Advanced Calculator Functions: SOLVE, Integrals, Complex
Advanced Mathematical Formulas on a Scientific Calculator
Casio fx-991 calculators can integrate, differentiate, sum series, solve equations, handle complex numbers and run statistics, but support varies by model. Each topic below has a worked example you can check, such as ∫ x² from 0 to 1 = 1/3.
Model support at a glance
Casio’s fx-991ES PLUS manual lists integration, differentiation, Σ calculations, SOLVE, CALC, complex numbers (CMPLX), statistics (STAT), equations (EQN), matrix and vector modes. The fx-991EX and fx-991CW are newer ClassWiz models with their own manuals; consult them for exact menu paths, and do not assume a function exists on a different model just because it exists here. Simpler Casio models, and the CalcNova online calculator, do not have all of these.
| Function | fx-991ES PLUS | Online CalcNova |
|---|---|---|
| Integration, differentiation, Σ | Listed in manual | No |
| SOLVE / EQN | Listed in manual | No equation solver |
| Complex numbers | CMPLX mode | No |
| Statistics | STAT mode | Standard deviation tool |
Integration
The definite integral ∫01 x² dx = [x³/3] from 0 to 1 = 1/3. On the calculator you enter the integrand, the lower and upper limits, and press =. The result is numerical, so expect an approximation such as 0.3333333333 rather than a symbolic answer. Use radians when integrating trigonometric functions.
Differentiation
The derivative of x³ at x = 2 is 3x² = 12. The calculator evaluates the derivative at a point numerically; it does not return a formula. It may be inaccurate at sharp corners, such as |x| at 0.
Summation
Σ of x² for x = 1 to 5 is 1 + 4 + 9 + 16 + 25 = 55. Enter the expression, then the start and end values of x.
Equations and SOLVE
EQN mode solves polynomial and simultaneous equations. Example: x² − 5x + 6 = 0 has roots 2 and 3, because (x − 2)(x − 3) = x² − 5x + 6. Enter a = 1, b = −5, c = 6.
SOLVE works numerically on any equation you type. Example: solve 2x + 3 = 11 for x gives x = 4. Because SOLVE depends on an initial guess, it can return one solution when several exist, or fail with an error. Substitute the result back in to check it.
Complex numbers
In CMPLX mode, type i with its key. (1 + 2i)(3 − i) = 3 − i + 6i − 2i² = 3 + 5i + 2 = 5 + 5i. The calculator can also show results in polar form r∠θ. For 1 + i, r = √2 ≈ 1.414 and θ = 45° (in degree mode).
Statistics
In STAT mode you choose a one-variable type, enter data into the table and read results from the STAT menu. For the data 2, 4, 6: mean = 4; population standard deviation = √(8/3) ≈ 1.633; sample standard deviation = 2. Know which one your question needs: sample and population differ.
Common mistakes
- Angle unit wrong when integrating or differentiating trig functions.
- Trusting one SOLVE root when the equation has more.
- Leaving a previous STAT table in memory; clear the data first.
- Mixing up sample and population standard deviation.
Frequently asked questions
- Does the calculator give symbolic integrals?
- No. It returns a numerical value for definite integrals.
- Which calculators have SOLVE?
- The ES PLUS manual lists it. For other models check the manual.
- Can I solve matrices with these functions?
- Yes, see the matrix guide.