Chapter 3 Numerical Calculations 3-1 3-2 3-3 3-4 3-5 3-6 Before Performing a Calculation Differential Calculations Quadratic Differential Calculations Integration Calculations Maximum/Minimum Value Calculations Summation (Σ) Calculations
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 Σ calculations. P.27 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} ...
-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.
3-2 Differential Calculations This average, which is called the central difference, is expressed as: 1 f (a + Ax) – f (a) f (a) – f (a – Ax) f '(a) = –– ––––––––––––– + ––––––––––––– 2 Ax Ax f (a + Ax) – f (a – Ax) = ––––––––––––––––– 2Ax uTo 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 Ax = 1E – 5 Input the function f(x).
Differential Calculations 3-2 k Applications of Differential Calculations • Differentials can be added, subtracted, multiplied or divided with each other. d d ––– f (a) = f '(a), ––– g (a) = g'(a) dx dx Therefore: f '(a) + g'(a), f '(a) × g'(a), etc. • 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. d ––– (sinx + cosx, sin0.5), etc.
3-3 Quadratic Differential Calculations [OPTN]-[CALC]-[d2/dx2] 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 3-3 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 n and the closing parenthesis can be omitted.
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.
Integration Calculations 3-4 uTo perform an integration calculation Example To perform the integration calculation for the function shown below, with a tolerance of “tol” = 1E - 4 ∫ 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 tolerance value. bE-e)w • In the function f(x), only X can be used as a variable in expressions.
3-4 Integration Calculations • Pressing A during 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 trigonometric 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.
3-5 Maximum/Minimum Value Calculations [OPTN]-[CALC]-[FMin]/[FMax] 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.
3-5 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 + 2 x + 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.
3-6 Summation (Σ) Calculations [OPTN]-[CALC]-[Σ(] 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 β (a k, k, α, β, n) ⇒ Σa k k=α Σ calculation is the calculation of the partial sum of sequence ak, using the following formula. β S = aα + aα +1 +........
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 ak and last term of sequence ak . • Input of n and the closing parentheses can be omitted. If you omit n, the calculator automatically uses n = 1. k Σ Calculation Applications • Arithmetic operations using Σ calculation expressions Expressions: Possible operations: n n k=1 k=1 Sn = Σ ak, Tn = Σ bk Sn + Tn, Sn – Tn, etc.
