How to find a numerical derivative and definite integral on the Casio fx-CG50
The gradient at a point with d/dx and a definite integral with ∫dx, plus areas between curves. Exact keystrokes for the fx-CG50 first, then the other calculators, a worked example you can check your answers against, the mistakes that cost marks, and the same thing in Desmos.
When you need it
IGCSE calculus is done by hand (differentiate xⁿ, find turning points). The calculator's d/dx is a good way to check a gradient you worked out, e.g. the gradient of a curve at x = 1.
Practise it: IGCSE units (calculus in Unit 3: Graphs).
Keystrokes on your calculator
Pick your calculator: the page remembers it next time.
Numerical derivative and integral on the Casio fx-CG50
Written for OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell). Casio fx-CG50: graphic (GDC).
- MENU 1 (RUN-MATRIX).
- Derivative at a point: OPTN F4 (CALC) F2 (d/dx). In the template type X² e^(−X) (e^ is SHIFT ln), move to the box after x = and type 1, then EXE.
- Definite integral: OPTN F4 (CALC) F4 (∫dx). Type the function, then the lower limit 0 and upper limit 2 in the boxes, then EXE.
- Area from the graph instead: in GRAPH, draw Y1, then SHIFT F5 (G-SOLVE) F6 (▷) → ∫dx, move to the lower limit and EXE, then the upper limit and EXE: the region is shaded and its value shown.
- In linear input mode the syntax is d/dx(f(X), 1) and ∫(f(X), 0, 2).
- For the distance travelled use ∫|v(t)|dt: Abs is in OPTN → NUMERIC → Abs (press F6 (▷) in the OPTN menu to reach NUMERIC).
- Area between two curves: ∫(Y1 − Y2) between the intersections, with Abs if the curves cross inside the interval. Y1 is typed with VARS F4 (GRAPH) F1 (Y) 1.
Numerical derivative and integral on the TI-84 Plus CE
Written for OS 5.8.x. TI-84 Plus CE: graphic (GDC); also the TI-84 Plus CE Python / CE-T Python Edition.
- Derivative: math 8: nDeriv( (on the CE with MathPrint it is a d/dx template). Type X² e^(−X) (e^ is 2nd ln), put X in the variable box and 1 as the value, enter.
- Integral: math 9: fnInt( (the ∫ template). Lower limit 0, upper limit 2, the function, and X after d. enter.
- From the graph: 2nd trace (CALC) 6: dy/dx at a point, or 7: ∫f(x)dx with lower and upper limits: the area is shaded.
- Using Y1 in the template (alpha trace (F4 shortcut) → Y1) saves retyping the function.
- abs( is math → NUM → 1: abs(.
Numerical derivative and integral on the TI-Nspire CX II
Written for OS 6.x. TI-Nspire CX II: graphic (GDC), non-CAS; the CX II CAS is not allowed in IB, A Level or IGCSE exams.
- Calculator page: menu → Calculus → Numerical Derivative at a Point. Choose the variable x and the value 1, OK, then type x²·e^(−x) in the template and enter.
- Integral: menu → Calculus → Numerical Integral, then fill in 0, 2 and x²·e^(−x) and enter. (The templates key templates also has the ∫ and d/dx templates.)
- From the graph: menu → Analyze Graph → Integral (click the lower and upper bounds) or → dy/dx.
- Defining f1(x) on a Graphs page lets you type f1(x) in the Calculator page instead of the whole function.
Numerical derivative and integral on the Casio fx-991EX ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991EX ClassWiz: scientific (no graphs).
- MENU 1 (Calculate).
- Integral: press ∫▭ (the integral key), type X² e^(−X) (e^ is SHIFT ln), then the limits 0 and 2 with the arrow keys, =.
- Derivative: SHIFT ∫▭ (d/dx). Type the function, move to x = and type 1, =.
- Both are numerical and can take a few seconds. Set radians (SHIFT MENU → Angle Unit → Radian) before any trig function.
Numerical derivative and integral on the Casio fx-991CW ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991CW ClassWiz: scientific (no graphs); the newer ClassWiz with a HOME menu of apps.
- HOME → Calculate.
- CATALOG → Func Analysis → Integration(∫). Type the function and the limits 0 and 2, EXE.
- CATALOG → Func Analysis → Derivative(d/dx). Type the function and x = 1, EXE.
- Angle unit is in SETTINGS → Calc Settings → Angle Unit.
Worked example
f(x) = x²e−x. Find (a) f′(1), (b) ∫₀² f(x) dx. (c) Why does ∫₀^π sin x dx give 0.0861 on some calculators?
Answers (checked with Python)
(a) f′(1) = 0.367879 (exactly e−1). (b) 0.646647 (exactly 2 − 10e−2). (c) The calculator is in degrees: the answer in radians is 2; in degree mode it integrates sin of x degrees and gives 0.0861.
Common calculator mistakes
- Degree mode: any trig integral or derivative is wrong unless the calculator is in radians. ∫₀^π sin x dx should be 2, not 0.0861.
- Area vs integral: if the curve goes below the x-axis, the integral subtracts that part. For an area use |f(x)| (Abs) or split the interval at the root.
- Wrong variable: on the fx-CG50 and TI-84 the variable in the function must be X; on the TI-Nspire it must match the variable in d/dx or dx.
- Rounding the limits: if a limit is an intersection you found earlier, use the stored full value, not 3 s.f.
- "Use calculus" or "show that" questions need the working: a calculator value alone scores little.
Do it in Desmos
Not in the exam. Desmos is not allowed in IGCSE exams. Use it to learn, to check an answer and to see the graph behind a question.
- Define f(x) = x^2 e^(−x).
- Type f′(1) (use the apostrophe for the dash): Desmos evaluates the derivative at 1.
- Type the integral with the int template: ∫ from 0 to 2 of f(x) dx. To shade the area, add the inequality 0 ≤ y ≤ f(x) {0 ≤ x ≤ 2}.
Open desmos.com/calculator (free, in any browser) and type:
f(x)=x^{2}e^{-x}f'(1)int_{0}^{2}f(x)dx0≤ y≤ f(x){0≤ x≤2}
More calculator skills
Sources
- Casio fx-CG50 Software User's Guide, OS 3.70 (PDF) : Casio fx-CG50, OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell).
- TI-84 Plus CE guidebook (Texas Instruments) : TI-84 Plus CE, OS 5.8.x.
- TI-Nspire CX II guidebooks (Texas Instruments) : TI-Nspire CX II, OS 6.x.
- Casio fx-991EX / fx-570EX Quick Start Guides : Casio fx-991EX ClassWiz, the fixed ROM (these models take no OS updates).
- Casio fx-991CW / fx-570CW User's Guide : Casio fx-991CW ClassWiz, the fixed ROM (these models take no OS updates).
- Desmos Help Center for the Desmos functions.
Menu names are taken from each manufacturer's manual for the OS versions shown. Menus can move slightly between OS updates: if a screen looks different on yours, the manual for your version is the reference, and please tell us so we can update this page. Worked-example answers were recomputed independently in Python.