Shunya / Calculus Lab

Shunya · Maths lab

Calculus lab

Type any function and watch calculus happen. Squeeze x towards a point to see a limit, shrink a secant into a tangent to get the derivative, find where a curve peaks and dips, solve real optimisation problems, watch rectangles add up to an integral, and follow a differential equation through its slope field.

Lessons

Pick your class, follow the steps in the lab, and check what you see against the working under the graph.

Class 11 — Limits

Concept: the limit of f(x) as x → a is the value f(x) is heading for, from both sides, even if f(a) is missing. In symbols, $\lim_{x \to a} f(x) = L$ .

Open Limits and press (x² − 1) ÷ (x − 1) at 1. The graph has a hole at x = 1, yet the table shows both sides heading to 2.
Drag the green point towards a from the left, then from the right, and watch f(x) settle.
Press |x| ÷ x at 0: the left limit is −1 and the right limit is +1, so the limit does not exist (a jump).
Try 1 ÷ x² (both sides run to +∞) and sin(1/x) (it never settles).
Press sin x ÷ x at 0 and read the famous limit 1. Then type your own function in the box.

Takeaway: a limit exists only when the left and right limits agree on a number.

Class 11 — The derivative as a slope

Concept: the derivative is the limit of the slope of a secant as its two points come together: $f'(x_0) = \lim_{h \to 0} \dfrac{f(x_0 + h) - f(x_0)}{h}$ .

Open Derivative with f(x) = x² and x₀ = 1. The orange secant joins x₀ and x₀ + h.
Press Shrink h towards 0: the secant turns into the teal tangent, and the secant slope approaches f′(1) = 2.
Read the table of quotients under the graph: 3.5, 3, 2.1, 2.01, 2.001 … → 2.
Tick show the derivative graph and drag x₀: the blue point is the slope at each place.
Try sin x, eˣ and ln x, and compare with the symbolic derivative shown at the top.

Takeaway: the derivative is the slope of the tangent, found by letting the second point of the secant slide into the first.

Class 12 — Continuity and differentiability

Concept: f is continuous at a if f(a) exists, the limit exists and they are equal. It is differentiable at a if the slopes from the left and the right agree. Differentiable always means continuous, but not the other way round.

Open Limits and press x² at 2: continuous, and both slopes agree, so differentiable.
Press |x| at 0: continuous (no break) but the slopes are −1 and +1, so not differentiable: a corner.
Press floor x at 1: a jump, so not continuous, and therefore not differentiable either.
Press (x² − 1) ÷ (x − 1) at 1: a removable discontinuity, because the limit exists but f(1) does not.
Open Derivative and choose |x| or x^(1/3): see a corner and a vertical tangent.

Takeaway: corners, jumps and vertical tangents are the three ways a derivative can fail.

Class 12 — Increasing, decreasing, maxima and minima

Concept: where f′ > 0 the function increases and where f′ < 0 it decreases. A local maximum is where f′ changes from + to −, and a local minimum from − to +. The second derivative test uses the sign of f″.

Open Maxima & minima with x³ − 3x. The curve is green where it rises and orange where it falls.
Read the working: f′ = 3x² − 3 = 0 at x = ±1. At x = −1, f″ = −6 < 0 (a local maximum); at x = 1, f″ = 6 > 0 (a local minimum).
Drag the green point and watch the sign of f′ and f″. Tick the concavity shading to see where the curve bends up or down.
Press x³ (no max or min): f′ = 0 at x = 0 but does not change sign, so it is a horizontal inflection.
Drag the interval dots to see the absolute maximum and minimum on a closed interval.

Takeaway: stationary points are where f′ = 0, and f′ or f″ tells you what each one is.

Class 12 — Optimisation

Concept: to find the best value of a quantity, write it as a function of one variable, differentiate, set the derivative to zero, and check with the second derivative.

Open Optimisation with the rectangle of fixed perimeter. Drag the point along the area graph: it peaks at x = P ÷ 4, a square.
Read the four steps in the working: A(x) = x(P/2 − x), A′(x) = P/2 − 2x, A′ = 0 → x = P ÷ 4, and A″ = −2 < 0.
Change the problem to an open box from a square sheet and check that the best cut is s ÷ 6.
Try the can of fixed volume: the least metal is used when the height equals the diameter.
Move the slider for the perimeter, sheet size or volume and see the best value change with it.

