Casio FX-9750G User manual

Solve, Differential/Quadratic

Differential, Integration,

Maximum/Minimum Value,

and ΣCalculations

3-1 Function Analysis Menu

3-2 Solve Calculations

3-3 Differential Calculations

3-4 Quadratic Differential Calculations

3-5 Integration Calculations

3-6 Maximum/Minimum Value Calculations

3-7 ΣCalculations

Chapter

3-1 Function Analysis Menu

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.

AK4(CALC)

1(Solve) ..... Used in Solve calculations

2(d/dx)........ Used in differential calculations

3(d2/dx2) ..... Used in quadratic differential calculations

4(∫dx).......... Used in integration calculations

6(g) ........... Previous menu

6(g)

1(FMin) ...... Used in minimum calculations

2(FMax) ..... Used in maximum calculations

3(Σ() ........... Used in Σcalculations

6(g) ........... Previous menu

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3-2 Solve Calculations

To solve calculations, first display the function analysis menu, and then input the

values shown in the formula below to determine root xvalues in the function f(x).

1(Solve) f(x),n,a,b)

With Solve calculations, the root of a function is determined using Newton’s method.

uNewton’s Method

This method is based on the assumption that f(x)can be approximated by a linear

expression within a very narrow range.

First, a starting value (predicted value) xois given. Using this starting value as a

base, approximate value x1is obtained, and then the left side and right side calcula-

tion results are compared. Next, approximate value x1is used as the initial value to

calculate the next approximate value x2. This procedure is repeated until the differ-

ence between the left side and right side calculated values is less than some minute

value.

uTo perform solve calculations

Example To calculate the value of root xin the following formula when

the initial estimated value is n = 1, the lower limit is a = 0, and the

upper limit is b = 1:

2x2+ 7x– 9 = 0

Input the function f(x).

AK4(CALC)1(Solve)

cvx+hv-j,

P.64

Initial estimate value Upper limit

Lower limit

123456

Input initial estimated value n.

Input lower limit aand upper limit b.

a,b)

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

ables (A through Z, r,

) are treated as constants, and the value currently as-

signed to that variable is applied during the calculation.

• Input of the closing parenthesis, lower limit aand upper limit bcan be omitted.

• Roots obtained using Solve may include errors.

• Since Solve uses Newton’s method, the following can sometimes occur.

—Certain initial estimated values can make it impossible to obtain roots. In this

case, try inputting another value that you assume to be near the root and per-

form the calculation again.

—The calculator may be unable to obtain a root, even though a root exists.

• Due to certain idiosyncrasies of Newton’s method, roots for the following types

of functions tend to be difficult to calculate.

—Periodic functions (i.e. sin x= 0)

—Functions whose graph produce sharp slopes (i.e. ex= 0, 1/x= 0)

—Discontinuous functions (i.e. x= 0)

• Note that you cannot use a Solve, differential, quadratic differential, integration,

maximum/minimum value or Σcalculation expression inside of a Solve calcula-

tion term.

3 - 2 Solve Calculations

3-3 Differential Calculations

• 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,∆x)

The differentiation for this type of calculation is defined as:

In this definition,

infinitesimal

is replaced by a

sufficiently small

∆x, 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+ ∆x, and of point aand point a– ∆xin function

y= f(x)are as follows:

In the above, ∆y/∆xis called the forward difference, while ∇y/∇xis the backward

difference.To calculate derivatives, the unit takes the average between the value of

∆y/∆xand ∇y/∇x, thereby providing higher precision for derivatives.

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Increase/decrease of

Point for which you want to determine the derivative

d/dx ( f (x), a, ∆x) ⇒––– f(a)

f (a+ ∆x) – f (a)

f '(a) = lim –––––––––––––

∆x∆x→0

f (a+ ∆x) – f (a)∆yf (a) – f (a– ∆x)∇y

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

∆x∆x∆x∇x

f (a+ ∆x) – f (a)

f '(a) –––––––––––––

∆x

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

∆x= 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 ∆x, 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 vari-

ables (A through Z, r,

) are treated as constants, and the value currently as-

signed to that variable is applied during the calculation.

• Input of ∆xand the closing parenthesis can be omitted. If you omit ∆x, the calcu-

lator automatically uses a value for ∆xthat is appropriate for the derivative value

you are trying to determine.

• Discontinuous points or sections with drastic fluctuation can adversely affect pre-

cision or even cause an error.

1f (a+ ∆x) – f (a)f (a) – f (a– ∆x)

f '(a) = –– ––––––––––––– +–––––––––––––

2∆x∆x

f (a+ ∆x) – f (a– ∆x)

= –––––––––––––––––

2∆x

3 - 3 Differential Calculations

kApplications of Differential Calculations

• Differentials can be added, subtracted, multiplied and divided with each other.

Example

• Differential results can be used in addition, subtraction, multiplication, and divi-

sion, and in functions.

Example 2 ×f'(a), log ( f'(a))

• Functions can be used in any of the terms (f(x), a, ∆x) of a differential.

Example

• Note that you cannot use a Solve, differential, quadratic differential, integration,

maximum/minimum value or Σcalculation expression inside of a differential cal-

culation term.

• Pressing Aduring calculation of a differential (while the cursor is not shown

on the display) interrupts the calculation.

• Always perform trigonometric differentials using radians (Rad Mode) as the

angle unit.

––– f (a) = f '(a), ––– g (a) = g'(a)

dx dx

Therefore:

f '(a) + g'(a), f '(a) ×g'(a)

––– (sinx+ cosx, sin0.5)

Differential Calculations 3 - 3

3-4 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 polyno-

mial 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 hcalcu-

lated using the last value of m, and the value of f"(x)based on the value of h

calculated 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,

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d2d2

––– ( f (x), a, n) ⇒––– f(a)

dx2dx2

Final boundary (

= 1 to 15)

Differential coefficient point

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Input 3 as point a, which is 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 vari-

ables (A through Z, r,

) are treated as constants, and the value currently as-

signed 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 pre-

cision 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)

• The result of a quadratic differential calculation can be used in a subsequent

arithmetic or function calculation.

2 ×f ''(a), log ( f ''(a) )

• 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 differ-

ential 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 Ma ERROR.

• You can interrupt an ongoing quadratic differential calculation by pressing the

Akey.

• You should always specify radians (Rad) as the unit of angle unit before per-

forming a quadratic differential calculation using trigonometric functions.

d2d2

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

dx2dx2

––– (sin x+ cos x, sin 0.5)

dx2

Quadratic Differential Calculations 3 - 4

3-5 Integration Calculations

To perform integration calculations, first display the function analysis menu, and then

input the values shown in the formula below.

4(∫dx) f(x),a,b,n)

∫

(f(x), a, b, n) ⇒

∫

bf(x)dx, N = 2n

Nnumber of divisions

Integration calculations are performed by applying Simpson’s Rule for the f(x)func-

tion you input. This method requires that the number divisions be defined as N = 2n,

where the value of nis an integer in the range of 1 through 9. If you do not specify a

value for n, the calculator automatically assigns a value in accordance with the inte-

gration being performed.

As shown in the illustration above, integration calculations are performed by calcu-

lating integral values from athrough 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 illustra-

tion.

* Iff(x) < 0where a <x <b, the surface area calculation produces negative values

(surface area ×– 1).

Number of Divisions (value for

in N = 2

is an integer from 1 through 9)

End Point

Start Point

Area of

∫

bf(x)dx is calculated

P.64

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