Casio CFX-9850G PLUS User manual

Numerical Calculations
3-1 Before Performing a Calculation
3-2 Differential Calculations
3-3 Quadratic Differential Calculations
3-4 Integration Calculations
3-5 Maximum/Minimum Value Calculations
3-6 Summation (Σ) Calculations
Chapter
3

54
3-1 Before Performing a Calculation
The following describes the items that are available in the menus you use when
performing Solve, differential/ quadratic differential, integration, maximum/
minimum value, and Σcalculations.
When the option menu is on the display, press 4(CALC) to display the function
analysis menu. The items of this menu are used when performing specific types of
calculations.
•{Solve}/{d/dx}/{d2/dx2}/{∫dx} ... {solve}/{differential}/{quadratic differential}/
{integration} calculations
•{FMin}/{FMax}/{Σ(} ... {minimum value}/{maximum value}/{Σ(sigma)}
calculations
Solve calculations
The following is the syntax for using the Solve function in a program.
Solve( f(x),n,a,b)
Upper limit
Lower limit
Initial estimated value
`•There are two different input methods that can be used for Solve calcula-
tions: direct assignment and variable table input.
With the direct assignment method (the one described here), you assign
values directly to variables. This type of input is identical to that used with
the Solve command used in the PRGM Mode.
Variable table input is used with the Solve function in the EQUA Mode. This
input method is recommend for most normal Solve function input.
P. 2 7
P. 394
P. 107

55
3-2 Differential Calculations [OPTN]-[CALC]-[d/dx]
To perform differential calculations, first display the function analysis menu, and
then input the values shown in the formula below.
2(d/dx)f(x),a,Ax)
The differentiation for this type of calculation is defined as:
In this definition, infinitesimal is replaced by a sufficiently small Ax, with the value
in the neighborhood of f ' (a)calculated as:
In order to provide the best precision possible, this unit employs central difference
to perform differential calculations. The following illustrates central difference.
A
A
AA
A
AA
The slopes of point aand point a+ Ax, and of point aand point a– Axin function
y= f(x)are as follows:
In the above, Ay/Axis called the forward difference, while ∇y/∇xis the backward
difference. To calculate derivatives, the unit takes the average between the value
of Ay/Axand ∇y/∇x, thereby providing higher precision for derivatives.
f (a+ Ax) – f (a)
f '(a) = lim –––––––––––––
Ax
Ax→0
f (a+ Ax) – f (a)Ayf (a) – f (a– Ax)∇y
––––––––––––– =––– ,––––––––––––– =–––
AxAxAx∇x
f (a+ Ax) – f (a)
f '(a) –––––––––––––
Ax
Increase/decrease of x
Point for which you want to determine the derivative
d
d/dx ( f (x), a, Ax) ⇒––– f(a)
dx

56
This average, which is called the
central difference , is expressed as:
uu
uu
uTo perform a differential calculation
Example To determine the derivative at point x= 3 for the function
y= x3+ 4x2+ x– 6, when the increase/decrease of xis defined
as AA
AA
Ax= 1E– 5
Input the function f(x).
AK4(CALC)2(d/dx)vMd+evx+v-g,
Input point x= afor which you want to determine the derivative.
d,
Input Ax, which is the increase/decrease of x.
bE-f)
w
•In the function f(x), only X can be used as a variable in expressions. Other
variables (A through Z, r, θ ) are treated as constants, and the value currently
assigned to that variable is applied during the calculation.
•Input of Axand the closing parenthesis can be omitted. If you omit Ax, the
calculator automatically uses a value for Axthat is appropriate for the deriva-
tive value you are trying to determine.
•Discontinuous points or sections with drastic fluctuation can adversely affect
precision or even cause an error.
3 - 2 Differential Calculations
1f (a+ Ax) – f (a)f (a) – f (a– Ax)
f '(a) = –– ––––––––––––– +–––––––––––––
2AxAx
f (a+ Ax) – f (a– Ax)
= –––––––––––––––––
2Ax