Takeaway: the maximum or minimum sits where the slope of the graph is zero.

Class 12 — Area under a curve and the definite integral

Concept: the area under a curve is approximated by thin rectangles, $\sum f(x_i)\,\Delta x$ , and the integral $\int_a^b f(x)\,dx$ is the limit as the strips become infinitely thin.

Open Integrals with x² from 0 to 2 and the Left sum with n = 8. Read the sum and the exact value 8/3 = 2.6667.
Press n = 20 then n = 100: the error shrinks in proportion to Δx.
Switch to Midpoint and Trapezoid: they are far more accurate with the same n.
Tick show the area function: its slope at b equals f(b). This is the fundamental theorem of calculus.
Press x³ − 2x (negative parts) and compare the signed integral with the total area.

Takeaway: the integral is a limit of sums, and it is computed as F(b) − F(a) where F′ = f.

Class 12 — Differential equations

Concept: a differential equation relates a function to its derivative. Its solution is a curve whose slope at every point matches the equation, and the initial point picks out one solution.

Open Differential equations with dy/dx = x + y. Every short line is the slope at that point.
Drag the green start point: the blue curve follows the slope field, and the working gives the general solution y = Ce^x − x − 1.
Tick Euler's method (orange). Lower the Euler step h and watch it close in on the blue curve.
Press dy/dx = y(1 − y), the logistic curve, and dy/dx = −x ÷ y, whose solutions are circles.
Compare the exact, Runge–Kutta and Euler values on the cards.

Takeaway: a first-order equation gives a slope at every point; separating the variables or using the slope field finds the curve.

The maths behind it

What each mode covers

Limits — type any f(x) and squeeze x towards a from both sides: one-sided limits, infinite limits, oscillation, and whether f is continuous or differentiable there. Class 11 limits and class 12 continuity.

Derivative — a secant you can shrink into a tangent, the difference quotient table, the symbolic derivative, tangent and normal lines and the graph of f′. Class 11 derivatives and class 12 tangents and normals.

Maxima & minima — the curve coloured where it rises and falls, stationary points classified by the first and second derivative tests, points of inflection, concavity and absolute extrema on an interval. Class 12 applications of derivatives.

Optimisation — five classic problems (fenced rectangle, pen against a wall, open box, can of least metal, rectangle in a semicircle) with the four-step working. Class 12 applications of derivatives.

Integrals — Riemann sums with left, right, midpoint and trapezoid rules, the exact integral, the error, signed against total area, and the area function that shows the fundamental theorem. Class 12 integrals.

Differential equations — slope fields, a draggable starting point, Runge–Kutta and Euler solutions and the exact solution for the standard first-order equations. Class 12 differential equations.

Part of Shunya. See the syllabus map for the chapters these modes map to. The Function & Graph Explorer is a good place to look at a function before you differentiate it. Numbers here are computed live from your function, so a few results are numerical estimates and are shown to four decimals.

Formulas used in this lab

Limits Class 11

Definition: $\lim_{x \to a} f(x) = L$
The limit exists when: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$
Algebra of limits: $\lim (f \pm g) = \lim f \pm \lim g,\qquad \lim (fg) = \lim f \cdot \lim g$
Standard limits: $\lim_{x \to 0}\dfrac{\sin x}{x} = 1,\qquad \lim_{x \to 0}\dfrac{e^x - 1}{x} = 1$
More standard limits: $\lim_{x \to 0}\dfrac{\ln(1 + x)}{x} = 1,\qquad \lim_{x \to a}\dfrac{x^n - a^n}{x - a} = na^{n-1}$
The number e: $\lim_{n \to \infty}\left(1 + \dfrac{1}{n}\right)^n = e$

Derivative Class 11

First principles: $f'(x) = \lim_{h \to 0}\dfrac{f(x + h) - f(x)}{h}$
Sum and product: $(u \pm v)' = u' \pm v',\qquad (uv)' = u'v + uv'$
Quotient: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$
Chain rule: $\dfrac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)$
Power and exponential: $(x^n)' = nx^{n-1},\qquad (e^x)' = e^x,\qquad (\ln x)' = \dfrac{1}{x}$
Trigonometric: $(\sin x)' = \cos x,\quad (\cos x)' = -\sin x,\quad (\tan x)' = \sec^2 x$

