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

64
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.
uu
uu
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.
uu
uu
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,
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Initial estimate value Upper limit
Lower limit
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Input initial estimated value n.
b,
Input lower limit aand upper limit b.
a,b)
w
• In the function f(x), only X can be used as a variable in expressions. Other vari-
ables (A through Z, r, θ ) 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

67
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 x
Point for which you want to determine the derivative
d
d/dx ( f (x), a, ∆x) ⇒––– f(a)
dx
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

68
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
∆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.
d,
Input ∆x, 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 vari-
ables (A through Z, r, θ ) 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

69
kk
kk
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.
dd
––– f (a) = f '(a), ––– g (a) = g'(a)
dx dx
Therefore:
f '(a) + g'(a), f '(a) ×g'(a)
d
––– (sinx+ cosx, sin0.5)
dx
Differential Calculations 3 - 3

70
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.
1
h= ––––
5m
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.
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,
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d2d2
––– ( f (x), a, n) ⇒––– f(a)
dx2dx2
Final boundary ( n = 1 to 15)
Differential coefficient point
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Input 3 as point a, which is 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 vari-
ables (A through Z, r, θ ) 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.
kk
kk
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
d2
––– (sin x+ cos x, sin 0.5)
dx2
Quadratic Differential Calculations 3 - 4

72
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) ⇒
∫
a
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 n in N = 2
n
,
n is an integer from 1 through 9)
End Point
Start Point
Area of
∫
a
bf(x)dx is calculated
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73
uu
uu
uTo perform an integration calculation
Example To perform the integration calculation for the function
∫
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 number of divisions.
g)
w
• In the function f(x), only X can be used as a variable in expressions. Other vari-
ables (A through Z, r, θ ) are treated as constants, and the value currently as-
signed to that variable is applied during the calculation.
• Input of nand the closing parenthesis can be omitted. If you omit n, the calculator
automatically selects the most appropriate value.
• Calculation precision is theoretically ±1 at the least significant digit of the dis-
played result.
kk
kk
kApplication of Integration Calculation
• Integrals can be used in addition, subtraction, multiplication and division.
Example
∫
a
bf(x) dx +
∫
c
dg(x) dx
• Integration results can be used in addition, subtraction, multiplication and divi-
sion, in functions.
Example 2 ×
∫
a
bf(x) dx, log (
∫
a
bf(x) dx)
• Functions can be used in any of the terms (f(x), a, b, n)of an integral.
Example
∫
cos 0.5
(sin x+ cos x) dx =
∫
(sin x+ cos x, sin 0.5, cos 0.5, 5)
sin 0.5
Integration Calculations 3 - 5

74
• Note that you cannot use a Solve, differential, quadratic differential, integration,
maximum/minimum value or Σcalculation expression inside of an integration cal-
culation term.
• Pressing Aduring calculation of an integral (while the cursor is not shown on
the display) interrupts the calculation.
• Always perform trigonometric integrations using radians (Rad Mode) as the
angle unit.
• This unit utilizes Simpson’s rule for integration calculation. As the number of
significant digits is increased, more calculation time is required. In some cases,
calculation results may be erroneous even after considerable time is spent per-
forming a calculation. In particular, when significant digits are less than 1, an
ERROR (Ma ERROR) sometimes occurs.
• Integration involving certain types of functions or ranges can result in relatively
large errors being generated in the values produced.
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 - 5 Integration Calculations
Positive
part (S)
Negative part (S)

75
3-6 Maximum/Minimum Value Calculations
After displaying the function analysis menu, you can input maximum/minimum cal-
culations 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
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Precision ( n = 1 ~ 9)
End point of interval
Start point of interval
Precision ( n = 1 ~ 9)
End point of interval
Start point of interval
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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 vari-
ables (A through Z, r, θ ) are treated as constants, and the value currently as-
signed to that variable is applied during the calculation.
• Input of nand the closing parenthesis following the precision value can be omit-
ted.
• Discontinuous points or sections with drastic fluctuation can adversely affect pre-
cision 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/mini-
mum 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 Ma 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 (Arg ERROR).
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3 - 6 Maximum/Minimum Value Calculations

77
3-7 ΣCalculations
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 follow-
ing formula.
kk
kk
kExample ΣCalculation
Example To calculate the following:
Use n= 1 as 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 {ak} and last term of sequence {ak}.
c,g,
Input n.
b)
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
6
Σ(k2– 3k+ 5)
k = 2

78
w
• You can use only one variable in the function for input sequence {ak}.
• Input integers only for the initial term of sequence {ak} and last term of sequence
{ak}.
• Input of nand the closing parentheses can be omitted. If you omit n, the calcula-
tor automatically uses n= 1.
kk
kk
kΣCalculation Applications
uu
uu
uArithmetic operations using Σcalculation expressions
Expressions:
Possible operations: Sn+ Tn, Sn– Tn, etc.
uu
uu
uArithmetic and function operations using Σcalculation results
2 ×Sn, log (Sn), etc.
uu
uu
uFunction 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 Σcalculation
term.
kk
kk
kΣCalculation Precautions
• Make sure that the value used as the final term β is greater than the value used
as the initial term α . Otherwise, an Ma ERROR will occur.
• To interrupt an ongoing Σcalculation (indicated when the cursor is not on the dis-
play), press the Akey.
nn
Sn= Σak, Tn= Σbk
k = 1k = 1
3 - 7 ΣCalculations