57
kk
kk
kApplications of Differential Calculations
•Differentials can be added, subtracted, multiplied or divided with each other.
Therefore:
•Differential results can be used in addition, subtraction, multiplication, and
division, and in functions.
2 ×f'(a), log ( f'(a)), etc.
•Functions can be used in any of the terms ( f(x), a, Ax) of a differential.
•Note that you cannot use a Solve, differential, quadratic differential, integration,
maximum/minimum value or Σcalculation expression inside a differential
calculation term.
•Pressing Aduring calculation of a differential (while the cursor is not shown
on the display) interrupts the calculation.
•Always use radians (Rad Mode) as the angle unit when performing trigono-
metric differentials.
Differential Calculations 3 - 2
dd
––– f(a) = f'(a), ––– g(a) = g'(a)
dx dx
f'(a) + g'(a), f'(a) ×g'(a), etc.
d
––– (sinx+ cosx, sin0.5), etc.
dx

58
3-3 Quadratic Differential Calculations
After displaying the function analysis menu, you can input quadratic differentials
using either of the two following formats.
3(d2/dx2)f(x),a,n)
Quadratic differential calculations produce an approximate differential value using
the following second order differential formula, which is based on Newton's
polynomial interpretation.
–f(x – 2h) + 16 f(x – h) – 30 f (x) + 16 f(x + h) – f(x + 2h)
f''(x)= –––––––––––––––––––––––––––––––––––––––––––––––
12h2
In this expression, values for “sufficiently small increments of x” are sequentially
calculated using the following formula, with the value of mbeing substituted as m
= 1, 2, 3 and so on.
1
h= ––––
5m
The calculation is finished when the value of f"(x)based on the value of h
calculated using the last value of m, and the value of f"(x)based on the value of
hcalculated using the current value of mare identical before the upper ndigit is
reached.
•Normally, you should not input a value for n. It is recommended that you only
input a value for nwhen required for calculation precision.
•Inputting a larger value for ndoes not necessarily produce greater precision.
uu
uu
uTo perform a quadratic differential calculation
Example To determine the quadratic differential coefficient at the point
where x= 3 for the function y= x3+ 4x2+ x– 6
Here we will use a final boundary value of n= 6.
Input the function f(x).
AK4(CALC)3(d2/dx2) vMd+
evx+v-g,
[OPTN]-[CALC]-[d2/dx2]
d2d2
––– ( f (x), a, n) ⇒––– f(a)
dx2dx2
Final boundary ( n = 1 to 15)
Differential coefficient point

59
Input 3 as point a, which is the differential coefficient point.
d,
Input 6 as n, which is final boundary.
g)
w
•In the function f(x), only X can be used as a variable in expressions. Other
variables (A through Z, r, θ ) are treated as constants, and the value currently
assigned to that variable is applied during the calculation.
•Input of the final boundary value nand the closing parenthesis can be omitted.
•Discontinuous points or sections with drastic fluctuation can adversely affect
precision or even cause an error.
kk
kk
kQuadratic Differential Applications
•Arithmetic operations can be performed using two quadratic differentials.
Therefore:
f ''(a)+ g''(a), f ''(a)×g''(a), etc.
•The result of a quadratic differential calculation can be used in a subsequent
arithmetic or function calculation.
2 ×f ''(a), log ( f ''(a) ), etc.
•Functions can be used within the terms ( f(x), a, n) of a quadratic differential
expression.
•Note that you cannot use a Solve, differential, quadratic differential, integration,
maximum/minimum value or Σcalculation expression inside of a quadratic
differential calculation term.
•Use only integers within the range of 1 to 15 for the value of final boundary n.
Use of a value outside this range produces an error.
•You can interrupt an ongoing quadratic differential calculation by pressing the
Akey.
•Always use radians (Rad Mode) as the angle unit when performing trigono-
metric quadratic differentials.
Quadratic Differential Calculations 3 - 3
d2
––– (sin x+ cos x, sin 0.5), etc.
dx2
d2d2
––– f (a) = f ''(a), ––– g (a) = g''(a)
dx2dx2

