Chapter 3 Solve, Differential/Quadratic Differential, Integration, Maximum/Minimum Value, and Σ Calculations 3-1 3-2 3-3 3-4 3-5 3-6 3-7 Function Analysis Menu Solve Calculations Differential Calculations Quadratic Differential Calculations Integration Calculations Maximum/Minimum Value Calculations Σ Calculations
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 Σ calculations. When the option menu is on the display, press 4 (CALC) to display the function analysis menu. AK4 (CALC) 1 2 3 4 5 6 1 (Solve) ..... Used in Solve calculations 2 (d/dx) ........ Used in differential calculations 3 (d2/dx 2) ..... Used in quadratic differential calculations 4 (∫dx) ..........
-2 Solve Calculations P.64 To solve calculations, first display the function analysis menu, and then input the values shown in the formula below to determine root x values in the function f(x). 1(Solve) f(x), n,a,b) Initial estimate value Upper limit Lower limit 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.
3-2 Solve Calculations Input initial estimated value n. b, Input lower limit a and upper limit b. a,b) 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 closing parenthesis, lower limit a and upper limit b can be omitted. • Roots obtained using Solve may include errors.
3-3 Differential Calculations P.64 • To perform differential calculations, first display the function analysis menu, and then input the values shown in the formula below.
3-3 Differential Calculations This average, which is called the central difference, is expressed as: 1 f (a + ∆x) – f (a) f (a) – f (a – ∆x) f '(a) = –– ––––––––––––– + ––––––––––––– 2 ∆x ∆x f (a + ∆x) – f (a – ∆x) = ––––––––––––––––– 2∆x u To perform a differential calculation Example To determine the derivative at point x = 3 for the function y = x3 + 4 x2 + x – 6, when the increase/decrease of x is defined as ∆ x = 1E – 5 Input the function f(x).
Differential Calculations 3-3 k Applications of Differential Calculations • Differentials can be added, subtracted, multiplied and divided with each other. Example d d ––– f (a) = f '(a), ––– g (a) = g'(a) dx dx Therefore: f '(a) + g'(a), f '(a) × g'(a) • Differential results can be used in addition, subtraction, multiplication, and division, 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.
3-4 Quadratic Differential Calculations P.64 After displaying the function analysis menu, you can input quadratic differentials using either of the two following formats. 3(d 2 /dx 2 ) f(x),a, n) Final boundary ( n = 1 to 15) Differential coefficient point d2 d2 –––2 ( f (x), a, n) ⇒ –––2 f (a) dx dx Quadratic differential calculations produce an approximate differential value using the following second order differential formula, which is based on Newton's polynomial interpretation.
Quadratic Differential Calculations 3-4 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 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 n and the closing parenthesis can be omitted.
3-5 Integration Calculations To perform integration calculations, first display the function analysis menu, and then input the values shown in the formula below. P.64 4(∫dx) f(x) , a , b , n ) Number of Divisions (value for n in N = 2 n, n is an integer from 1 through 9) End Point Start Point ∫( f (x), a, b, n) ⇒ ∫ b a f (x)dx, N = 2n Area of ∫ b a f (x)dx is calculated N number of divisions Integration calculations are performed by applying Simpson’s Rule for the f (x) function you input.
Integration Calculations 3-5 u To perform an integration calculation Example To perform the integration calculation for the function ∫ 5 1 (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.
3-5 Integration Calculations • Note that you cannot use a Solve, differential, quadratic differential, integration, maximum/minimum value or Σ calculation expression inside of an integration calculation term. • Pressing A during 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.
3-6 Maximum/Minimum Value Calculations P.64 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.
3-6 Maximum/Minimum Value Calculations 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 . 1 2 3 4 5 6 a,d, Input the precision n = 6. g) w • In the function f(x), only X can be used as a variable in expressions.
3-7 Σ Calculations To perform Σ calculations, first display the function analysis menu, and then input the values shown in the formula below. 6(g)3(Σ() a k , k , α , β , n ) Distance between partitions Last term of sequence {ak} Initial term of sequence {ak} Variable used by sequence { ak} Σ ( a k, k, α, β, n) = β Σa k k=α Σ calculation is the calculation of the partial sum of sequence { a k}, using the following formula. S = aα + aα +1 +........
3-7 Σ Calculations w • You can use only one variable in the function for input sequence {ak}. • Input integers only for the initial term of sequence {a k} and last term of sequence {a k}. • Input of n and the closing parentheses can be omitted. If you omit n, the calculator automatically uses n = 1. k Σ Calculation Applications u Arithmetic operations using Σ calculation expressions Expressions: n n k=1 k=1 Sn = Σ ak, Tn = Σ bk Possible operations: Sn + Tn , Sn – Tn, etc.