Continuity and differentiability Class 12

Continuous at a: $\lim_{x \to a} f(x) = f(a)$
Differentiable at a: $\lim_{h \to 0^-}\dfrac{f(a + h) - f(a)}{h} = \lim_{h \to 0^+}\dfrac{f(a + h) - f(a)}{h}$
Differentiable implies continuous: $f'(a) \text{ exists} \Rightarrow f \text{ is continuous at } a$
The reverse is false (|x| is continuous at 0 but not differentiable): $f(x) = |x|$

Tangent and normal Class 12

Slope of the tangent: $m = f'(x_0)$
Tangent line: $y - y_0 = f'(x_0)\,(x - x_0)$
Normal line: $y - y_0 = -\dfrac{1}{f'(x_0)}\,(x - x_0)$
Rate of change of y with respect to x: $\dfrac{dy}{dx}$

Increasing, decreasing, maxima and minima Class 12

Increasing and decreasing: $f'(x) > 0 \Rightarrow \text{increasing},\qquad f'(x) < 0 \Rightarrow \text{decreasing}$
Stationary points: $f'(x) = 0$
First derivative test: $f' \text{ changes } + \to - \Rightarrow \max,\qquad - \to + \Rightarrow \min$
Second derivative test: $f'(c) = 0,\ f''(c) < 0 \Rightarrow \max;\quad f''(c) > 0 \Rightarrow \min$
Concavity and inflection: $f'' > 0:\ \text{concave up},\quad f'' < 0:\ \text{concave down}$
Absolute extrema on [a, b]: compare the ends and every stationary point c: $f(a),\ f(b),\ f(c)$

Optimisation Class 12

Fixed perimeter P: $A(x) = x\left(\dfrac{P}{2} - x\right),\qquad x = \dfrac{P}{4}$
Pen against a wall, fencing L: $A(x) = x(L - 2x),\qquad x = \dfrac{L}{4}$
Open box from a square sheet s: $V(x) = x(s - 2x)^2,\qquad x = \dfrac{s}{6}$
Can of volume V, least metal: $S(r) = 2\pi r^2 + \dfrac{2V}{r},\qquad r = \sqrt[3]{\dfrac{V}{2\pi}},\ h = 2r$
Rectangle in a semicircle of radius R: $A(x) = 2x\sqrt{R^2 - x^2},\qquad x = \dfrac{R}{\sqrt{2}}$
The four steps: $\text{write } f(x);\ f'(x) = 0;\ \text{check } f''(x);\ \text{state the answer}$

Integrals Class 12

Riemann sum: $\sum_{i=1}^{n} f(x_i)\,\Delta x,\qquad \Delta x = \dfrac{b - a}{n}$
Definite integral: $\int_a^b f(x)\,dx = \lim_{n \to \infty}\sum_{i=1}^{n} f(x_i)\,\Delta x$
Fundamental theorem: $\int_a^b f(x)\,dx = F(b) - F(a),\qquad F'(x) = f(x)$
Standard integrals: $\int x^n dx = \dfrac{x^{n+1}}{n + 1},\quad \int e^x dx = e^x,\quad \int \dfrac{dx}{x} = \ln x$
Trigonometric: $\int \sin x\,dx = -\cos x,\qquad \int \cos x\,dx = \sin x$
Properties: $\int_a^b f = -\int_b^a f,\qquad \int_a^b f = \int_a^c f + \int_c^b f$
Trapezoid rule: $\dfrac{\Delta x}{2}\,\bigl[f(x_0) + 2f(x_1) + \dots + f(x_n)\bigr]$

Differential equations Class 12

Order and degree: $\text{order} = \text{highest derivative},\qquad \text{degree} = \text{its power}$
Variable separable: $\dfrac{dy}{dx} = f(x)g(y) \Rightarrow \int \dfrac{dy}{g(y)} = \int f(x)\,dx$
Growth and decay: $\dfrac{dy}{dx} = ky \Rightarrow y = Ce^{kx}$
Linear first order: $\dfrac{dy}{dx} + Py = Q,\qquad \text{integrating factor } e^{\int P\,dx}$
Euler's method: $y_{n+1} = y_n + h\,f(x_n, y_n)$
Particular solution: $\text{use } y(x_0) = y_0 \text{ to find the constant } C$