60
3-4 Integration Calculations [OPTN]-[CALC]-[ ∫ dx]
To perform integration calculations, first display the function analysis menu and
then input the values in one of the formulas shown below.
Gauss-Kronrod Rule
4(∫dx) f(x),a,b,tol )
∫
(f(x), a,b,tol) ⇒
∫
a
bf(x)dx
Simpson’s Rule
4(∫dx) f(x),a,b,n)
∫
(f(x), a, b, n) ⇒
∫
a
bf(x)dx, N = 2n
As shown in the illustration above, integration calculations are performed by
calculating integral values from ato bfor the function y= f(x)where a<x<b, and
f(x)>0*. This in effect calculates the surface area of the shaded area in the
illustration.
*When f(x) < 0on a<x<b, the surface area calculation produces negative
values (surface area below the x-axis).
kChanging Integration Calculation Methods
This calculator can use either Gauss-Kronrod Rule or Simpson’s Rule to perform
integration calculations. To select a method, display the set up screen and select
either “Gaus” (for Gauss-Kronrod Rule) or “Simp” (for Simpson’s Rule) for the
Integration item.
All of the explanations in this manual use Gauss-Kronrod Rule.
Area of
∫
a
bf(x)dx is calculated
Tolerance
End point
Start point
Number of divisions (value for n in N = 2
n
,
n is an integer from 1 through 9)
End point
Start point
P. 6

61
uu
uu
uTo perform an integration calculation
Example To perform the integration calculation for the function shown
below, with a tolerance of “tol” = 1E - 4
∫
1
5(2x2+ 3x+ 4) dx
Input the function f(x).
AK4(CALC)4(∫dx)cvx+dv+e,
Input the start point and end point.
b,f,
Input the tolerance value.
bE-e)w
•In the function f(x), only X can be used as a variable in expressions. Other
variables (A through Z, r, θ ) are treated as constants, and the value currently
assigned to that variable is applied during the calculation.
•Input of “tol” in Gauss-Kronrod Rule, “n” in Simpson’s Rule, and closing
parenthesis with both rules can be omitted. If you omit “tol”, the calculator
automatically uses a value of 1E - 5. In the case of “n”, the calculator automati-
cally selects the most appropriate value.
•Integration calculations can take a long time to complete.
kk
kk
kApplication of Integration Calculation
•Integrals can be used in addition, subtraction, multiplication or division.
∫
a
bf(x) dx +
∫
c
dg(x) dx, etc.
•Integration results can be used in addition, subtraction, multiplication or
division, in functions.
2 ×
∫
a
bf(x) dx, etc. log (
∫
a
bf(x) dx), etc.
•Functions can be used in any of the terms (f(x), a, b, n)of an integral.
∫
cos 0.5
(sin x+ cos x) dx =
∫
(sin x+ cos x, sin 0.5, cos 0.5, 5)
sin 0.5
•Note that you cannot use a Solve, differential, quadratic differential, integration,
maximum/minimum value or Σcalculation expression inside of an integration
calculation term.
Integration Calculations 3 - 4

62
•Pressing Aduring calculation of an integral (while the cursor is not shown
on the display) interrupts the calculation.
•Always use radians (Rad Mode) as the angle unit when performing trigono-
metric integrations.
•Factors such as the type of function being used, positive and negative values
within divisions, and the division where integration is being performed can
cause significant error in integration values and erroneous calculation results.
Note the following points to ensure correct integration values.
(1) When cyclical functions for integration values become positive or negative for
different divisions, perform the calculation for single cycles, or divide between
negative and positive, and then add the results together.
∫
a
bf(x)dx =
∫
a
cf(x)dx + (–
∫
c
bf(x)dx)
Positive part ( S )Negative part ( S )
(2) When minute fluctuations in integration divisions produce large fluctuations in
integration values, calculate the integration divisions separately (divide the
large fluctuation areas into smaller divisions), and then add the results
together.
∫
a
bf(x)dx =
∫
a
x1f(x)dx +
∫
x1
x2f(x)dx +.....+
∫
x4
b
f(x)dx
3 - 4 Integration Calculations
Negative part (S)
Positive
part (S)

