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
Checking a definite integral you have done by hand, the trapezium rule answer against the true value, and the gradient at a point. Questions that say "show that" or "use calculus" need the algebra: the calculator only confirms.
Practise it: Differentiation · Integration · Numerical methods.
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 A Level exams. Use it to learn, to check a calculator answer, and to see what a function or data set looks like.
- 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}.
The interactive graph loads here when you scroll to it.
Or open desmos.com/calculator 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.