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

3-1 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

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.

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→0

f (a+ Ax) – f (a)Ayf (a) – f (a– Ax)∇y

––––––––––––– =––– ,––––––––––––– =–––

AxAxAx∇x

f (a+ Ax) – f (a)

f '(a) –––––––––––––

Increase/decrease of

Point for which you want to determine the derivative

d/dx ( f (x), a, Ax) ⇒––– f(a)

This average, which is called the

central difference

, is expressed as:

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

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.

Input Ax, which is the increase/decrease of x.

bE-f)

•In the function f(x), only X can be used as a variable in expressions. Other

variables (A through Z, r,

) 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

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

––– f(a) = f'(a), ––– g(a) = g'(a)

dx dx

f'(a) + g'(a), f'(a) ×g'(a), etc.

––– (sinx+ cosx, sin0.5), etc.

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.

h= ––––

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.

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 (

= 1 to 15)

Differential coefficient point

Input 3 as point a, which is the differential coefficient point.

Input 6 as n, which is final boundary.

•In the function f(x), only X can be used as a variable in expressions. Other

variables (A through Z, r,

) 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.

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

––– (sin x+ cos x, sin 0.5), etc.

dx2

d2d2

––– f (a) = f ''(a), ––– g (a) = g''(a)

dx2dx2

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) ⇒

∫

bf(x)dx

Simpson’s Rule

4(∫dx) f(x),a,b,n)

∫

(f(x), a, b, n) ⇒

∫

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

∫

bf(x)dx is calculated

Tolerance

End point

Start point

Number of divisions (value for

in N = 2

is an integer from 1 through 9)

End point

Start point

P. 6

uTo perform an integration calculation

Example To perform the integration calculation for the function shown

below, with a tolerance of “tol” = 1E - 4

∫

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,

) 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.

kApplication of Integration Calculation

•Integrals can be used in addition, subtraction, multiplication or division.

∫

bf(x) dx +

∫

dg(x) dx, etc.

•Integration results can be used in addition, subtraction, multiplication or

division, in functions.

2 ×

∫

bf(x) dx, etc. log (

∫

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

•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.

∫

bf(x)dx =

∫

cf(x)dx + (–

∫

bf(x)dx)

Positive part (

)Negative part (

)

(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.

∫

bf(x)dx =

∫

x1f(x)dx +

∫

x2f(x)dx +.....+

∫

f(x)dx

3 - 4 Integration Calculations

Negative part (S)

Positive

part (S)

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