deviation for the population is standard deviation when
the sample data is deemed a population (full data).
Let’s check the results based on the previous data.
“ANS” FOR 1-VARIABLE STATISTICS
= 50 (
number of input data
)
= 69 (average value)
= 17.7568613 (standard deviation)
= 17.5783958 (standard deviation of the population)
= 3450 (
sum of the data
)
Operation Display
DATA (Close the input table.)
STAT
Calculates Statistical values.
For examples,
APPLICATIONS:
Single-variable statistical calculations are used in a broad range of
fields, including engineering, business, and economics. They are
most often applied to analysis in atmospheric observations and
physics experiments, as well as for quality control in factories.
61
<Example 2>
When the weight of a calculator was measured,
the results at left were obtained.
Find the average and standard deviation of the weight.
Average = 96.884
Operation
97.27
No
1 97.27
96.83
96.65
2
3
4 96.90
5 96.77
Weight [g]
96.83
96.77
...
Standard deviation = 0.209723627
Display
DATA
DATA (Close the input table.)
STAT
(Display the input table.)
62
<Example 3>
Operation Display
• To
insert a line in front of the cursor position,
press .
• To delete the entire line where cursor is positioned,
press .
Data table 2
X: 30, 40, 40, 50
X: 30, 45, 45, 45, 60
40 2
50
30
DATA CORRECTION
Move the cursor ( ) to the data that you want to correct,
enter the numeric value, and press .
INS-D
45 3
60
INS-D
DATA
63
<Example 4>
Data table 3
Operation Display
6.2 13
8.2 7
The table below summarizes the dates in April when cherry
blossoms bloom, and the average temperature for March in
that same area. Determine basic statistical quantities for
data X and data Y based on the data table.
Select two-variable statistics mode and
linear regression calculation in sub-mode.
(The input table is displayed.)
2010 2011 2012 2013 2014 2015 2016 2017
6.2 7.0 6.8 8.7 7.9 6.5 6.1 8.2
13 9 11 5 7 12 15 7
Year
x Average temperature
y Date blossoms bloom
.
.
.
DATA INPUT FOR 2-VARIABLE STATISTICS
64
“ANS” FOR 2-VARIABLE STATISTICS
In addition to the 1-variable statistic keys, the following keys have been added for
calculating 2-variable statistics.
NOTE:
The codes for basic statistical quantities of sample data x and their meanings
are the same as those for single-variable statistical calculations.
Let’ s check the results based on the previous data.
= 8 (Total count of data)
= 7.175 (Average for data x)
= 0.973579551 (Standard deviation for data x)
= 0.91070028 (Standard deviation of the population for data x)
= 57.4 (Sum of data x)
= 418.48 (Sum of data x raised to the second power)
= 9.875 (Average for data y)
= 3.44082631 (Standard deviation for data y)
= 3.2185983 (Standard deviation of the population for data y)
= 79 (Sum of data y)
= 863 (Sum of data y raised to the second power)
= 544.1 (Sum of the product of data x and data y)
Operation Display
DATA (Close the input table.)
STAT
Calculates Statistical values.
For examples,
.
.
.
.
.
.
65
<Example 5>
When a weight was hung on a spring, the following
relation was obtained for the extension of the spring and
the force applied to the spring. Use linear regression to
find the coefficients a and b of the relational expression
y = a + bx, and the correlation coefficient r.
Operation
0.028 0.20
0.073
0.207
...
Spring extension x [m]
Force F [N]
0.028
0.073
0.207
0.118
0.16
0.2
0.39
1
0.6
0.77
0.39
1.00
Display
DATA
DATA (Close the input table.)
STAT
66
<Example 6>
The hot water inside an electric pot is maintained at 92 C.
When a thermometer is placed in this hot water, the values indicated by the
thermometer at times x and the differences y between these values and the
temperature of the hot water are shown below. Using Euler's exponential
regression, find the formula that expresses the relation between each time x
and the temperature difference y.
Operation
0 67
4
40
...
37
1
Temperature difference y [ C] from liquid
(Room temperature 25 C, hot water temperature 92 C)
Time x [S]
Thermometer temperature [ C]
0
4
8
12
16
20
24
28
32
36
40
25
55
71
79
85
88
90
90
91
91
91
67
37
21
13
7
4
2
2
1
1
1
e: Napier's constant
e=2.718281828…
Display
When x and y are in the following relationship, use Euler's exponential regression to find the
coefficients a and b of the relational expression y = aebx, and the correlation coefficient r.
0
x y
4
8
12
16
20
24
28
32
36
40
67
37
21
13
7
4
2
2
1
1
1
Correlation coefficient
r 1
Correlation exists
Fig. 1 Fig. 2 Fig. 3
Correlation exists No correlation
r -1 r = 0
y
x
xxxxxxxxxx
xxx
xx
xxx
xx
x
x
x
x
xx
x
x
x
x
x
x
x x
y y
DATA
DATA (Close the input table.)
STAT
67
Matrix Calculation
In a certain year (year 0), the share of manufacturer A is 10% and the
share of manufacturer B is 90%. Manufacturer A then releases a new
product, and each following year it maintains 90% of the share ak it had
the previous year (year k), and usurps 20% of the share bk of
manufacturer B.
Find the transition matrix for this process and the shares of
manufacturers A and B after 2 years.
The share of each company after one year is expressed as follows using
a0 and b0.
Thus, a1 and b1 are
The transition matrix is
: This is equal to matA2.
Answer
0.9 0.2
0.1 0.8
A =
0.83 0.34
0.17 0.66
A
2
=
a1 = 0.9a0 + 0.2b0
b1 = (1-0.9)a0 + (1-0.2)b0
Expressing a2 and b2 using a0 and b0 gives
a2 = 0.9(0.9a0 + 0.2b0) + 0.2(0.1a0 + 0.8b0)
= (0.9 x 0.9 + 0.2 x 0.1)a0 + (0.9 x 0.2 + 0.2 x 0.8)b0
= 0.83a0 + 0.34b0
b2 = 0.1(0.9a0 + 0.2b0) + 0.8(0.1a0 + 0.8b0)
= (0.1 x 0.9 + 0.8 x 0.1)a0 + (0.1 x 0.2 + 0.8 x 0.8)b0
= 0.17a0 + 0.66b0
In the same way, after two years
a2 = 0.9a1 + 0.2b1
b2 = 0.1a1 + 0.8b1
a1 = 0.9a0 + 0.2b0
b1 = 0.1a0 + 0.8b0
In summary,
a2 = 0.83a0 + 0.34b0
b2 = 0.17a0 + 0.66b0
Manufacturer A
Share 10%
20%
10%
Manufacturer B
Share 90%
<Example>
(This example is for EL-W506T only.)
68
<Calculate the square>
(MATRIX)
Calculate
Enter matA
Set the mode to Matrix
Matrix mode
<2 x 2 Matrix>
<0: Save to matA>
<Enter numeric values>
Operation Display
(MATRIX)
(STORE)
(EDIT)
0.9 0.2
0.1 0.8
Matrix Calculation
Find the shares of manufacturers A and B after 2 years.
0.83 10
0.34 90
0.17 10
0.66 90
(A: 38.9%)
(B: 61.1%)
69 SHARP CORPORATION (FEB. 2019)
19BSC39E1
… 51 …

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