63
[OPTN]-[CALC]-[FMin]/[FMax]
3-5 Maximum/Minimum Value Calculations
After displaying the function analysis menu, you can input maximum/minimum
calculations using the formats below, and solve for the maximum and minimum of
a function within interval a<x<b.
uu
uu
uMinimum Value
6(g)1(FMin) f(x),a,b,n)
uu
uu
uMaximum Value
6(g)2(FMax) f(x),a,b,n)
uu
uu
uTo perform maximum/minimum value calculations
Example 1 To determine the minimum value for the interval defined by start
point a= 0 and end point b= 3, with a precision of n= 6 for the
function y= x2– 4x+ 9
Input f(x).
AK4(CALC)6(g)1(FMin) vx-ev+j,
Input the interval a= 0, b= 3.
a,d,
Input the precision n= 6.
g)
w
Precision ( n = 1 to 9)
End point of interval
Start point of interval
Precision ( n = 1 to 9)
End point of interval
Start point of interval

64
Example 2 To determine the maximum value for the interval defined by start
point a= 0 and end point b= 3, with a precision of n= 6 for the
function y= –x2+ 2x+ 2
Input f(x).
AK4(CALC)6(g)2(FMax) -vx+cv+c,
Input the interval a= 0,b= 3.
a,d,
Input the precision n= 6.
g)
w
•In the function f(x), only X can be used as a variable in expressions. Other
variables (A through Z, r, θ ) are treated as constants, and the value currently
assigned to that variable is applied during the calculation.
•Input of nand the closing parenthesis following the precision value can be
omitted.
•Discontinuous points or sections with drastic fluctuation can adversely affect
precision or even cause an error.
•Note that you cannot use a Solve, differential, quadratic differential, integration,
maximum/minimum value or Σcalculation expression inside of a maximum/
minimum calculation term.
•Inputting a larger value for nincreases the precision of the calculation, but it
also increases the amount of time required to perform the calculation.
•The value you input for the end point of the interval (b) must be greater than
the value you input for the start point (a). Otherwise an error is generated.
•You can interrupt an ongoing maximum/minimum calculation by pressing the
Akey.
•You can input an integer in the range of 1 to 9 for the value of n. Using any
value outside this range causes an error.
3 - 5 Maximum/Minimum Value Calculations

65
6
Σ(k2– 3k+ 5)
k = 2
3-6 Summation (Σ) Calculations [OPTN]-[CALC]-[Σ(]
To perform Σcalculations, first display the function analysis menu, and then input
the values shown in the formula below.
6(g)3(Σ() ak,k, α , β ,n)
Σcalculation is the calculation of the partial sum of sequence ak, using the
following formula.
kk
kk
kExample ΣCalculation
Example To calculate the following:
Use n= 1as the distance between partitions.
Input sequence ak.
AK4(CALC)6(g)3(Σ()aKx-daK+f,
Input variable used by sequence ak.
aK,
Input the initial term of sequence akand last term of sequence ak.
c,g,
Input n.
b)
w
Distance between partitions
Last term of sequence ak
Initial term of sequence ak
Variable used by sequence ak
β
Σ(ak, k,
α , β , n) ⇒Σak
k = α
β
S = a α + a α
+1 +........+ a
β = Σak
k = α

66
3 - 6 Summation (Σ)Calculations
•You can use only one variable in the function for input sequence ak.
•Input integers only for the initial term of sequence akand last term of sequence
ak.
•Input of nand the closing parentheses can be omitted. If you omit n, the
calculator automatically uses n= 1.
kk
kk
kΣCalculation Applications
•Arithmetic operations using Σcalculation expressions
Expressions:
Possible operations: Sn+ Tn, Sn– Tn, etc.
•Arithmetic and function operations using Σcalculation results
2 ×Sn, log (Sn), etc.
•Function operations using Σcalculation terms (ak, k)
Σ(sink, k, 1, 5), etc.
•Note that you cannot use a Solve, differential, quadratic differential,
integration, maximum/minimum value or Σcalculation expression inside of a Σ
calculation term.
•Make sure that the value used as the final term β is greater than the value
used as the initial term α . Otherwise, an error will occur.
•To interrupt an ongoing Σcalculation (indicated when the cursor is not on the
display), press the Akey.
nn
Sn= Σak, Tn= Σbk
k = 1k = 1